الإحصاء-الوصفي-والاستدلالي

‫ المانيا‬- ‫برلين‬ ‫اإلحصاء الوصفي واالستداللي‬ ‫المركر الديمقراطي العربي‬ ‫المركر الديمقراطي العربي‬ ‫علـي أحمـد السقـاف‬ Prof.Dr. Ali Ahmed Alsagf ‫أستـاذ اإلحصـاء‬ ‫قسم اإلحصـاء والمعلوماتيـة‬ DESCRIPTIVE AND INFERENTIAL STATISTICS ‫األستاذ الدكتور‬ ‫اإلحصـاء الـوصفي واالستـداللي‬ DESCRIPTIVE AND INFERENTIAL STATISTICS Democratic Arabic Center Berlin – Germany ‫ووح‬ ‫هذا الكراب‬ ‫هو رووخلج ووتد ي ذوواق ا حصوواا اتدالوايف اتمووراتن ي لاح ووح ووا‬ .‫ال م اتداق ح قصم اتدالايف وامل حاذ ح ل شخلج من اخ حضد‬ ‫او و وواو اتدالو و ووايف ال و و و واتدالو و ووايف‬ ‫أمام و و و‬ ‫و و ووخل‬ ‫نقصو و ووم و و ووم اتدالو و ووايف ا‬ ‫ا ر وو‬ ‫اتمو ووراتند وان و ووا ح و ووه اتدال و ووايف اتم و ووراتن ه و و ام و وورنرا ال و ووا‬ .‫نح محثد حنه‬ ‫حه الا‬ ‫وو و و و الث ا و و وواخ‬ ‫حير و و و ا اتدال و و ووايف ال و و و اتم و و ووال ة املص و و وور احح لر و و و‬ ‫ و ووخلس ذص و وول ذ ص و و هاد اتدال و ووايف اتم و ووراتن حير و و ا ذ و ووري‬.‫الخلق و ووح و ل و ووري ت‬ ‫اتموال ة والوون حوه ينووا ورم ارووا القوخلاقاخ دو ا ا ر و اتدالووا و لوري حووه‬ . ‫واق ال نح املصح تح حه هذا ا ر‬ This book is the fruit of my experience teaching the so called subject inferential statistics at the University of Aden, Faculty of Administrative Sciences, Department of Statistics and Informatics, for the past ten years. Statistics is divided into two main branches: Descriptive Statistics and Inferential Statistics. The aim of inferential statistics is to deduce the characteristics of a population from the characteristics of a sample drawn from it. Descriptive statistics contain the methods used to summarize and describe numerical data in order to facilitate its interpretation. Inferential statistics includes those methods through which decisions are made about the statistical population, based on the reality of the sample drawn from it. ‫الطبعـة األولـى‬ 2020 DEMOCRATIC ARABIC CENTER Germany : Berlin 10315 Gensinger- Str : 112 http://democraticac.de VR.3383-6414.B :‫رقم التسجيل‬ TEL: 0049-CODE 030-89005468/030-898999419/030-57348845 MOBILTELEFON: 0049174274278717 ƛ ܸߢࢽ ࡦƁ߁ ዖዄ Ž૊ સၫƋࣝެ࠾ኃ ቮƁƊܸެࢲ ଘଐƁ 1 ። ፎକଐ߁ ެࢲ କଐƁƢ। ्࢟ ࣝࡦƁŻܸࢤ ߾ ଔଐƁ ܸୃกܸ଍ଉŽ–ໝ ແࡦफउ –ࠣ‫ݶ‬ Ɖܸࢤ ެߕକଐƁƢࠣ‫ࡦ ࢽޟܸ ࢲޟ‬ƁƢࠣ‫ޟ‬පඓ ൷Ɓ౭౑ࢲ କଐƁƄ ܸࢲ ƁƋኃ ᒔᐺ። ጏ࢓‫ݕ‬ ࡦƁ࣬ ࣊ Ɓ࢓ߢೕಹኃ ቮƁ೾೥࢓଍ଉƁƄ ƁƋƁ߁ ࢟ ſ ƛ ܸߢࢽ ࡦƁ߁ ዖዄ Ž૊ સၫƋࣝެ࠾ኃ ቮƁƊܸެࢲ ଘଐƁ ። ፎକଐ߁ ެࢲ କଐƁƢ। ्࢟ ࣝࡦƁŻܸࢤ ߾ ଔଐƁ ļ Ļ ļ ʆû Ū ŉ ʫŐ ż ŏƫńŎŅŉŔ ſ ń Ļ ļ Ȝ û ɟŠ ʫŢ Ņťȑ ŮńŏƸ ńžĿ û Ļ ļ ʫÍ Ņçřǔ ʆſ ńŎŅŉŔ ń 2 ܸୃกܸ଍ଉŽˍ ໝ ແࡦफउˍ ࠣ‫ݶ‬ Ɖܸࢤ ެߕକଐƁƢࠣ‫ࡦ ࢽޟܸ ࢲޟ‬ƁƢࠣ‫ޟ‬පඓ ൷Ɓ౭౑ࢲ କଐƁƄ ܸࢲ ƁƋኃ ᒔᐺ። ጏ࢓‫ݕ‬ ࡦƁ࣬ ࣊ Ɓ࢓ߢೕಹኃ ቮƁ೾೥࢓଍ଉƁƄ ƁƋƁ߁ ࢟ ſ ƛ ܸߢࢽ ࡦƁ߁ ዖዄ Ž૊ સၫƋࣝެ࠾ኃ ቮƁƊܸެࢲ ଘଐƁ 3 ። ፎକଐ߁ ެࢲ କଐƁƢ। ्࢟ ࣝࡦƁŻܸࢤ ߾ ଔଐƁ ܸୃกܸ଍ଉŽ–ໝ ແࡦफउ –ࠣ‫ݶ‬ Ɖܸࢤ ެߕକଐƁƢࠣ‫ࡦ ࢽޟܸ ࢲޟ‬ƁƢࠣ‫ޟ‬පඓ ൷Ɓ౭౑ࢲ କଐƁƄ ܸࢲ ƁƋኃ ᒔᐺ። ጏ࢓‫ݕ‬ ࡦƁ࣬ ࣊ Ɓ࢓ߢೕಹኃ ቮƁ೾೥࢓଍ଉƁƄ ƁƋƁ߁ ࢟ ſ ‫‪። ፎକଐ߁ ެࢲ କଐƁƢ। ्࢟ ࣝࡦƁŻܸࢤ ߾ ଔଐƁ‬‬ ‫‪ƛ ܸߢࢽ ࡦƁ߁ ዖዄ Ž૊ સၫƋࣝެ࠾ኃ ቮƁƊܸެࢲ ଘଐƁ‬‬ ‫اﻟﻨـ ـﺎﺷ ـﺮ‬ ‫اﻟﻤﺮﻛﺰ اﻟﺪﻳﻤﻘﺮاﻃﻲ اﻟﻌﺮﺑﻲ‬ ‫ﻟﻠﺪراﺳﺎت اﻻﺳﺘﺮاﺗﻴﺠﻴﺔ واﻟﺴﻴﺎﺳﻴﺔ واﻻﻗﺘﺼﺎدﻳﺔ‬ ‫أﻟﻤﺎﻧﻴﺎ ‪ /‬ﺑﺮﻟﻴﻦ‬ ‫‪Democratic Arabic Center‬‬ ‫‪Berlin / Germany‬‬ ‫ﻻ ﻳﺴﻤﺢ ﺑﺈﻋﺎدة إﺻﺪار ﻫﺬا اﻟﻜﺘﺎب أو أي ﺟﺰء ﻣﻨﻪ أو ﺗﺨﺰﻳﻨﻪ‬ ‫ﻓﻲ ﻧﻄﺎق اﺳﺘﻌﺎدة اﻟﻤﻌﻠﻮﻣﺎت أو ﻧﻘﻠﻪ ﺑﺄي ﺷﻜﻞ ﻣﻦ اﻷﺷﻜﺎل‪ ،‬دون إذن ﻣﺴﺒﻖ ﺧﻄﻲ ﻣﻦ اﻟﻨﺎﺷﺮ‪.‬‬ ‫ﺟﻤﻴﻊ ﺣﻘﻮق اﻟﻄﺒﻊ ﻣﺤﻔﻮﻇﺔ‪ :‬اﻟﻤﺮﻛﺰ اﻟﺪﻳﻤﻘﺮاﻃﻲ اﻟﻌﺮﺑﻲ ﺑﺮﻟﻴﻦ‪ -‬أﻟﻤﺎﻧﻴﺎ‬ ‫‪All rights reserved No part of this book may by reproducted.‬‬ ‫‪Stored in a retrieval system or transmitted in any from or by any means‬‬ ‫‪without prior permission in writing of the published‬‬ ‫اﻟﻤـﺮﻛـﺰ اﻟﺪﻳﻤﻘ ـﺮاﻃـ ـﻲ اﻟﻌ ـﺮﺑـﻲ‬ ‫ﻟﻠﺪراﺳﺎت اﻻﺳﺘﺮاﺗﻴﺠﻴﺔ واﻟﺴﻴﺎﺳﻴﺔ واﻻﻗﺘﺼﺎدﻳﺔ أﻟﻤﺎﻧﻴﺎ‪/‬ﺑﺮﻟﻴﻦ‬ ‫‪Berlin10315 Gensingerstr :112‬‬ ‫‪Tel :0049-code Germany‬‬ ‫‪54884375-030‬‬ ‫‪91499898-030‬‬ ‫‪86450098-030‬‬ ‫اﻟﺒ ـﺮﻳﺪ اﻹﻟﻜﺘ ـﺮوﻧــﻲ‬ ‫‪book@democraticac.de‬‬ ‫ݶࠣ– ‪ܸୃกܸ଍ଉŽ–ໝ ແࡦफउ‬‬ ‫ݕ࢓‪පඓ ൷Ɓ౭౑ࢲ କଐƁƄ ܸࢲ ƁƋኃ ᒔᐺ። ጏ‬ޟࠣ‪ࢽ ࡦƁƢ‬ޟܸ ࢲޟࠣ‪Ɖܸࢤ ެߕକଐƁƢ‬‬ ‫‪ࡦƁ࣬ ࣊ Ɓ࢓ߢೕಹኃ ቮƁ೾೥࢓଍ଉƁƄ ƁƋƁ߁ ࢟ ſ‬‬ ‫‪4‬‬ ‫‪። ፎକଐ߁ ެࢲ କଐƁƢ। ्࢟ ࣝࡦƁŻܸࢤ ߾ ଔଐƁ‬‬ ‫‪ƛ ܸߢࢽ ࡦƁ߁ ዖዄ Ž૊ સၫƋࣝެ࠾ኃ ቮƁƊܸެࢲ ଘଐƁ‬‬ ‫رﺋﻴﺲ اﻟﻤﺮﻛﺰ اﻟﺪﻳﻤﻘﺮاﻃﻲ اﻟﻌﺮﺑﻲ‪ :‬أ‪ .‬ﻋـﻤــﺎر ﺷــﺮﻋــﺎن‬ ‫اﺳﻢ اﻟﻜﺘﺎب‪ :‬اﻹﺣﺼﺎء اﻟﻮﺻﻔﻲ واﻻﺳﺘﺪﻻﻟﻲ‬ ‫اﻷﺳﺘﺎذ اﻟﺪﻛﺘﻮر ‪ /‬ﻋﻠﻲ أﺣﻤﺪ اﻟﺴﻘﺎف‬ ‫ﺿﺒﻂ وﺗﺪﻗﻴـﻖ‪ :‬د‪ .‬ﺳــﺎﻟـﻢ ﺑﻦ ﻟﺒـ ـ ــﺎد‬ ‫اﻟﺘﺼﻤﻴﻢ واﻹﺧـﺮاج‪ :‬أ‪ .‬د‪ .‬ﺑﺪراﻟـﺪﻳﻦ ﺷﻌﺒـﺎﻧﻲ‬ ‫رﻗﻢ ﺗﺴﺠﻴﻞ اﻟﻜﺘﺎب‪VR . 3383 - 6414 . B :‬‬ ‫ﻋﺪد اﻟﺼﻔﺤﺎت‪174:‬‬ ‫اﻟﻄﺒﻌﺔ اﻷوﻟﻰ‬ ‫ﺳﺒﺘﻤﺒﺮ ‪ 2020‬م‬ ‫ݶࠣ ˍ‪ܸୃกܸ଍ଉŽˍ ໝ ແࡦफउ‬‬ ‫ݕ࢓‪පඓ ൷Ɓ౭౑ࢲ କଐƁƄ ܸࢲ ƁƋኃ ᒔᐺ። ጏ‬ޟࠣ‪ࢽ ࡦƁƢ‬ޟܸ ࢲޟࠣ‪Ɖܸࢤ ެߕକଐƁƢ‬‬ ‫‪ࡦƁ࣬ ࣊ Ɓ࢓ߢೕಹኃ ቮƁ೾೥࢓଍ଉƁƄ ƁƋƁ߁ ࢟ ſ‬‬ ‫‪5‬‬ ። ፎକଐ߁ ެࢲ କଐƁƢ। ्࢟ ࣝࡦƁŻܸࢤ ߾ ଔଐƁ ƛ ܸߢࢽ ࡦƁ߁ ዖዄ Ž૊ સၫƋࣝެ࠾ኃ ቮƁƊܸެࢲ ଘଐƁ ‫اﻟﻤـﺤﺘــﻮﻳـﺎت‬ ࠦႁ ߦࢧ ࡩƄ 8-6 20-9 Ɩ ࣠ࢠ࣠ଐଌƄ Ƈ ܻ‫ޯ࣠ݹ‬ഷവറƄ Ʀ ༇໩ ଘଓƄƢॅळ‫ߘޯࠢހܻࡩݘ‬ൎ൉ၮƥࠢߦ‫ݶ‬ ܻ࣌ ƥūžܻࢧ ࠁ ଘଓƄࣷࣳၮযওܻࠧߦࡷƞ ƢକଐƁࡠ ࢤ ߣࡦƁ žܻࢧ ࠁ ଘଓƄࣷࣳၮߎ ‫ۑ ކݘ࢖ݹ‬ žܻࢧ ࠁ ଘଓƄࣷࣳၮযওܻࠧߦࡷ ‫ۑ‬ žܻࢧ ࠁ ଘଓƄࣷࣳၮߎ ‫ݶ‬ ܻ࣌ ƥ ‫ۑ‬ žܻࢧ ࠁ ଘଓƄࣷࣳၮƇ ଘଓܻ༬ீ ‫ۑ‬ 39-21 ܻᐳႩ ୹ ‫ݑ‬ƥࠦ୆෷ܻࢧ ࠁ ଘଓƄƇ ܻ‫ࡩީޢܻރ‬Ƅߎ ‫ޢ‬ჱࢧ ‫ ކ‬። ጒܸޯࡦƁࡠ ࢤ ߣࡦƁ Ƈ ܻ‫ࡩީޢܻރ‬Ƅߎ ‫ޢ‬ჱࢧ ‫ۑ ކ‬ Ƈ ܻ‫ࡩީޢܻރ‬ƄƓ ୹ ‫ۑ ݑ‬ ࠦ‫ݹ‬ഁ೨࢖ଐଌƄࠦၮ౮౑ࡩƄࢪ ყႾܻߥࡷ‫ࡦޯܸࡦݜ‬Ɓࡠ ࢤ ߣࡦƁ 51-40 ፥ ጒܻࣀ ஊै Ƅ࣊ ࢵ ࣠ࡩƄ ‫ۑ‬ ࿃ ႓߄ ެಛಗƄ࣊ ࢵ ࣠ࡩƄ ‫ۑ‬ १ ॏߙƄ࣠ޯࡩƄ࣊ ࢵ ࣠ࡩƄ ‫ۑ‬ ࣊ ‫ࡩ࣠ ࢵޢ‬Ƅ ‫ۑ‬ ơ Ƅ࣠ެଐଌƄ ‫ۑ‬ ‫ ݝ‬ჴࣃ ჴࡩƄࢪ ყႾܻߥࡷ݇ ‫ݽ‬ Ɓ༜་Ɓࡠ ࢤ ߣࡦƁ 63-52 Ʀ ߄ ଐଌƄ ‫ۑ‬ ഭ ୨୆ก༟༎ƄƦ ߄ ଐଌƄߎ ࢧ ‫ۑ ރ‬ ࣊ ࢵ ࣠ޯଐଌƄƞ Ƅ࢖༨စ ଘଓƄ ‫ۑ‬ Ƨ Ǝܻ‫ݘޢ‬ଐଌƄƞ Ƅ࢖༨စ ଘଓƄ ‫ۑ‬ 75-64 6 Ƈ ଘଓܻࡺޯࠁ ଘଓƄࢧ ࡴܸஆॅ Ɓࡠ ࢤ ߣࡦƁ Ƈ ଘଓܻࡺޯࠁ ଘଓƄ፥ ጿࠦ‫ ࢵܻ ࢵޢ‬ƀযওܻࠧߦࡷ ‫ۑ‬ Ƈ ଘଓܻࡺޯࠁ ଘଓƄ߄ ၮƄ࣠ߘ ‫ۑ‬ ܸୃกܸ଍ଉŽ–ໝ ແࡦफउ –ࠣ‫ݶ‬ Ɖܸࢤ ެߕକଐƁƢࠣ‫ࡦ ࢽޟܸ ࢲޟ‬ƁƢࠣ‫ޟ‬පඓ ൷Ɓ౭౑ࢲ କଐƁƄ ܸࢲ ƁƋኃ ᒔᐺ። ጏ࢓‫ݕ‬ ࡦƁ࣬ ࣊ Ɓ࢓ߢೕಹኃ ቮƁ೾೥࢓଍ଉƁƄ ƁƋƁ߁ ࢟ ſ ƛ ܸߢࢽ ࡦƁ߁ ዖዄ Ž૊ સၫƋࣝެ࠾ኃ ቮƁƊܸެࢲ ଘଐƁ ࠦႁ ߦࢧ ࡩƄ 84-76 99-85 ። ፎକଐ߁ ެࢲ କଐƁƢ। ्࢟ ࣝࡦƁŻܸࢤ ߾ ଔଐƁ Ɩ ࣠ࢠ࣠ଐଌƄ Ɣ ୍ܻงƎଘଓƄ ƍ Ɖܸࢽ ࡦƁࡠ ࢤ ߣࡦƁ ƣ ࣠ࢵ ౰ే‫ހ‬Ɣ ୍ܻงƎƄࡣ ࡷܻ‫ۑ ࡷݘ‬ ƣ ܻࡷ౰౎‫ ࢵޢ‬ว ง༟༎ƄƔ ୍ܻงƎƄࡣ ࡷܻ‫ۑ ࡷݘ‬ ƎƄ߄ ༨စ ଘଓƄ݇ ‫ݽ‬ ܸࢽ ࡦƁࡠ ࢤ ߣࡦƁ Ʀ ࢖‫ݓ‬ ࢧ ࡩƄƇ ܻ‫࢖ހݘ‬ଐଌƄࠦߥ‫ۑ ࣍࢖ݹ‬ ࣊ ‫ ࣀޢ‬ხࡩƄ࣯ ࣐ உै ƄƎƄ߄ ༨စ ଘଓƄቿቱƌܻ‫ۑ ࡷݘ‬ ਔ ৲߄ ႁ ޯࡩƄࡣ ࡷܻ‫ۑ ࡷݘ‬ ࠦ‫ ࠁޯࡺܻࡩޢ‬ଘଓƄƇ ܻ‫ݹݘ‬Ə࣠ޯࡩƄࡹ ࡴܸޯࡦƁࡠ ࢤ ߣࡦƁ ௪ ௎߄ ஊै ƄƧ ƍ݊ ‫ݹ‬Ə࣠‫ۑ ކ‬ 113-100 ƣ ࣠ࢵ ࣠‫ݹ ݊ހ‬Ə࣠‫ۑ ކ‬ ഭ ୨‫ࡩ ࣐ީޢ‬Ƅ݊ ‫ݹ‬Ə࣠ޯࡩƄ ‫ۑ‬ t݊ ‫ݹ‬Ə࣠‫ۑ ކ‬ Ƨ ᇷᆝ ݊ ‫ ୹ހ‬୪݊ ‫ݹ‬Ə࣠‫ކ‬8.5 F݊ ‫ݹ‬Ə࣠‫ۑ ކ‬ मअ߄ ߥޯࡩƄ݇ ࢲ ܸެࡦƁࡠ ࢤ ߣࡦƁ 135-114 ߄ ‫ޢ‬எै Ƅमअ߄ ߥޯࡩƄ࢝ ‫ݶ‬ ܻࢧ ࠀ ‫ۑ‬ ௪ ௎ܻީޯࡩƄƢॅळ‫ޯࡺ ݊ࡷݘ‬഻വറ፥ ጒܻࣀ ஊै Ƅ࣊ ࢵ ࣠ޯଐଌƄमअ߄ ߥ‫ۑ ކ‬ t ݊ ‫ݹ‬Ə࣠‫ކ‬ƢƄ߄ ႀ ޯࢵ ܻ‫ހ‬፥ ጒܻࣀ ஊै Ƅ࣊ ࢵ ॅळࡩࠦߥ޲ࡩƄƆ౰౔ߙ ‫ۑ‬ ໠ ໄ࣐ ࢵ ࣠ޯࡷ໠ ໄ‫ހ‬Ɵ ࢖ߦᑃᐽࠦߥ޲ࡩƄƆ౰౔ߙƎ߄ ߥ‫ۑ ކ‬ ໠ ໄޯხఱ ఝ໠ ໄ‫ހ‬Ɵ ࢖ߦࡩƄƥࠦީࣀ ჱᑃᐽࠦߥ޲ࡩƄƌƥ߄ ၰ ‫ۑ‬ Ƈ ܻ‫ࡩߦ࢖ࢠޢ‬ƄƎܻީޯࠀ Ƅྯ ႓ܸ‫ݕ‬ ࡦƁࡠ ࢤ ߣࡦƁ 165-136 Ƈ ܻ‫ࡩߦ࢖ࢠޢ‬ƄƎܻީޯࠀ ƄƢ࣠ࠩߦࡷ ‫ۑ‬ Ɠ ƥ࢖ߦࡩƄƎܻީޯࠀ ƄƇ ܻ‫ۑ ߙ࢖ࢠޢ‬ Ƈ ܻ‫ࡩߦ࢖ࢠޢ‬ƄƎܻީޯࠀ Ƅ፥ ጿžܻ࣐ ࠀ ଘଓƄߎ ‫ޢ‬ჱࢧ ‫ۑ ކ‬ Ƈ ܻ‫ࡩߦ࢖ࢠޢ‬ƄƎܻީޯࠀ ƄƖ Ƅ࣠‫ރ‬ƀߎ ‫ޢ‬ჱࢧ ‫ۑ ކ‬ ‫ ࣀ ݪ‬ჱࡩƄƥƇ ܻ࣐ ࢵ ࣠ޯ൞ᒑᐽƇ ܻ‫ࡩߦ࢖ࢠޢ‬ƄƎܻީޯࠀ Ƅ ‫ۑ‬ 7 ܸୃกܸ଍ଉŽ–ໝ ແࡦफउ –ࠣ‫ݶ‬ Ɖܸࢤ ެߕକଐƁƢࠣ‫ࡦ ࢽޟܸ ࢲޟ‬ƁƢࠣ‫ޟ‬පඓ ൷Ɓ౭౑ࢲ କଐƁƄ ܸࢲ ƁƋኃ ᒔᐺ። ጏ࢓‫ݕ‬ ࡦƁ࣬ ࣊ Ɓ࢓ߢೕಹኃ ቮƁ೾೥࢓଍ଉƁƄ ƁƋƁ߁ ࢟ ſ ƛ ܸߢࢽ ࡦƁ߁ ዖዄ Ž૊ સၫƋࣝެ࠾ኃ ቮƁƊܸެࢲ ଘଐƁ ࠦႁ ߦࢧ ࡩƄ ። ፎକଐ߁ ެࢲ କଐƁƢ। ्࢟ ࣝࡦƁŻܸࢤ ߾ ଔଐƁ Ɩ ࣠ࢠ࣠ଐଌƄ ‫ ࣀ ݪ‬ჱࡩƄƥ Ƈ ܻ࣐ ࢵ ࣠ޯ൞ᒑᐽƓ ƥ࢖ߦࡩƄƎܻީޯࠀ Ƅે ઻ၮƇ ܻߥ‫ۑ ކ ࣐ީޢ‬ ơ ൎ൉ߥޯࢵ ଘଓƄƥࢇ ‫ࡩޯ࣠ߙޢ‬ƄƆƌ࣠எै Ƨ ᇷᆝ ݊ ‫ ୹ހ‬୪Ǝܻީޯࠀ Ƅ ‫ۑ‬ 170-166 173-171 8 ௪ ௎ܻީޯࡩƄࡣ ‫࡯ޢ‬༨ဈ ໠ ໄ࣐ ࢵ ࣠ޯࡷࡼ ࡷ౰౗ࡁ ଛଓƇ ܻ‫ࡩߦ࢖ࢠޢ‬ƄƎܻީޯࠀ Ƅ ‫ۑ‬ ࢇ ࠁ ൎ൉ଐଌƄ ݊ ࠅ Ƅ࢖ଐଌƄ ܸୃกܸ଍ଉŽ–ໝ ແࡦफउ –ࠣ‫ݶ‬ Ɖܸࢤ ެߕକଐƁƢࠣ‫ࡦ ࢽޟܸ ࢲޟ‬ƁƢࠣ‫ޟ‬පඓ ൷Ɓ౭౑ࢲ କଐƁƄ ܸࢲ ƁƋኃ ᒔᐺ። ጏ࢓‫ݕ‬ ࡦƁ࣬ ࣊ Ɓ࢓ߢೕಹኃ ቮƁ೾೥࢓଍ଉƁƄ ƁƋƁ߁ ࢟ ſ ƛ ܸߢࢽ ࡦƁ߁ ዖዄ Ž૊ સၫƋࣝެ࠾ኃ ቮƁƊܸެࢲ ଘଐƁ ። ፎକଐ߁ ެࢲ କଐƁƢ। ्࢟ ࣝࡦƁŻܸࢤ ߾ ଔଐƁ ŸĿ Ļ ȧ ȯŠ ʆ ʫÍ Ņřǔ ſ ńű ű ŅťȌ Ļ û ǳ ļļ Ļû Ļ ůš ʆ ʆ Ƽ ʫŽ ő ſ ńŰż Ņɩ ľŷŉŤƀ Š Ż ŷťȃ ŅŜ Ż 9 ܸୃกܸ଍ଉŽˍ ໝ ແࡦफउˍ ࠣ‫ݶ‬ Ɖܸࢤ ެߕକଐƁƢࠣ‫ࡦ ࢽޟܸ ࢲޟ‬ƁƢࠣ‫ޟ‬පඓ ൷Ɓ౭౑ࢲ କଐƁƄ ܸࢲ ƁƋኃ ᒔᐺ። ጏ࢓‫ݕ‬ ࡦƁ࣬ ࣊ Ɓ࢓ߢೕಹኃ ቮƁ೾೥࢓଍ଉƁƄ ƁƋƁ߁ ࢟ ſ ƛ ܸߢࢽ ࡦƁ߁ ዖዄ Ž૊ સၫƋࣝެ࠾ኃ ቮƁƊܸެࢲ ଘଐƁ ። ፎକଐ߁ ެࢲ କଐƁƢ। ्࢟ ࣝࡦƁŻܸࢤ ߾ ଔଐƁ ྽ Ⴐ ࠁ ࡣ ‫ߦݘ‬ᑃᐽࠦߦ࡯ޯ༧ீ Ƈ ܻߘܻߥޯࢸ Ƅơ ൎ൉ၯ ࡼ ࡷƨࠦၰ Ƅྲ Ⴋ Ꭸᎌ ಒ౷ᚻࡩƄƣ ſ࢖ߥࡩƄ፥ ጿžܻࢧ ࠁ ଗଓƄࠦ൞ᇅᆝ Ƈ ƌƎƥ ‫ ۓ‬ᐟࡷࠦެࡓ ࡷƀƆ߄ ၮ፥ ጿ ፟ ጘ࣠ଐଌƄ཯ ƭཅƯ༨စƬ Ƭࡼ Ư༨စƫ ܻƮᄦ‫ۓ‬ Ƃƥ 94Ꭸᎌ୹ ୪ƆƎ࣠ࢵ Ƅ߄Ʈ‫ے‬ ƫၮƯࡴƬࠧ ߄Ʈ‫ۓ‬ ƫၮƫƥƯࡴƬܻࠧƫࢧ ࠁƯ ƀƯ ᚰ ߄ ƫߥƮࡩܻᐱ ۗ ә ܻࢥ ‫ݹ‬ƀƥ 12໠ ໄࢵ ܻ‫ݹ‬ƆƎ࣠ࢵ ƪ໠ ໄƭީƮࡷ‫ە‬ƢƪܻƫࡷƂ፥ Ƭᆝƫƥ ƯࡴƬࠧƫƎܻƫᄬſƫƥ Ƅ࣠Ƭࡷ߄Ʈ‫ۓ‬ ƫߘܻƫࡷ Ƭ ‫ݪ‬ƬޯƯࡓƫ‫ރ‬ƫƥ ۗ ƭጿƬƤܻƫެƯყࢧƫ ࠁƯ ƀᚰƪžƯ࿃ ƫ႖ᇯƮ‫ۓ‬ ƭ໠Ưໄƫ‫ހ‬Ư࢔ƭஊैƯ ƄƧƮ‫ە‬ƀƫ ƫ‫ހ‬ƮᎨ‫ۓ‬ Ƭ᎜Ә ࡣ ႁ ެࡩƄ18 Ʃযওƭࠁ ƫƎ ƩƎ ࣠Ƭߦƫ‫ݓ‬ ƫࡩƫቻƮƫᒗᐽƄƮƫƣ Ƃܻ ƫ‫ކ‬Ưƣ ᓘƫۗ ƥ ᚰᚷƫࣳƯ‫ݘ‬ƫެƭࡩƯࡴƬܻࠧƫެƯ޲ƫ‫ݘ‬ ۗ ƫࠧ࣠Ƭࢧ Ư༨ဈƬ ଘƫଓƭቻƫƮᒗᐽƄƫࠦƫࡺ Ư‫ݘ‬ƭ‫ރ‬ƄƥƮƬ߄Ƭ‫ݘ‬ ࡴƬᐱƬᐒƭƮީƫჱƬ‫ޢ‬ƫߙܻƨ‫ޢݘ‬ƭዙƫዋ Ƭ‫ٷ‬Ʈ‫ٵٶ‬ƄƬ ଐ Ⴐ ࠁ ƀ ᚰ Ƭ޲ƭީƫࡩܻƫƭଌ྽ ‫ࡴۓ‬ƬᐱƬᐥƫ‫ݘ‬Ư୍ƫ෺ƫƢ Ư࣠ƫ‫ݹ‬ቿቱƌ഻ܻവറƄƆƎ࣠ࢵ ፥ ጿƇ ƌƎƥሯሡሁ Ә 12ߎ ࠩࡓ ᚻࡩƄә Ƅƨ߄ƫࡷƀƄ࣠ (6ቿቱƌ഻ܻവറƄƆƎ࣠ࢵ Ʃ߄ ‫ޢ‬ƭᐳƫ႖ƪžƯ࿃ ƫ႖ᇯƭƮ Ƭᆝ ે ઻ၮƬ‫ٷ‬Ʈ‫ٵٶ‬Ƅ ƫƥ ƬƤ࣠Ƭఱƫఝƫƥ Ƭ‫ٷ‬Ʈ‫ٵٶ‬ƄƬ ᚰ Ƭळƭዙƫኸ ܻƫ೘ƭೃ ‫ۓ‬ ‫ۓ‬Ƥܻƫࢧ ࠁƯ ƀƄॅ ߃ ެࡷ Ƣ߄ ႀ ޯࢵ Ƅ߄ ߥࡩƥ߄ ‫ࡩݘ‬Ƅ཯ཏ‫ݹݘ‬žܻࢧ ࠁ ଘଓƄƢ ࣠ࠩߦࡷƣ ܺߙ ،ܻߦ࡯ࢵ ƆƎ࣠ࡁ߃ ଐଌƄࠦ୆คſ࢖ߥࡩƄƇ ܻ‫ݹ‬ଜଓƄᇯ ᆝ ፥ ጿ Ƈ ࢖ࠩ࣌ ཯དྷࡩƄơ ƥኆ ᒗᐽࠦߦ࡯ޯശവറƄࠦ‫ݹ‬ƎƄƌଘଓƄƇ ܻࣀ ࢵ ࣓ଐଌƄƥƇ ܻࡷ࣠ࡓ ஊै Ƅࡴ ᐱᏹ཯དྷࡩƄƇ Ƅžܻࢧ ࠁ ଘଓƄ፥ ጿƢ߄ ߥࡩƄ ƌ࣠ެஎै Ƅƌ߄ၮƥƣ ᇴᇍ ࣀ ࡩƄ߄ ၮƥྲ Ⴐ ࠁ ፥ ጿžܻࢧ ࠁ ଘଓƄƢ߄ ႀ ޯࢵ Ƅƥ‫ޯ࡯ߦࠦۑ‬༧ீ ࠦ‫ޢ‬༧࿻ Ǝܻ‫ ࡣކ‬ၰ Ƅ୹ ୪፥ ጿ . Ʀ ༇໩ ଘଓƄƇ Ƅƥ౰౗ࡩƄ݊ ‫ޢ‬ዙዋ ƥƇ ܻ‫ރ‬Ƅ࣠‫ޢ‬ஊै Ƅƥ፥ ፎ ƄƎ༝༎ƄƉ ܻ୓คଘଓƄƥƠ ൎ൉ࡷଘଓƄƥ stics ƥቿቱƥኆ ቱƄ཯ཏ‫ކݘ‬ƥ State ܻዙ዗໠ ໄ‫ [ ࡷ ࡓ࣠ރࠦࡷ ࡼࡷߥ ࣐ݘ‬Statistics ] žܻࢧ ࠁ Ƃࠦ൞ᇅᆝ ࡣ ࢢ Ƅ ‫أن‬ [Statistics žܻࢧ ࠁ ଘଓƄࠦ൞ᇅᆝ ‫ۑ‬ቿቱƥኆ ቱƄƇ ܻߥ࡯‫཯ࡷޯݘ‬ཏ‫ݹݘ‬ཀྵཏ‫ݘ‬ଐଌƄƄ߃ ᐱᏳžܻࢧ ࠁ ଘଓƄƥ‫ۑ‬Ƈ ܻߥ࡯‫ޯݘ‬ଐଌƄܻܻࠧެ‫ࡷݘ‬ అ௷ ஊै Ƅࡼ ߙ߄‫ޢ‬༬࿻ Ƨ ኅ ቱƄቿቱƥኆ ቱƄࡣ ၴ Ǝ཯ཏ‫ކݘ‬ƥ { Statista } ࠦ‫ެޢ‬ყ჋ൎᑉᐽƄࠦሤሡሓ ᚻࡩƄࡼ ࡷบ งƀ] Ƨ ኅ ቱƄ^ 1719-1772،Gottfried`Ǝ࣠ࣀ ߙƥ౰౎ࡩƄࠦ൞ᇅᇍ ࡩƄƤ߃ ࠧƢ߄ ႀ ޯࢵ Ƅ߄ ߥࡩƥ {Statesman } ፥ ጿƆ୹ ୪ơ ƥଛଓƇ ࢖ࠩ࣌ žܻࢧ ࠁ ଘଓƄࠦ൞ᇅᆝ ƥƆƌ߄‫ࡷޯݘ‬ƣ Ƅኆ ቷީࡩࠦ‫ࡩ ࣀޢܻ ࢵޢ‬ƄƢॅळ‫ࡩݘ‬Ƅࠢ‫ހܾރ‬žܻࢧ ࠁ ଘଓƄƞ ୹ ‫ݑ‬ ƣ Ƅ࣠ެ‫ހݘ‬Ǝ࣠ᐳႡଐଌƄƅ ܻຨ຀ࡩƄ Ʀ ߄ ၰ Ƅ፥ ጿ " Elements of Universal Erudition " (Baran J.F.Von Bielfed1770) ࿃ ႓ܻ‫ࡩ ࣀޢ‬ƄƢܻ࣏ ެࡩƄࠦ‫ެܻݑ ࡼࡷܻࠧޢ‬൞൙‫ݹݘ‬Ƨ ኅ ቱƄࣷࣳ‫ࡩݘ‬Ƅࠢ‫ކݘ࢖ݹߦܻހܾރ‬Ʀ ࣠ޯࠁ ƀžܻࢧ ࠁ ଘଓƄƣ Ƅ࣠ެ‫ހݘ‬ƅ ܻຨ຀ࡩƄ ơ ࣠ࢧ ߙ "ࠦୖ෺߄ ஊै Ƅơ ƥኆ ᒗᐽ ࠦߥ࡯‫ޯݘ‬ଐଌƄ ᑻ ᑧ‫ܻ ࢢࠦ ކ‬ၯ ƥ ࠦ୆෷ܻࢧ ࠁ ଘଓƄ Ɵ ࢖࣐ ࡩƄ ƥ ‫ࡩޯ ࣐ީޢߥޢࠦ ۑ‬Ƅ Ƈ ܻ‫ݹܻࢠޢ‬༟༎Ƅ ࡼ ࡷ Ɩ ࢖ߙ ࣠ࠧ žܻࢧ ࠁ ଘଓƄ ‫ࡷ ࡼ رواد‬ƥ‫ۑ‬Ƈ ଘଓܻࡺޯࠁ ଘଓƄơ ࣠ࠁ Ƈ ܻ‫ݹܻࢠޢ‬༟༎Ƅࠦ‫઻ ેރ ࣏࢖ݹ‬ၮཀྵཏხ჋ࠦެ‫ࡩݘޢ‬Ƅࡼ ࡷ݊ ࡺޯ഻വറƄࡼ ‫ݑ‬ơ ଘଓ߄ ޯࢵ ଘଓܻ‫ހ‬ Karl ، Laplace ، ( James Bernoulli ơ ܻ޲ࡷƀࡼ ࡷƇ ܻ‫ݹܻࢠޢ‬༟༎Ƅžܻ൞൙ၮƇ ଘଓܻࡺޯࠁ ଘଓƄࠦ‫ރ ࣏࢖ݹ‬ . Ƈ ଘଓܻࡺޯࠁ ଘଓƄࠦ‫ރ ࣏࢖ݹ‬मअ࣐࣠ ‫ކ‬፥ ጿƄ࣠ࡺᐳ႓ƀ௪ ௎ኅ ቱƄ ،Gauss - 1777 Ɛ ƥܻၴ ࠯ ‫ݹ‬Ǝਔ ৲࢖ߙ፥ ጕܻଐଌଛଓƄࡴ ࠧžܻࢧ ࠁ ଗଓƄࣷࣳၮƎ࣐࣠ ‫ކ‬፥ ጿƨƄ౰ేީࡁ ƄƎƥƌƄ࣠ީ‫ࡩݘ‬௪ ௎ኅ ቱƄžܻ൞൙‫ࡩݘ‬Ƅ 1857 ƣ ࣠ࢵ ౰ే‫ހ‬ơ Ǝᇴᆝ Ƨ ౮ే࡯༬စ ଗଓƄᎨ፻ܻ‫ݘ‬ᑃᐽƥ‫ ۑ‬1827 – 1749 Ɛ ൎ൉‫ހ‬ଘଓ࿃ ྜྈ࢖ߦࡩƄƥū 1858 Ɣ ୍ܻงƎଘଓƄࡣ ࡷܻ‫ࡷݘ‬ƥƔ ୍ܻงƎଘଓƄࡣ ࡷܻ‫ܻކݘ࢖ݹ ߎࡷݘ‬ᐱᐟࡷžܻࢧ ࠁ ଗଓƄࣷࣳၮ፥ ጿƆ౰ే޲ࡁ Ƈ ܻࡷܻᐳ႓Ƃ 1936 – 10 ܸୃกܸ଍ଉŽ–ໝ ແࡦफउ –ࠣ‫ݶ‬ Ɖܸࢤ ެߕକଐƁƢࠣ‫ࡦ ࢽޟܸ ࢲޟ‬ƁƢࠣ‫ޟ‬පඓ ൷Ɓ౭౑ࢲ କଐƁƄ ܸࢲ ƁƋኃ ᒔᐺ። ጏ࢓‫ݕ‬ ࡦƁ࣬ ࣊ Ɓ࢓ߢೕಹኃ ቮƁ೾೥࢓଍ଉƁƄ ƁƋƁ߁ ࢟ ſ ƛ ܸߢࢽ ࡦƁ߁ ዖዄ Ž૊ સၫƋࣝެ࠾ኃ ቮƁƊܸެࢲ ଘଐƁ ። ፎକଐ߁ ެࢲ କଐƁƢ। ्࢟ ࣝࡦƁŻܸࢤ ߾ ଔଐƁ ౰౎ޯ‫ݹݘ‬ƥ‫ࡩޢࠦۑ‬ൎ൉ߥޯࢵ ଘଓƄƥࢇ ‫ࡩޯ࣠ߙޢ‬ƄƆƌ࣠ࠅ Ƈ ƄƎܻީޯࠀ ଘଓƧ ᇴᆝ ݊ ‫ ୹ހ‬୪Ǝܻީޯࠀ ƄƢƄ߄ ႀ ޯࢵ ƄƥƤमअ߄ ߥ‫ކ‬ƥ፥ ገ࢔எै Ƅ ūžܻࢧ ࠁ ଗଓƄࣷࣳ‫ࡩݘ‬౰ే޲ࡓ ᚻࡩƄƄ࣠ߙܻࢠƀ௪ ௎ኅ ቱƄࡼ ࡷ ( 1962 – 1890 ྲ Ⴁყߙኆ ቱܻ‫ރ‬ƥƎ፥ ገܻࢧ ࠁ ଗଓƄᎨ፻ܻ‫ࡩݘ‬Ƅ ࣷࣳၮ፥ ጿƇ ܻࡷܻᐳ႓ଗଓƄࡼ ࡷܻࠧ౰ేၩƥ௪ ௎ܻީޯࡩƄࡣ ‫࡯ޢ‬༨ဈ ƥƅ Ǝܻႅ ޯࡩƄযওࡺࢧ ‫ࣳࣷކ‬ၮƇ ܻ‫ ࢵܻ ࢵޢ‬ƀ݊ ࢠƥƧ ኅ ቱƄ࣠ࠧƥ . žܻࢧ ࠁ ଗଓƄ Żܸࢤ ߾ କଐƁࣰࣴ ၫߋ ‫ݕ࢓ݶ‬ ‫ ރ‬1.1 Ƈ ଘଓ഻ܻവറƄᇯ ᆝ ፥ ጿࠢީ‫ ࢵܻࡩޢ‬ƀƥࠢ‫ݶ‬ ƌܻީࡷƥūࠢ‫ࠦކ ࣐ީޢ ࢇރ ࣏࢖ݹܻކ‬୆คᇴᇍ ࡷƂࡼ ࡷࠢޯ‫ޢ‬ዙ዗ƀžܻࢧ ࠁ ଘଓƄࣷࣳၮ‫ ࣀ ݪ‬ჴࡁ Ƅ ‫ ࢵܻࡩޢ ݪ‬ଛଓƄƢƄ߄ ႀ ޯࢵ Ƅƣ ᇴᇍ ࡷଗଓܻ‫ ࢢީ ߰ހ‬ƀ߄ ߥߙ‫ޢݘܻࠩۑ‬ዙ၅ေࡼ ࡓ ೘಼Ƈ ܻ‫ޢܻރ‬ხჅƤ࢖ࠧƄ࣠࣌ ࡼ ‫ݑ‬౰ేީ‫ࡩޯݘ‬Ƅࡼ ࡓ ೘಼཯དྷࡩƄ .Ƣॅळ‫ࡩݘ‬Ƅߎ ࡯ޯ༧ீ ፥ ጿܻᐱᏹܻߥ‫ކ ࣐ީޢ‬ƥࠦ୆෷ܻࢧ ࠁ ଗଓƄ Ƣॅळ‫ࡩݘ‬Ƅƞ ൎ൉ޯࠀ ଘଓܻߥީ࣍ ࠦߦ࡯ޯ༧ீ Ɵ ࢖࣐ ‫ހ‬ƞ ࢖‫ݹݘ‬Ɔƌ߄ ‫ࡷޯݘ‬Ʀ ༇໩ ƀƢॅळ‫ߘޯࠢހݘ‬ൎ൉‫ࠦࡩݘ‬ႅ ‫ޢ‬ჴ჈ƥࣷࣳ‫ ࡁݘ‬žܻࢧ ࠁ ଗଓƄ ഃ೨ƍ ࡼ ࡓ ೘಼ƥ໠ ໄ୆෷ܻࢧ ࠁ ଘଓƄࡼ ࡷਔ ৲߄ ‫ࡩݘ‬Ƅࡣ ީߘࡼ ࡷƆƌ߄‫ࡷޯݘ‬Ƈ ܻߦ‫ކݘ࢖ݹ‬Ƈ ࢖ࠩ࣌ ƥ‫ ࢵࠦۑ‬ƄƎኆ ቱƄƇ ଘଓܻ༬ீ ƥ : ፥ ጘଘଓƄ࣠ࠒ ެࡩƄે ઻ၮƇ ܻߦ‫ࡩޯݘ࢖ݹ‬Ƅ࢛ ‫ހݘ‬ Ɛ ܻࢵ ᇷᆝࠦ‫ޢ‬ዙኼ༟༎Ƅࢇ ‫ݶ‬ ܻߥஊै Ƅว ෺୍࣠งƥߎ ‫ޢ‬ჱࢧ ‫ ކ‬،݊ ዙዋ ݊ ࡷࡣ ࡷܻ‫୓ݘ‬෺Ƨ ኅ ቱƄࣷࣳ‫ࡩݘ‬Ƅ࣠ࠧžܻࢧ ࠁ ଘଓƄ .i .࢖ࠧƄ࣏࣠ ࡩƄࠦ‫ރ‬Ǝܻߥࡷƥߎ ࢢ ƥƥ౰ేࣀ ߦޯࡩ Ƈ ଘଓܻࡺޯࠁ ଘଓƄƥƇ Ƅमअ߄ ߥޯࡩƄࣷࣳၮ࣠ࠧžܻࢧ ࠁ ଘଓƄࣷࣳၮ .ii Ƈ ܻ‫ࡩީޢܻރ‬Ƅ ౰ేࣀ ߦ‫ކ‬ƥࡣ ‫࡯ޢ‬༨ဈ ƥযও࣏ ୐งƥ ،݊ ସବଣ ࠦࡷ߄ ႀ ޯࣀ ଐଌƄƟ ࢖࣐ ࡩƄ፟ ፑƄ౰ేఴ ఓžܻࢧ ࠁ ଘଓƄ .iii ƣ ࣠ߥߦ୓෺໠ ໄ୆෷ܻࢧ ࠁ ଘଓƄƣ ƄଘଓƄžܻࢧ ࠁ ଘଓƄࣷࣳၮߎ ‫ކݘ࢖ݹ‬፥ ጿࠦߦ‫ࡩ ࣐ߦޢ‬ƄƇ ܻߙൎ൉ޯࠀ ଘଓƄࡼ ࡷࡴ ‫ ݌‬༟༎ܻ‫ހ‬ ."Ƈ ܻ‫ࡩީޢܻރ‬Ƅ౰ేࣀ ߦ‫ކ‬ƥว ෺୍࣠งƥߎ ‫ޢ‬ჱࢧ ‫ކ‬ƥ݊ ዙዋ ࣷࣳၮࠢ‫ހ ࡓ࣠ރ‬ ። ፎକଐ߁ެࢲ କଐƁƢ। ्࢟ ࣝࡦƁŻܸࢤ ߾ କଐƁ  ࢝ ‫ ߄ࡷࠦࡩޯ࡯ ࠑޢ‬ႀ ޯࣀ ଐଌƄ Ɵ ࢖࣐ ࡩƄ ‫ ࢵܻࡩޢ ݪ‬ଘଓƄ१ ॐࢢ ࣠ࡩƄžܻࢧ ࠁ ଘଓƄƧ ࣠ޯ༨࿻ १ ॐࢢ ࣠ࡩƄžܻࢧ ࠁ ଘଓƄ .I [ ܻ୆คܻ୆กƣ ࣠ࡓ ‫ކ‬ƣ Ƅࡼ ࡓ ೘಼‫ ࢵܻࡩޢ ݪ‬ଘଓƄƤ߃ ࠧƥ‫ܻࠧۑ‬౰ేࣀ ߦ‫ޢ ࡣކ‬ᐳྜྋƓ ࢖‫ݓ‬ ‫ހ‬ᑑ ᐽƍƥƇ ܻ‫ࡩީޢܻރ‬Ƅߎ ࢢ ƥƥ . ࠦީࢵ ܻެࡷơ ƥƄ߄ ၴ ፥ ጿܻࠩ‫ࢠݘ‬ƥƥ፥ ጒܻࣀ ஊै Ƅࡣ ‫࡯ޢ‬ႁ ޯࡩƄƥƀgraphical ] Ƅ߃ ࠧࡼ ࡷࠦ‫ ࣀ ࠒ࣠ހ‬ଐଌƄࠦެ‫ࡩݘޢ‬Ƅ݊ ߘƄƥࡼ ࡷᑑ ᐽƍƥ፥ ገܻࢧ ࠁ ଘଓƄ݊ ࡺޯ഻വറƄơ ࣠ࠁ Ƈ ƄƎƄ࢖ߥࡩƄƍܻ༧ဈ Ƅܻಛಗൎ൉ၯ .II ࠦެ‫ࡩݘޢ‬Ƅߎ ࢢ ƥࣩ ࡺఱ ఠƥ‫ ࠁޯࡺܻࡩޢࠦۑ‬ƄƔ ƥྲ ႖‫ ݝ‬༨ဈ ܻࠧƍܻ༧ဈ Ƅযঠ‫ݹ‬Ƈ ƄƎƄ࢖ߥࡩƄƤ߃ ࠧƥ‫ۑ‬፥ ገܻࢧ ࠁ ଘଓƄ݊ ࡺޯ഻വറƄ ݊ ࡺޯ഻വറƄߎ ࢧ ‫཯ކ‬དྷࡩƄ࢝ ‫ݶ‬ ܻࢧ உै ƄܻࡺެყჅ [ sample statistics] [ ࠦެ‫ࡩݘޢ‬Ƅࠦ୆෷ܻࢧ ࠁ ܺᄣܻᐳႰ ‫ݶ‬ ܻࢧ ࠀ ƥ [ Parameters] .Ꭸ፻ܻ‫ࡺ ࣩࡷݘ‬ఱ ఠ Żܸࢤ ߾ କଐƁࣰࣴ ၫবঐࠤܸߣࡴ 11 1.2 ܸୃกܸ଍ଉŽ–ໝ ແࡦफउ –ࠣ‫ݶ‬ Ɖܸࢤ ެߕକଐƁƢࠣ‫ࡦ ࢽޟܸ ࢲޟ‬ƁƢࠣ‫ޟ‬පඓ ൷Ɓ౭౑ࢲ କଐƁƄ ܸࢲ ƁƋኃ ᒔᐺ። ጏ࢓‫ݕ‬ ࡦƁ࣬ ࣊ Ɓ࢓ߢೕಹኃ ቮƁ೾೥࢓଍ଉƁƄ ƁƋƁ߁ ࢟ ſ ƛ ܸߢࢽ ࡦƁ߁ ዖዄ Ž૊ સၫƋࣝެ࠾ኃ ቮƁƊܸެࢲ ଘଐƁ ። ፎକଐ߁ ެࢲ କଐƁƢ। ्࢟ ࣝࡦƁŻܸࢤ ߾ ଔଐƁ ። ጅܸࢤ ߾ କଐƁ݇ ࡷެസലമƁƢࠣީ‫ݕޟ‬ ࡦƁ .1.2.1 ࣠ࡩൎ൉޲ዙኽ‫ۑ‬Ƅ߄ ‫ ࠅޢ‬ൎ൉‫޲ޢ‬೘೉ቻቱቿቷ޲ያዪƣ ࣠ࡓ ‫ޢ ݟކ‬༨ဂ ،݊ ࡺޯ഻വറƄࡼ ࡷž༅໮ ፥ ፕ [ Sample ] : ࠣީ‫ݕޟ‬ ࡦƁ  ࣏ࠦ ߙܻ༨ீ ፥ ጿƆ࢖ࡺ޲ଐଌƄࡣ ‫ࡩެ ࠑޢ‬ƄƎܻ༬ೞ Ƅƥƀƣ ߄ၮࠦ‫ܻࡷݘ‬ၴ ፥ ጿƅ ƄƌଜଓƄࠦ‫ޢ‬ᆿᆝ ƅ ൎ൉࣍ ࡼ ‫୐ܻݑ‬ช߄ ༨ဈ ƣ ߄ ၮ፥ ጿƢܻ‫ࡩݘ‬ƄƖ ܻ࣐ ߥࡩƄ१ ॐ࣌ ࣠ࡷࡼ ࡷž࢔எᆋ ƣ ߄ ၮ࣏ࠦ ߙܻ༨ீ ፥ ጿƧ ഁ೨࢖ଐଌƄ࠯ ެީࡩƄ१ ॐ࣌ ࣠ࡷƥƀ ūƇ ࣠ࡷྲ Ⴎ ࠁ [ Statistic] ࠦ୆෷ܻࢧ ࠁ Ƃࣩ ࡺఱ ఠࠦެ‫ࡩݘޢ‬Ƅ࢝ ‫ݶ‬ ܻࢧ ࠀ ƥ‫ࡩݘޢެࠦۑ‬Ƅ፟ ፑƄ౰ేఴ ఝܻ୐คܾߙ‫ۑۑ‬ ᇯ ᆝ ፟ ፑƄƆƎܻࢸ ്ᑉᐽ໠ ໄ୆෷ܻࢧ ࠁ ଘଓƄࡣ ީߘࡼ ࡷƢ߄ႀ ޯఱ ఠ݊ ࡺޯ༬ீ ࠦ൞ᇅᆝ [ Population ] : ݇ ࡷެസലമƁ  Ǝܻ༬ೞ Ƅᇯ ᆝ ƥƀƣ ߄ ၮࠦ‫ܻࡷݘ‬ၴ ፥ ጿƅ ൎ൉࣐ ࡩƄᇯ ᆝ ơ ࣠ߥ‫ݑެ߄ࡷܻރ‬ൎ൉޲ࡷ‫ ࢵࠦۑ‬ƄƎኆ ᒗᐽܻࠧƎܻ‫ ࠀޯޢ‬ƄᎨ᎙཯དྷࡩƄྲ Ⴋ ܻެ‫ࡩݘ‬Ƅ ߄ࢧ ߥ‫୐ܻރ‬คܾߙ‫ۑ‬ƣ ߄ ၮ࣏ࠦ ߙܻ༨ீ ፥ ጿƢܻ‫ࡩݘ‬ƄƖ ܻ࣐ ߥᑃᐽ໠ ໄ‫ࡩޯܻހݘ‬Ƅ໠ ໄߦ࣌ ࣠ଐଌƄᇯ ᆝ ƥƀࡼ ৤ৈষƄ፥ ጿƆ࢖ࡺ޲ଐଌƄࡣ ‫ࡩެ ࠑޢ‬Ƅ [ Parameter ]ࠦ൞൙‫ࡺ ࣩࡷݘ‬ఱ ఠ݊ ࡺޯ഻വറƄ࢝ ‫ݶ‬ ܻࢧ ࠀ ƥ‫ۑ‬፥ ገܻࢧ ࠁ ଘଓƄ݊ ࡺޯ഻വറƄ Ƅ ܸ‫ࡦަޟܸހ‬Ɓ݇ ዖወ .1.2.2 ፥ ገܻࢧ ࠁ Ƃ݊ ࡺޯ༬ீ ࡼ ࡷܻࠩ‫ݘ‬ዙዋ যঠ‫཯ݹ‬དྷࡩƄƇ Ƅ߄ ܻࠧࣃ ଐଌƄƥࢇ ‫ݶ‬ ܻߥஊै Ƅࡼ ࡷࠦၮ࣠ዙಣಝ፥ ፕ [ Data ] Ƅ ܸ‫ࡦަޟܸހ‬Ɓ ቿቱܻஊै Ƅ ،ࡣ ၯ ኆ ቱƄ࣊ ࢵ ࣠ޯࡷƥūযও࡯‫ࡩޯݘ‬ƄƦ ࣠ޯࣀ ࡷƥࠦެࠩଐଌƄƥࡼ ࣀ ࡩƄƥᄎ ႓ଘଓƄƇ ܻ‫ࡩީޢܻރ‬Ƅቿቷ޲ࡷƀࡼ ࡷƥ‫ۑ‬໠ ໄ‫ࡷݘ‬ ᅵै Ƅ‫ ࠅޢࠦۑۑۑۑ‬ƥ༝༎Ƅ ࠦ‫࡯ޢ‬ዙኸ ፥ ጿࠦߘኆ ቱƄƥࠦ‫ࡩݘެܻݹ‬Ƅࡼ ࡷਔ ৹ଘଓƥ‫ۑ‬፥ ገܻࢧ ࠁ ଘଓƄ‫ࡩީ ࠒ ݟ‬Ƅ፥ ጿ፟ ፑƥଘଓƄƆ࣐࣠ உै Ƅᇯ ᇍ ఴ ఓƇ ܻ‫ࡩީޢܻރ‬Ƅ݊ ዙዋ ࠦ‫ܻ ࣍ޟ‬உै ƄƇ ܻ‫ࡩީޢܻރ‬Ƅƣ ଘଓᑑ ᐽƍƥ፥ ገܻࢧ ࠁ ଘଓƄࡣ ‫࡯ޢ‬ႁ ޯࡩƄ፥ ጿƐ ܻࢵ ଘଓƄ౰౎ޯ‫ܻކݘ‬ᐱ᏶ଘଓࠦ୆෷ܻࢧ ࠁ ଘଓƄƇ ܻ‫ࡩީޢܻރ‬Ƅ݊ ዙዋ . ܻᐱᏳƟ ࣠‫ࡷ࣠މ‬౰ేၩƥࠦ‫ܻ ࣍ޟ‬ၯ Ƈ ܻၴ ܻޯެჴࢵ ƄƆƎƥྲ Ⴎ ࡩܻ‫ޯ ߴހ‬ჱ჋ ‫ࡩޢࠦ و‬ƥƀƇ ܻ‫ܻރ‬୆ก፟ ፑƄߎ ެࢧ ‫ࠦކ‬୆෷ܻࢧ ࠁ ଘଓƄƇ ܻ‫ࡩީޢܻރ‬Ƅƣ ܻߙƎƌܻࢧ ଐଌƄ‫ ࠁ ࣀ ݪ‬Ƅ ܸ‫ࡦަޟܸހ‬ƁƋƉܸࢤ ࡴ 1.2.3 ࠦ‫މܻރ࣠ݹ‬Ƈ ܻ‫ܻރ‬୆ก ཯ དྷࡩƄࠦࢵ ƄƎኆ ቱƄƓ ࢖‫ݓ‬ ࡩࠢࣀ ߦ‫ࡩީܻ ࠁ ݟރ‬Ƅࡣ ީߘࡼ ࡷ‫ީ ࠒ ݟ‬ᑃᐽࡣ ࢢ ଘଓƄ፥ ጿ݊ ዙ၅ေƇ ܻ‫ܻރ‬୆ก ࠣ‫ࡦޟ‬ƢକଐƁƄ ܸ‫ࡦަޟܸހ‬Ɓ ࡼ ࡷƦ ࢖༬ဈ ཯དྷࡩƄƆਔ ৲߄ ‫ࡩݘ‬Ƅ Ƈ ܻၰ ࣠ࣀ ଐଌƄࢇ ‫ ݊ݑ ࡼ ࣍࢖ݹ‬ዙ၅ေƥࠦߦࢧ ࡩƄࠦ‫ ࢢ࡯ޢ‬ƄƇ ܻ‫ࡩީޢܻރ‬ƄƤ߃ ࠧࡣ ޲ࡷ‫ܻۑ‬ᐱᏳƢ࣠ߥ‫ݹ‬ Ǝ߄ ࢧ ଐଌƄࡼ ࡷॡ ॏޯࣀ ࡷ፥ ፕ ࠦ‫ࡩޢ‬ƥଘଓƄƇ ܻ‫ࡩީޢܻރ‬Ƅ‫ۑ‬ƈ ܻ༨ဂ ଘଓƄഁ೨ Ƅ୹ ୪ƥƇ ܻࣀ ࢵ ࣓ଐଌƄ࢛ ‫ހݘ‬ƥࠦࡷ࣠ࡓ ஊै Ƅࡣ ީߘ ‫ ݝ‬༨ဈ ƣ ᇴᇍ ࣀ ᑃᐽ፥ ፎ ܻࡺޯࠅ ଘଓƄ -Ƨ ƌܻࢧ ޯߘଘଓƄ݊ ࢠ࣠ࡩƄࠦࢵ ƄƎƌࡴ ࠧ߄ ၰ ƄƌƄƎƄƄƍƄơ ܻ޲ଐଌƄࡣ ‫ޢ‬ხࢵ ે ઻ၮ‫ۑ‬ơ ƥଘଓƄ ūƆྲ ႓ଘଓƄࡣ ၯ ƌū࣯ ൞൙‫ࡩݘ‬Ƅࡣ ‫ࡩޯܾࠧޢ‬Ƅ ،ƣ ᇴᇍ ࣀ ࡩƄࠦ‫࡯ߦޢ‬ၯ ࡼ ‫ݑ‬Ƈ ܻ‫ܻރ‬୆ก݊ ዙ၅ဤࠢ‫ ߙܾރ‬،ࠦެ‫ߘ࢖ݹࠦࡷݘޢ‬፥ ጿ࢖ߥߦࡩƄ࣊ ࠀ ƢܻߘƧ ኅ ቱƄơ ƥଘଓƄ࢝ ࠑ ࣃ ࡩƄࠢ‫ރ‬ଘଓ‫ࡩޢࠦۑ‬ƥƀƇ ܻ‫ܻރ‬୆กࣩ ࡺఱ ఠƇ ܻࡷॅळ‫ݘ‬ଐଌƄƤ߃ ࠧᇯ ᆝ ،ᅵै Ƅ ،໠ ໄࡩܻ‫ݘ‬ଐଌƄƌ߄ ၮ ࠦ‫ࡩޢ‬ƥଘଓƄƇ ܻ‫ࡩީޢܻރ‬Ƅ‫ۑ‬Ƈ ܻ‫ࡩީޢܻރ‬ƄƤ߃ ࠧ݊ ዙ၅ါ ౰ేၩƥƀ ቿቷ‫ߥܻހ‬ଐଌƀ Ɔྲ ႖ܻީࡷࠦߥ‫ࡷެࠢݑޢެࠦ ހ ࣐࢖ݹ‬ž༅໮ ƥƄቻᇝᆝ ݊ ࡺޯ഻വറƄƌƄ࢖ߙƄࡼ ࡷܻᐱᐕ࡯ၮơ ࣠ࢧ ஊै Ƅࡼ ࡓ ೘಼ (፥ ፎ ܻࡺޯࠅ ଘଓƄࡣ ࢢ Ƅ࣠ޯࡩƄ บ ค౰౔‫ރ‬ଘଓƄƥ ƣ ࣠ߦ࡯ޯࡩƄƥਔ ৲౰౎ࡩᇴᆝ ،Ɔྲ ႖ܻީࡷ 12 ܸୃกܸ଍ଉŽ–ໝ ແࡦफउ –ࠣ‫ݶ‬ Ɖܸࢤ ެߕକଐƁƢࠣ‫ࡦ ࢽޟܸ ࢲޟ‬ƁƢࠣ‫ޟ‬පඓ ൷Ɓ౭౑ࢲ କଐƁƄ ܸࢲ ƁƋኃ ᒔᐺ። ጏ࢓‫ݕ‬ ࡦƁ࣬ ࣊ Ɓ࢓ߢೕಹኃ ቮƁ೾೥࢓଍ଉƁƄ ƁƋƁ߁ ࢟ ſ ƛ ܸߢࢽ ࡦƁ߁ ዖዄ Ž૊ સၫƋࣝެ࠾ኃ ቮƁƊܸެࢲ ଘଐƁ ። ፎକଐ߁ ެࢲ କଐƁƢ। ्࢟ ࣝࡦƁŻܸࢤ ߾ ଔଐƁ ൎ൉޲ࡷ‫ۑ‬Ƈ ൎ൉഻വറƄƥमअƎܻߥޯࡩƄƥ‫ ࡓޯ ݪ‬ᚻࡩᇴᆝ ،ƆƎ࣠ࣃ ჱଐଌƄ౰ేၩƥƄƆƎ࣠ࣃ ჱଐଌƄƇ ܻ‫ࡩީޢܻރ‬Ƅ፥ ፕ ࠣ‫ࡦޯܸހࣝݶ‬ƁƄ ܸ‫ࡦަޟܸހ‬Ɓ • ࡼ ࡷ߃ ၯ ࣓‫ޢ‬ᚻܻࡷᑑ ᐽቦሼƥ ،ܻᐱᐕࡷ߄ ႀ ޯࣀ ଐଌࠦ‫މܻރ࣠ݹ‬Ǝƌܻࢧ ࡷ౰౎ޯ‫ކࠢކݘ‬ƄƎ࣠ࣃ ჱࡷƥ ،žܻࢧ ࠁ ്ᑉᐽƧ ഁ೨࢖ଐଌƄƏܻࠩஎै Ƅ Ƈ ܻࡺ࣏ ެଐଌƄƇ ƄƎ࣠ࣃ ჱࡷᑑ ᐽቦሼ ‫ ( و‬Ɵ ൎ൉࣐ ࡩƄƥƉ Ƅƥ༝༎Ƅ ،߄ ‫ࡩޢ‬Ƅ࣠ଐଌƄƥƇ ܻ‫ࡩ࣠ߙޢ‬ƄƇ ଘଓܻၰ ፥ ጕ߄ ଐଌƄࡣ ႅ ࣀ ࡩƄ .Ɔ߄ ႁ ޯଐଌƄࡴ ࡷଛଓᇴᆝࠦ‫ࡩޢ‬ƥኆ ቱƄ ࡼ ࡷࠢ‫ݶ‬ Ƅ༇໮ Ƅࡼ ࡓ ೘಼ƈ ࣠ࠒ ީࡩƄࡼ ࡷƆƌ߄ ‫ޯݘ‬ଐଌƄƖ Ƅ࣠‫ރ‬൑ᑉᐽƇ ܻ‫ࡩީޢܻރ‬Ƅ݊ ዙዋ Ƅ ܸ‫ࡦަޟܸހ‬Ɓ݇ ዖወ Ɯ ࢓࣊ .1.2.4 :໠ ໄޯ‫ࡩޯܻࡩޢ‬Ƅ໠ ໄޯߥ‫ࡩ ࣐࢖ݹ‬Ƅơ ൎ൉ၯ [ census method] ࡣ ࡷܻࣃ ࡩƄྲ Ⴐ ஊै Ƅࠦߥ‫ ( ࣍࢖ݹ‬i ) [sample method ] ࠦެ‫ࡩݘޢ‬Ƅࠦߥ‫( ࣍࢖ݹ‬ii ) Ƨ ኅ ቱƄƥ፥ ገܻࢧ ࠁ ଘଓƄ݊ ࡺޯ഻വറƄࡼ ࡷࠦ‫࢖ ݌࣠ހ‬ଐଌƄƇ ܻ‫ࡩީޢܻރ‬Ƅ݊ ዙዋ ࠦ‫࡯ޢ‬ዙኸ Ƨ ࣠ޯ༨࿻ ߰ ࣀ ࡷᇯ ᆝ ࡠ ࡴܸࣀ ࡦƁྯ Ⴍ இॅ ƁI . ƣ ᇴᆝ ƄƍƄ‫ ࢵࠦۑ‬ƄƎኆ ቱƄ‫ ݝ‬༨ဈ ્ ᇣᇍ ࡩƄ݊ ࡺޯ഻വറƄࣩ ࡺఱ ఓƥ‫ ۑ‬ᅵै Ƅ‫ۑۑۑ‬žܻ‫ ࢸޢ‬Ƅ ،ƒ ܻ༧ೞ Ƅ Ƈ Ƅƌ࢖ߦࡷࡼ ࡷƣ ࣠ࡓ ୓෺ ࢇ ࣍ ܻެࡷ፥ ጿྲ ႓ଘଓƄᇯ ᆝࠦ೘ಹܻߘᇯ ᆝ ƅ Ƅ࣠ࠖ ޯࢵ Ƅƥ࢝ ஊ஀ ᑑ ᐽƍ‫୓ ࣐࡯ ݪ‬෺ ،ܻࡷኆ ቷ‫ހ‬ƣ ᇴᇍ ࢵ ࠦࢵ ƄƎƌƅ ॅळ࣐ ଐଌƄ ໠ ໄ‫ݑޢ‬ƄƎ༝༎Ƅơ ܻࡺ‫ݘ‬ᑃᐽƧ ƌܻࢧ ޯߘଘଓƄ݊ ࢠ࣠ࡩƄࠦࢵ ƄƎƌƅ ॅळ࣐ ଐଌƄƣ ᇴᆝ ƄƍƄƥ‫ۑ‬ኆ ቷީࡩƄᑑ ᐽƍ፥ ጿ ߎ ‫ݹ‬༟༎Ƅƥྲ Ⴎ ஊै Ƅ ፥ ጿ໠ ໄ‫ݑޢ‬ƄƎ༝༎Ƅ ơ ܻࡺ‫ࡩݘ‬Ƅ࣊ ߥߙࡼ ࡺࢥ ୓ข໠ ໄ‫ީ ࠒ࣠މ‬ଐଌƄࠦ೘ಹܻߘƣ ܾߙ،،ኆ ቷީࡩƄࡼ ࡷࠦެ‫ܻߙ ࣏ࠦࡷݘޢ‬༨ீ ࡼ ࡷܻࡷࠦ‫ݹ‬मअ߄ ࡷ፥ ጿ Ƈ Ƅƌ࢖ߦࡷࡼ ࡷƆƌ࢖ߦࡷᇯ ᆝ ƣ ଘଓࠦߘኆ ቱƄƥࠦ‫࣠މ࣠ߘޢ‬ଐଌƄ፥ ፕ ࡣ ࡷܻࣃ ࡩƄྲ Ⴐ ஊै Ƅࠦߥ‫ݹܻ ࣍࢖ݹ‬Ƅ୷ ୪ƣ ƀ‫ݹࠦۑ‬मअ߄ ଐଌƄᑻ ᑧ‫ކ‬ ߎ ‫ࡩޢ‬ᇴᇍ ‫ކ‬Ɩ ܻߦ‫ކ‬ƎƄ፥ ጿࡣ ޲๻๬ใࡣ ࡷܻࣃ ࡩƄྲ Ⴐ ஊै Ƅƅ ࣠‫ݑޢ‬ƣ ܾߙ༇໩ Ƅว คܻၴ ࡼ ࡷ‫ܻۑ‬ᐱᐢࢵ ƄƎƌযঠ‫ޯࡺ ݊ ࠧެܻݹ‬഻വറƄ .Ƣ߄ ႀ ޯࣀ ଐଌƄ‫ࡩ࣠ߘ ݝ‬Ƅơ ࣠࣍ ƥࠦެ‫ࡩݘޢ‬Ƅƅ ॅळࢵ ܾ‫ރࠦހ‬Ǝܻߥࡷ ྲ Ⴐ ஊै Ƅ :ࠦ‫ࡩޯܻࡩޢ‬ƄƇ ଘଓܻஊै Ƅ፥ ጿƢ߄ ႀ ޯఱ ఠࡣ ࡷܻࣃ ࡩƄྲ Ⴐ ஊै Ƅࠦߥ‫ ࣍࢖ݹ‬ƣ Ƃ Ƅƌƥ߄ ༨ீ ࠦࢵ ƄƎኆ ቱƄࡣ ༨ீ ݊ ࡺޯ഻വറƄƣ ᇴᆝ ƄƍƄ ࠦߘኆ ቱƄࡼ ࡷࠦ‫ܻࡩޢ‬ၮࠦၴ Ǝƌਔ ৲मएܻࡷ߄ ެ‫ݑ‬ ܻߘ࣠‫ࡷݘ‬ᇯ ᇍ ఴ ఓଘଓቻቱƌ߄ ഷവറƄࡼ ࡷ༝༎Ƅƥࡣ ࡷܻࣃ ࡩƄྲ Ⴐ ஊै Ƅે ઻ၮƟ ܻߦ‫ރ‬ଘଓƄƖ ࣠ࢠ࣠ࡷƣ ᇴᆝ Ƅ‫اذ‬ i) ii) iii) ࠦࢵ ƄƎƌযঠ‫ࠧެܻݹ‬ƥ‫ࡩݘޢެࠦۑ‬Ƅࠦߥ‫ ࣍࢖ݹ‬፥ ጿܻᐱᐢஎै ܻ‫ࡷݘ‬Ꭸ᎙ࡣ ࡷܻࣃ ࡩƄྲ Ⴐ ஊै Ƅࠦߥ‫ ࣍࢖ݹ‬ƅ ࣠‫ݕޟީࠣ ݑޢ‬ ࡦƁࠣߢ‫ێ ࣊࢓ݶ‬ii ƣ ܻߙ ࠢ‫࡯ޢ‬ၮ žܻ୐ก ƥ ،݊ ࡺޯ഻വറƄ ࡼ ࡷ ž༅໮ Ƈ ܻ‫ܻރ‬୆ก Ǝܻީޯ‫ݑ‬ଘଓƄ ໠ ໄ‫ ߃ ހݘ‬ၯ ଘଓƄ যঠ‫ݹ‬ƥ ،݊ ࡺޯ഻വറƄ ࡼ ࡷ ž༅໮ Ƅ߄ ‫ ࠅޢ‬ൎ൉‫޲ޢ‬೘೉݊ ࡺޯ഻വറƄࡣ ޲೘೉ƣ Ƅ‫ ݪ‬༬࿻ ࠦެ‫ࡩݘޢ‬Ƅƥ‫ࡩݘޢެࠦۑ‬Ƅࡼ ࡷܻࠩ࣍ ܻީެჴࢵ Ƅযঠ‫ޯࡺ ݊ݹ‬഻വറƄ ơ ࣠ࠁ Ƈ ܻၴ ܻޯެჴࢵ ଘଓƄ ݊ ࡺޯࠖ ൞ᒑᐽࠦ‫ ࡓ࣠ރ‬ᅚଌƄƇ Ƅƌ࢖ߦଐଌƄࠦࢵ ƄƎƌࡼ ࡷଘଓਔ ৹ ،ࠦެ‫ࡩݘޢ‬Ƅࠦߥ‫ ࣍࢖ݹ‬፥ ጿƥ‫ޯࡺ ݊ۑ‬഻വറƄƇ ܻߦࢢ ࢪ ࡓ ‫ކݘ‬ƥ ū݊ ࡺޯࠖ ൞ᒑᐽ ቿቷ޲ያዪࠦެ‫ࡩݘޢ‬Ƅบ คᇴᆝ ƄƍƄƥ ‫ࡺ ࣩݑޢެࠦۑ‬ఱ ఠƇ Ƅƌ࢖ߦଐଌƄࡼ ࡷž༅໮ ࠦࢵ ƄƎƌযঠ‫ ݹ‬،ᇯ ᇍ ࡁ ፥ ገܻࢧ ࠁ ଘଓƄ . ࠦެ‫ࡩݘޢ‬Ƅࡣ ‫࡯ޢ‬༨ဈ ࡼ ࡷܻࠩࠅ ܻޯެჴࢵ Ƅࡼ ࡓ ೘಼݊ ࡺޯ഻വറƄƇ ܻߦࢢ ࡴ ࠧƀƣ ܻߙ :ࠦ୆งଘଓƄƇ ଘଓܻஊै Ƅ፥ ጿƢ߄ ႀ ޯఱ ఠࠦ୐෺ܻ‫ݘ‬ଐଌƄࠦߥ‫ ࣍࢖ݹ‬ƣ Ƃ 13 ܸୃกܸ଍ଉŽ–ໝ ແࡦफउ –ࠣ‫ݶ‬ Ɖܸࢤ ެߕକଐƁƢࠣ‫ࡦ ࢽޟܸ ࢲޟ‬ƁƢࠣ‫ޟ‬පඓ ൷Ɓ౭౑ࢲ କଐƁƄ ܸࢲ ƁƋኃ ᒔᐺ። ጏ࢓‫ݕ‬ ࡦƁ࣬ ࣊ Ɓ࢓ߢೕಹኃ ቮƁ೾೥࢓଍ଉƁƄ ƁƋƁ߁ ࢟ ſ ƛ ܸߢࢽ ࡦƁ߁ ዖዄ Ž૊ સၫƋࣝެ࠾ኃ ቮƁƊܸެࢲ ଘଐƁ ። ፎକଐ߁ ެࢲ କଐƁƢ। ्࢟ ࣝࡦƁŻܸࢤ ߾ ଔଐƁ ƌƥ߄ ༨ீ ౰ేၩ‫ࡩީ ࠒ ݟ‬Ƅࡣ ༨ீ ݊ ࡺޯ഻വറƄƣ ᇴᆝ ƄƍƄ ࠢ‫ݶ‬ ܻެߙ፟ ፑƄƧ ƌ࣓‫ࡩީ ࠒ ݟݹ‬Ƅࡣ ༨ீ ݊ ࡺޯ഻വറƄƣ ᇴᆝ ƄƍƄ ‫ࡩީ ࠒ ݟ‬Ƅࠦ‫࡯ޢ‬ዙኸ ፥ ጿƎܻީޯ‫ݑ‬ଘଓƄ໠ ໄ‫ ߃ހݘ‬ၯ ࣓‫ݹ‬፥ ፑܻଐଌƄว คܻஎै Ƅƣ ᇴᆝ ƄƍƄ ࠣީ‫ݕޟ‬ ᑀᐺࠣ‫ ࢲܸ ࢲޟ‬କଐƁƄ Ɓ౫ౄࡷ଍ଉƁ i) ii) iii) ܻ‫ ߄ݹ‬ၴ ൎ൉‫޲ޢ‬೘೉፥ ገܻࢧ ࠁ ଘଓƄ݊ ࡺޯࠖ ൞ᒑᐽቿቷ޲ያዪƣ ࣠ࡓ ‫ޢ ݟކ‬༨ဂ ࠦ‫ހݘެܻݹ‬Ǝܻޯ༧ဈ ƣ Ƅ‫ ݪ‬༬࿻ ࠦެ‫ࡩݘޢ‬Ƅࡣ ‫޲ޢ‬৤৕ষƄ .i • ࢪ ࡓ ‫ކݘ‬ଘଓܻᐱ᏶ƀơ ܻࡺޯࠁ Ƅ ،ᑑ ᐽኅ ቱܻߙൎ൉ၯ ،ܻ‫ߙޢ‬ᇴᆝ ƣ ࣠ࡓ ‫ݹ‬ƣ Ƅ‫ ݪ‬༬࿻ ࠦެ‫ࡩݘޢ‬Ƅࡴ ᄽ஛ ࠦ‫ ࡓߦܻݹ‬ᚻࡩƄ .i ݊ ࡺޯ഻വറƄƇ ܻߦࢢ ቿቷߥޯࣀ ࡷƣ ࣠ࡓ ‫ޢ ݟކ‬༨ဂ ܻࠧƎܻ‫ ࠀޯޢ‬Ƅযঠ‫ݹ‬ƣ Ƅ‫ ݪ‬༬࿻ ࠦެ‫ࡩݘޢ‬ƄƇ Ƅƌ࢖ߦࡷࡼ ࡷƆƌ࢖ߦࡷᇯ ᆝ ơ ൎ൉ߥޯࢵ ଘଓƄ .ii Ʀ ༇໩ ଘଓƄࡼ ‫ݑ‬ ݊ ࡺޯࠖ ൞ᒑᐽࠦ‫ ࡓ࣠ރ‬ᅚଌƄƇ Ƅƌ࢖ߦଐଌƄ໠ ໄ‫ހ‬Ɵ ƥ࢖ߙ߄ ၴ ࣠‫ކ‬ଘଓƠ ܻެࠧƣ ܾ‫ހ‬ᄅ ྈܻႅ ޯࡩܻ‫ݹߥ ࢧ ߄ހ‬ᄅ ྈܻႅ ޯࡩƄ .iii ࣯ ࣐ ‫ކݘ‬ƣ Ƅ‫ ݪ‬༬࿻ ݊ ࡺޯ഻വറƄࢪ ߦ‫ࡷ ࡼރ‬ƣ ܻޯჱ‫ݑޢ‬Ƈ ߃ ၯ ƄƄƍƄ‫ݘޢެࠦۑ‬ᑃᐽᑑ ᐽቦሼƥ .ࠦႅ ‫ޢ‬ჴެࡩƄࢪ ߦ‫ރ‬ Ƅ ܸީ‫ݕޟ‬ ࡦƁƓ Ɓࣝ‫ހ‬Ž.1.2.5 :ܻዙ዗໠ ໄ‫ ࢵܻ ࢵޢ‬Ƅ໠ ໄၮ࣠‫ރ‬፟ ፑƄƇ ܻެ‫ࡩݘޢ‬Ƅᄎ ႞ߥ୐ง [ Random Sampling] ࠣୃ෴Ɓࣝࣀ ‫ݕ‬ ࡦƁࠣީ‫ݕޟ‬ ࡦƁ  [ Non-random sampling] ࠣୃ෴Ɓࣝࣀ ‫ݕ‬ ࡦƁ౭ౄၦࠣީ‫ݕޟ‬ ࡦƁ  :ࠦ୆งଘଓƄƖ Ƅ࣠‫ރ‬ଘଓƄࡼ ࡷƣ ࣠ࡓ ୓ข ࠣୃ෴Ɓࣝࣀ ‫ݕ‬ ࡦƁࠣީ‫ݕޟ‬ ࡦƁ  ࡼ ࡷࠦ୆෷༅໮ ࠦၮ࣠ዙಣಝ፥ ፕ [ Random Simple Sampling ] ࣐ࠦ ‫ ࣀޢ‬ხࡩƄࠦ୆෷Ƅ࣠ࣃ ‫ࡩݘ‬Ƅࠦެ‫ࡩݘޢ‬Ƅ .i ƌƄ࢖ߙƀ݊ ‫ޢ‬ዙዋ ƣ ƀཀྵཏ‫ݘ‬೘ೃƧ ƀū݊ ࡺޯ഻വറƄᑑ ᐽƍࡼ ࡷࠦެ‫ ࡁݘޢ‬Ǝܻޯࠑ ޯࡩࠦࢢ ࢖ߦࡩƄࢪ ߦ‫ܻރ‬ಛಗ፥ ገܻࢧ ࠁ ଘଓƄ݊ ࡺޯ഻വറƄ ࠦެ‫ࡷެࠢݑޢ‬Ƈ ౰ేޯࠀ ƄƄƍƂᄅ ྈܻႅ ޯࡷ݊ ࡺޯ഻വറƄƣ ƀ፟ ፑƂᑑ ᐽƍ݊ ࠅ मअƥūƄƥƎܻޯ༧࿻Ƭ ƣ ƀ፥ ጿࠦࢢ ࢖ߙࡴ ಛಗ݊ ࡺޯ഻വറƄ ƣ ܾ‫ࠧެܻހ‬ƆƎܻࢸ ଗଓܻ‫ ހ‬मअ߃ எै Ƅࡼ ࡷƥ‫ܻކࠢۑ‬ያዜ ƥࠢࢧ ‫ݶ‬ ܻࢧ ࠀ ݊ ‫ޢ‬ዙዋ ܻᐱᐕߙ࢖࣏ࠩ ‫ކ‬ƥቻቷ‫޲ޢ‬೘೉݊ ‫ޯ ࣐ޢ‬ఱ ఠࠦߥ‫ ࣍࢖ݹ‬Ƨ ܾ‫ހ‬ƥ ƣ ࣠ࡓ ‫ކ‬ƣ ଘଓ݊ ࡺޯ഻വറƄ፥ ጿƤ߄ ၰ ƥᇯ ᇍ ࡩࠦ‫ݹ‬ƥܻࣀ ჴࡷƒ ࢖ߙžܻ࣐ ‫ݑ‬Ƅ཯ཏ‫ހ ࡣކݘ‬،،ࠦ‫ࡩݘߦ࣠ݹ‬Ƅ཯ཏ‫ކݘ‬ଘଓࠦ୆෷Ƅ࣠ࣃ ‫ࠦݑ‬൞ᇅᆝ . ࠦެ‫ࡩݘޢ‬Ƅ፥ ጿƆƎܻޯ༧ீ Ɔžܻߦࡁ ౰౗ࡁ ƀࠦެ‫ݑޢ‬যওࡺࢧ ‫ކ‬፥ ፕ ࠦ‫ࡩ ࣐ީߥޢ‬Ƅࠦެ‫ࡩݘޢ‬Ƅ [ Stratified Sampling ] ࠣ‫ࡦ ࣍ަߢޟ‬Ɓࠣީ‫ݕޟ‬ ࡦƁ .II ‫ و‬،࣐ࠦ ‫ ࣀޢ‬ხࡩƄࠦ୆෷Ƅ࣠ࣃ ‫ࡩݘ‬Ƅࠦެ‫ࡩݘޢ‬Ƅƅ ॅळࢵ ƀƢƄ߄ ႀ ޯࢵ Ƅࢇ ‫ܻݑ ࡼ ࣍࢖ݹ‬ᐱᐕ࡯ၮơ ࣠ࢧ ஊै Ƅযঠ‫཯ݹ‬དྷࡩƄࠦެ‫ࡩݘޢ‬Ƅࡼ ࡷ ፥ ገܻࢧ ࠁ ଘଓƄ݊ ࡺޯ഻വറƄ፥ ጿࠦఱ ఝܻႅ ޯࡷ౰ేၩࠦ୆෷ܻࢧ ࠁ ଘଓƄƇ ܻ‫ޯࡺݘ‬഻വറƄบ คᇴᆝ ƄƍƄƅ ॅळࢵ ଘଓƄƄ߃ ࠧƢ߄ ႀ ޯఱ ఓ ౮ే๻๬ใƆ౰ే‫ݓ‬ ࢧ ࡩƄƇ ܻ‫ޯࡺݘ‬഻വറƄࡼ ࡷࠦၮ࣠ዙಣಝ፟ ፑƄ፥ ገܻࢧ ࠁ ଘଓƄ݊ ࡺޯ഻വറƄযওࣀ ߥ‫ކ‬যঠ‫ࡩ ࣐ީߥޢࠦݹ‬Ƅࠦެ‫ࡩݘޢ‬Ƅ፥ ጿ‫ ߄ۑ‬ၰ Ƅ࣠ࡩƄ 14 ܸୃกܸ଍ଉŽ–ໝ ແࡦफउ –ࠣ‫ݶ‬ Ɖܸࢤ ެߕକଐƁƢࠣ‫ࡦ ࢽޟܸ ࢲޟ‬ƁƢࠣ‫ޟ‬පඓ ൷Ɓ౭౑ࢲ କଐƁƄ ܸࢲ ƁƋኃ ᒔᐺ። ጏ࢓‫ݕ‬ ࡦƁ࣬ ࣊ Ɓ࢓ߢೕಹኃ ቮƁ೾೥࢓଍ଉƁƄ ƁƋƁ߁ ࢟ ſ ƛ ܸߢࢽ ࡦƁ߁ ዖዄ Ž૊ સၫƋࣝެ࠾ኃ ቮƁƊܸެࢲ ଘଐƁ ። ፎକଐ߁ ެࢲ କଐƁƢ। ्࢟ ࣝࡦƁŻܸࢤ ߾ ଔଐƁ ‫ۑ‬Ƈ ܻ‫ޯࡺݘ‬഻വറƄƤ߃ ࠧƢ࣠ᄽ஛ ‫ ࣀ ݪ‬ჱჅƌƄ߄ ၮƄ፟ ፑƄƅ ॅळ࣐ ଐଌƄࠦެ‫ࡩݘޢ‬Ƅࡴ ᄽ஛ ᄎ ႞ߥ‫ݹ‬Ꭸ᎜‫ܻۑ‬ᐱᐟყჅܻࡺ‫ߙޢ‬Ƈ Ƅ߄ ၰ ࣠ࡩƄࡣ ‫ܻމ‬๻๬฽ Ƥ߃ ࠧƖ ࣠ዙಣಝƥ౰ే‫ݓ‬ ࢧ ࡩƄ݊ ࡺޯ഻വറƄᑑ ᐽƍƇ Ƅ߄ ၰ ƥ፥ ጿ߄ ၴ ࣠‫ࠦކ‬୆෷༅໮ ࠦ୆෷Ƅ࣠ࣃ ‫ ࡴݑޢެࠦݑ‬ᄽ஛ ࡣ ޲೘಼ƌ߄ၮᇯ ᆝƥ ࠦ‫ॅހ‬ळ࣐ ଐଌƄࠦެ‫ࡩݘޢ‬Ƅࡣ ޲೘಼Ƈ ܻެ‫ࡩݘޢ‬Ƅ ࠦ‫ݹ‬ƎƄƌଘଓƄƢॅळ‫ࡩݘ‬Ƅࠦ‫ޢ‬ᆿᆝࠦީ࡯࣍ ݊ ࡺޯ༬ீ ࡼ ࡷ‫ ܻ࣍ࡩ ݪ‬200ࡼ ࡷࠦ‫ݑޢެࠦࡷ ࡓ࣠ރ‬Ǝܻ‫ ࠀޯޢ‬Ƅ ƅ ॅळ࣐ ࡷࠢ‫ރ‬ƀƓ ࢖ߦެࡩ Ƅƍܻߙ ،ቿቷ‫ࡷޯࡺܻމ‬ƆƎ࣠ࢧ ‫ࡩ ࣐࡯ީࠦހ‬Ƅ݊ ‫ޢ‬ዙዋ ࡣ ޲೘೉‫ޢ ݟ‬༨ဂ ‫ ܻ࣍ࡩ ݪۑ‬5000ࡴ ࠩ‫࣠ݑ‬ዙಣಝ݅ ࡩܻީࡩƄƣ ߄ ၮࠦ‫ܻࡷݘ‬ၴ ፥ ጿ : ፥ ፕ ƥࠦ‫ޢ‬൞൙ၮƢܻࣀ ߘƄࠦ‫ހݘ‬ƎƄࡼ ࡷƣ ࣠ࡓ ୓ขࠦ‫ݹ‬ƎƄƌଘଓƄƢॅळ‫ࡩݘ‬Ƅࠦ‫ޢ‬ᆿᆝ ƣ Ƅܻެ൞൙ၮ ‫ ܻ࣍ࡩ ݪ‬1800 ࠦީࢵ ܻഷവറƄᄎ ႞ߘ  1600 ơ ܻዙኸ ଘଓƄƆƎƄƌƄᄎ ႞ߘ  1200 žܻࢧ ࠁ ଘଓƄᄎ ႞ߘ  400 ࠦ‫ࡩ ࢧ ࠒޢ‬ƄƆƎƄƌଘଓƄᄎ ႞ߘ  72 = 200 x ( 5000 /1800 ࠦެ‫ࡩݘޢ‬Ƅ፥ ጿࠦީࢵ ܻഷവറƄᄎ ႞ߘࠦީ࡯࣍ ‫ ݪ‬ყࢧ ‫ރ‬ƣ ƍƀ  64 = 200 x (5000 /1600 ࠦެ‫ࡩݘޢ‬Ƅ፥ ጿƆƎƄƌଘଓƄᄎ ႞ߘࠦީ࡯࣍ ‫ ݪ‬ყࢧ ‫ ރ‬ 48 = 200 x ( 5000 /1200 ࠦެ‫ࡩݘޢ‬Ƅ፥ ጿžܻࢧ ࠁ ଘଓƄᄎ ႞ߘࠦީ࡯࣍ ‫ ݪ‬ყࢧ ‫ ރ‬ 16 = 200 x ( 5000 /400 ࠦ‫ࡩ ࢧ ࠒޢ‬ƄƆƎƄƌଘଓƄᄎ ႞ߘࠦީ࡯࣍ ‫ ݪ‬ყࢧ ‫ރ‬ ࠦީ࡯࣐ ࡩƄযওߘ౰౔‫ހ‬ᑑ ᐽƍƥࠦީࢵ ܻഷവറƄᄎ ႞ߘࠦީ࡯࣍ ࡼ ࡷࠦ୆෷Ƅ࣠ࣃ ‫ݑ‬ƆƎ࣠ࢧ ‫ ܻ࣍ࡩީܻހ‬72Ǝܻ‫ހܻ ࠀޯޢ‬Ƣ࣠ߥ‫ރ‬ᑑ ᐽƍ߄ ‫ހݘ‬ ࢪ ߦ୐กᑑ ᐽቦሼƥ‫ࠦۑ‬୆෷Ƅ࣠ࣃ ‫ݑ‬ƢܻߘƎƄࠦ‫ހݘ‬ƎƄࡼ ࡷƣ ࣠ࡓ ᅚଌƄƥܻ୆෷Ƅ࣠ࣃ ‫ܻݑ‬ዙኼƎ 72Ǝܻޯ༧စ Ꭸ᎜ 1800፟ ፑƄ 1ࡼ ࡷ 16ƥžܻࢧ ࠁ ଘଓƄᄎ ႞ߘࡼ ࡷܻީࡩܻ࣍ 48 ‫ و‬ƆƎƄƌଘଓƄᄎ ႞ߘࡼ ࡷܻީࡩܻ࣍ 64Ǝܻ‫ހܻ ࠀޯޢ‬Ƣ࣠ߥ‫ࡩ ࣐࢖ݹߥࠦރ‬Ƅ ࠦ‫ࡩ ࢧ ࠒޢ‬ƄƆƎƄƌଘଓƄᄎ ႞ߘࡼ ࡷܻީࡩܻ࣍ . 200 = 16+ 48 + 64+ 72 ƆƎܻޯശവറƄࠦެ‫ࡩݘޢ‬Ƅࡴ ᄽ஛ ƣ ܻߙƄቦ቎ࠧƥ [ Systematic Sampling ] ࠣࡷ࣌ ެީ଍ଉƁࠣީ‫ݕޟ‬ ࡦƁ .III ፥ ገܻࢧ ࠁ ଘଓƄ݊ ࡺޯࠖ ൞ᒑᐽࠦ‫ ࡓ࣠ރ‬ᅚଌƄƇ Ƅƌ࢖ߦଐଌܻ‫ࠦހ‬೘ಹܻߘ࢖ߙ࣠୓ขܻࡷ߄ެ‫ݑ‬Ɔƌܻၮࠦࡺ࣏ ޯެଐଌƄࠦެ‫ࡩݘޢ‬Ƅࠦߥ‫ ࣍࢖ݹ‬Ƣ߄ ႀ ޯఱ ఠ ࡴ ᄽ஛ ે ઻ၮ݊ ࡺޯࠖ ൞ᒑᐽ્ ᇣᇍ ࡩƄƌ߄ ‫ࡩݘ‬Ƅযওࣀ ߥ‫ކ‬Ɛ ܻࢵ ƀે ઻ၮܻᐱᐟࡷƇ Ƅ߄ ၰ ࣠ࡩƄƎܻ‫ ࠀޯޢ‬Ƅƣ ࣠ࡓ ‫ݹ‬ƥ‫ࡩީ ࠒ ݟۑ‬Ƅࡣ ༨ீ ၖ࿟ ܻެࡩƄࡴ ߘ༟༎Ƅે ઻ၮࡴ ࣏ ޯެࡷƥƪƥ ܻࣀ ჴࡷᇯ ᇍ ఴ చƥ્ ઻ࢢ ଛଓƄ݊ ࡺޯ഻വറƄƇ Ƅ߄ ၰ ƥ݊ ‫ݹ‬Ə࣠‫ކ‬Ꭸ᎜ࡼ ࡷƥūࠦ‫ॅހ‬ळ࣐ ଐଌƄࠦެ‫ࡩݘޢ‬Ƅ ƌ߄ ၮࡣ ޲೘಼ࡴ ߘƎ࣠ࠧƥ‫ ܻ࣍ࡩ ݪ‬500 ࣠ࠧ݊ ࡺޯࠖ ൞ᒑᐽ્ ᇣᇍ ࡩƄƌ߄ ‫ࡩݘ‬Ƅƣ ᇴᆝ ƄƍƂƨൎ൉޲ࡷ‫ۑ‬যওࣀ ߥޯࡩƄᑑ ᐽƍࡼ ࡷ ࠦ‫ޢ‬ᆿᇍ ࡩƄƇ Ƅ߄ ၰ ࣠ࡩƄ݊ ‫ݹ‬Ə࣠‫ކ‬ƣ ࣠ࡓ ‫ ߙޢ‬،ܻީࡩܻ࣍ 20 ፥ ፕ ࠦ‫ॅހ‬ळ࣐ ଐଌƄࠦެ‫ࡩݘޢ‬Ƅบ คᇴᆝƥūܻࡷࠦ‫ޢ‬ᆿᆝ ፥ ጿࠦީ࡯࣐ ࡩƄ ࠦެ‫ࡩݘޢ‬Ƅࡴ ߘƎƌ߄ ඳඣ ൭Ɛ ܻࢵ ଛଓƄƄ߃ ࠧે ઻ၮƥ‫ ۑ‬25=20÷500 ፥ ጘଜଓƄᇯ ᇍ ࣃ ࡩƄે ઻ၮ݊ ࡺޯࠖ ൞ᒑᐽࠦ‫ ࢢ࡯ޢ‬ଛଓƄ 15 ܸୃกܸ଍ଉŽ–ໝ ແࡦफउ –ࠣ‫ݶ‬ Ɖܸࢤ ެߕକଐƁƢࠣ‫ࡦ ࢽޟܸ ࢲޟ‬ƁƢࠣ‫ޟ‬පඓ ൷Ɓ౭౑ࢲ କଐƁƄ ܸࢲ ƁƋኃ ᒔᐺ። ጏ࢓‫ݕ‬ ࡦƁ࣬ ࣊ Ɓ࢓ߢೕಹኃ ቮƁ೾೥࢓଍ଉƁƄ ƁƋƁ߁ ࢟ ſ ። ፎକଐ߁ ެࢲ କଐƁƢ। ्࢟ ࣝࡦƁŻܸࢤ ߾ ଔଐƁ ƛ ܸߢࢽ ࡦƁ߁ ዖዄ Ž૊ સၫƋࣝެ࠾ኃ ቮƁƊܸެࢲ ଘଐƁ ࢖࣏ࠩ ‫ކ‬ƥ‫ۑۑ‬ƨൎ൉޲ࡷ 10 ࡼ ࡓ ‫ࡩޢ‬ƥ 25 ࡴ ߘ༟༎Ƅࡼ ࡷࡣ ߘƀƣ ࣠ࡓ ‫ޢ ݟݹ‬༨ဂ -ơ ƥଛଓƄ‫ࡩ ࣐ܻࡩ ݪ‬Ƅᄎ ႓ƄƧ ƀ 235 210 185 160 135 110 :፥ ጘଘଓƄ࣠ࠒ ެࡩƄે ઻ၮࠦެ‫ࡩݘޢ‬ƄƇ Ƅƌ࢖ߦࡷ 85 60 35 10 [ Cluster Sampling ] ࠣ‫ݶ‬ Ɖࣝߢީ‫ݕ‬ ࡦƁࠣީ‫ݕޟ‬ ࡦƁ IV. 485 460 435 410 385 360 335 310 285 260 ࠦ୐෺ܻީޯࡷ߄ ‫ࡩݘެܻߘޢ‬Ƅƣ ࣠ࡓ ‫ކ‬ƣ ƀ‫ ݪ‬༬࿻ ‫ࡩݘެܻߘޢ ߄ۑ‬Ƅƀ߄ ީࡷ፥ ጿࠦ‫ࡩ ࣐ީߥޢ‬Ƅࠦ୐෺ܻ‫ݘ‬ଐଌƄࡼ ‫ޯ࡯ ߎݑ‬༧ဈ ࠦ‫ݹ‬ƌ࣠ߥެ‫ࡩݘ‬Ƅࠦެ‫ࡩݘޢ‬Ƅ ፥ ጿযঠ‫ݹ‬ƥ‫ࡩ ࣐ީߥޢࠦۑ‬Ƅࠦެ‫ݘޢ‬ᑃᐽࠢ‫ࡷ ࣃܻހ‬ᇯ ᇍ ࢸ ‫ࡩ ࣐ީߥޢࠦۑ‬Ƅࠦެ‫ࡩݘޢ‬Ƅࢪ ࡓ ၮƧ ƀܻᐱᐟყჅܻࡺ‫ࠦߙޢ‬ఱ ఝܻႅ ޯࡷ ،ܻࠩ࡯ၯ Ƅƌ፥ ጿ ߄‫ࡺ ࣩݑެܻߘޢ‬ఱ ఠƇ ܻၮ࣠ዙಣಝ፟ ፑƂ‫ࡩީ ࠒ ݟ‬Ƅ݊ ࡺޯ༬ீ যওࣀ ߥ‫ࡩ ࣐࢖ݹߥࠦކ‬ƄƤ߃ ࠧ Ƈ ܻၮ࣠ࡺ഻വറƄƤ߃ ࠧ‫ࠦۑ‬ᐱᏳܻࣃ ࡷƟ ࢖࣐ ‫ހ‬ƥƀ‫ࡩީ ࠒ ݟ‬Ƅ݊ ࡺޯ഻വറ፥ ጿƄ࢖‫ݓ‬ எै Ƅ݊ ‫ݹ‬Ə࣠ޯࡩƄ‫ ࠁ ࣀ ݪ‬žƄ࣠ࢵ [Clusters ] Ƈ ܻၮ࣠ࡺ഻വറƄ žƄ࣠ޯࠁ Ƅ‫ ݪ‬ხఱ చƧ ƌ࣠ߥެ‫ހܻࡩݘ‬Ɩ ࣠ެࡩƄે ઻ၮள ஭࣍ ƀ‫ ݪ‬ხࣀ ࡩƄƄ߃ ಛಗƥࠦ‫ࢠܻߙޢ‬ƂƇ ܻၮ࣠ዙಣಝ፟ ፑƂᄎ ႞ߥ‫ކ‬ ᇯ ᇍ ఴ చܻᐱᐕ࡯ၮࡣ ࢧ ࠒ ޯଐଌƄƇ ܻၮ࣠ࡺ഻വറƄ࢛ ‫ހݘ‬Ǝܻ‫ࡩީܻ ࠁ ݟހܻ ࠀޯޢ‬ƄƢ࣠ߥ‫ݹ‬ūযওࣀ ߥޯࡩƄƄ߃ ࠧ߄ ‫ۑހݘ‬Ƈ ܻၮ࣠ዙಣಝે ઻ၮ Ƈ ܻࡷॅळ‫ݘ‬ଐଌƄ݊ ዙዋ যঠ‫ݹ‬ƥࠦެ‫ࡩݘޢ‬Ƅ፥ ጿž༅໮ ߰ ީࢧ ޯࡩƆƎܻޯശവറƄࠦၮ࣠ࡺ഻വറƄƌƄ࢖ߙƀ݊ ‫ޢ‬ዙዋ ߃ ၯ ƀযঠ‫ޢ ݟݹ‬༨ဂ ū፥ ገƄ࣠ࣃ ‫ݑ‬ ‫ۑ‬ƆƎܻޯശവറƄƇ ܻၮ࣠ࡺ഻വറƄƤ߃ ࠧƌƄ࢖ߙƀࡼ ࡷ Ơ ܻያዜ ଘଓƄƣ ॅळࢥ ߦ‫ݹ‬Ƅ࣠‫ރ‬ᇴᆝ ƄƍƄܻࡺ‫ߙޢ‬ƣ ߄ ၮࠦ୐෺߄ࡷƣ ᇴᇍ ࣀ ࡩ፥ ደൎ൉ᐱᐢࢵ ଘଓƄ࣊ ৤৒ষƄે ઻ၮƞ ࢖‫ࡩޯݘ‬Ƅਔ ৲ƎƄƄƍƄൎ൉޲ࡷ Ǝܻ‫ ࠀޯޢ‬ଘଓƄࠦ୆คܻ޲ࡩƄቿቷၰ ࢖ଐଌƄƥƇ ܻ‫ݹ‬मअ߄ ൞ᒑᐽ፥ ገƄ࣠ࣃ ‫ࡩݘ‬ƄƎܻ‫ ࠀޯޢ‬ଘଓƄযঠ‫ݹ‬፟ ፑƥଘଓƄቿቷၰ ࢖ଐଌƄ፥ ጿࠢ‫ۑۑߙܻރ‬Ƣ࣠ॶॱ५ƄƢƀ ࠦ‫ހݘ‬Ƅ༟༎Ƅቿ ቷၰ ࢖ଐଌƄƥࡴ ᐱᐟࡁ ܻࣀ ࡷ‫ ྲ ࠁ ࣀ ݪ‬႓൑ᑉᐽ፥ ገƄ࣠ࣃ ‫ࡩݘ‬ƄƎܻ‫ ࠀޯޢ‬ଘଓƄࠦ޲ࡩܻ޲ࡩƄቿቷၰ ࢖ଐଌƄᎨ᎜ഁ೨ Ƅ࢖൞ᒑᐽ፥ ገƄ࣠ࣃ ‫ࡩݘ‬Ƅ ƣ ߄ ၮ࣏ࠦ ߙܻ༨ீ ƣ ᇴᇍ ࣀ ࡩቿቷ޲ያዪƆƎܻޯശവറƄࠦެ‫ࡩݘޢ‬Ƅƣ ࣠ࡓ ‫ކ‬ƥ‫ ྲۑ‬႓ଘଓƄƤ߃ ࠧࡼ ࡷƌƄ࢖ߙ൑ᑉᐽ፥ ገƄ࣠ࣃ ‫ࡩݘ‬ƄƎܻ‫ ࠀޯޢ‬ଘଓƄ ‫ݹࠦۑ‬ƌ࣠ߥެ‫ࡩݘ‬Ƅࠦެ‫ࡩݘޢ‬Ƅƅ ॅळࢵ ƀƢƄ߄ ႀ ޯࢵ ܻ‫ހ‬ ࠣ‫ࡦޟ‬ƢକଐƁƄ ܸ‫ࡦަޟܸހ‬Ɓ݇ ዖወ Ɯ ࢓࣊ ‫ێ ێ ێ‬ ࠦ‫ࡩޯܻࡩޢ‬ƄƟ ࢖࣐ ࡩƄƢƄ߄ ႀ ޯࢵ ܻ‫ހ‬ᑑ ᐽƍƥܻᐱᐕ࡯ၮơ ࣠ࢧ ஊै Ƅࡼ ࡓ ೘಼ࠦ‫ࡩޢ‬ƥଘଓƄƇ ܻ‫ࡩީޢܻރ‬Ƅ ௪ ௎ƌ Ƅ߄ ‫ࡩݘ‬Ƅࡼ ࡷࡴ ᐱᐕ࡯޲ያዪƥƀƣ ࣠޲ࠁ ܻީࡩƄƢ࣠ߥ‫ࡩ ࣐࢖ݹߥࠦݹ‬ƄƤ߃ ࠧ፥ ጿ ƃྯ ႓ܸަ଍ଉƁࠣ‫ࡦ ࣀ ࠎ ࢤޟ‬Ɓቼቴ‫ݽ‬ ܸߢ଍ଉƁ ‫ۑ‬i ཯དྷࡩƄƥࠦ୆෷ܻࢧ ࠁ ଘଓƄƆƎܻ෬ෝᛂ ଘଓƄࢇ ‫ݑ ࡼ ࣍࢖ݹ‬Ƈ ܻ‫ࡩީޢܻރ‬Ƅ߃ ၯ ƀࠦ‫࡯ޢ‬ዙኸ যঠ‫ކ‬ƥ‫ۑ‬໠ ໄ‫ީ ࠒ࣠މ‬൞ᒑᐽƆྲ ႖ܻީଐଌƄቿቷ‫ߥܻހ‬ଐଌܻ‫ހ‬ ‫ࡩީ ࠒ ݟۑ‬ƄƖ ࣠ࢠ࣠ࡷ࣯ ࣐ ‫ݓ‬ ‫ކ‬Ɔƌ߄ ༨ீ ቿቷ‫ ࢵޟ‬ƀે ઻ၮƧ ࣠ޯ༨ဈ ፥ ጿࡣ ޲๻๬ึܻᐱᏳ ࣠‫ ࡓ ࡼݑޢ‬ᚻࡩ‫ܻۑ‬ᐱᐢ᠃ܻߦࡁ ƥܻᐱᐕ࡯ၮơ ࣠ࢧ ஊै Ƅযঠ‫཯ݹ‬དྷࡩƄƇ ܻ‫ࡩީޢܻރ‬Ƅࠦߘƌ ،ࠦߥ‫ࡩ ࣐࢖ݹ‬ƄƤ߃ ࠧƇ Ƅ౮ేያዪࡼ ࡷƥ ࠦ‫ހݘࡺ࡯ޢ‬ƣ ࣠ࡷ࣠ߥ‫ݹ‬௪ ௎ኅ ቱƄ௪ ௎ƌƄ߄ ‫ࡩݘ‬Ƅ፥ ጿƓ ౰౔ߦ‫ݹ‬ƥ‫୓ ࣐࡯ީࠢۑ‬ขƧ ኅ ቱƄ౰ేީࡓ ᚻࡩƄࡼ ࡷ༝༎Ƅƥࠦ‫ࡩݘܻࡩޢ‬Ƅߎ ‫ࡩޢ‬ᇴᇍ ޯࡩƄ ‫ ߄م‬ၮว คܻၴ ፟ ፑƄƇ ൎ൉‫ߥܻހ‬ଐଌƄࠦ‫࡯ޢ‬ዙኸ ፥ ጿƆ౰౎ၯ ࡴ ᐱᏬኆ ቱƥܻ‫ܻࡩޢ‬ၮ୍ܻ෺Ǝਔ ৿໠ ໄ‫ހ‬Ǝ߄ ࡷƄ࣠‫ݹ ࡓ࣠ރ‬ƣ ƄƇ ܻ‫ࡩީޢܻރ‬Ƅžܻߦ‫ޢ‬ჴࢵ Ƅ ‫ۑ‬ቿᒩᑧࢥ ࡷƇ ܻ‫ܻރ‬୆ก࣯ ࣐ ‫ݹݘ‬ƣ Ƅࡼ ࡓ ೘಼Ƨ ኅ ቱƄƥ໠ ໄ‫ީ ࠒ࣠މ‬ଐଌƄે ઻ၮ౰ే‫ࡩޯܾމ‬Ƅ፥ ጿ౮ేႁ ޯࡩƄƥƇ ܻ‫ܻހ‬ၴ ଘଓƄ፥ ጿࡣ ၯ ߄ ޯࡩƄ 16 ܸୃกܸ଍ଉŽ–ໝ ແࡦफउ –ࠣ‫ݶ‬ Ɖܸࢤ ެߕକଐƁƢࠣ‫ࡦ ࢽޟܸ ࢲޟ‬ƁƢࠣ‫ޟ‬පඓ ൷Ɓ౭౑ࢲ କଐƁƄ ܸࢲ ƁƋኃ ᒔᐺ። ጏ࢓‫ݕ‬ ࡦƁ࣬ ࣊ Ɓ࢓ߢೕಹኃ ቮƁ೾೥࢓଍ଉƁƄ ƁƋƁ߁ ࢟ ſ ƛ ܸߢࢽ ࡦƁ߁ ዖዄ Ž૊ સၫƋࣝެ࠾ኃ ቮƁƊܸެࢲ ଘଐƁ ። ፎକଐ߁ ެࢲ କଐƁƢ। ्࢟ ࣝࡦƁŻܸࢤ ߾ ଔଐƁ ࡼ ࡷƆྲ ႖ܻީଐଌƄ ࠦ‫ࡩ ࣃ ࠑ ࢧޢ‬Ƅቿቷ‫ߥܻހ‬ଐଌƄࡼ ‫ࡩ ࣐࢖ݹߥࠦݑ‬ƄƤ߃ ࠧߎ ࡯ޯ༧ဈ ଘଓƃྯ ႓ܸަ଍ଉƁ౭ౄၦࠣ‫ࡦ ࣀߣࣝݶ‬Ɓቼቴ‫ݽ‬ ܸߢ଍ଉƁ ‫ۑ‬ii ߎ ‫ܻކ‬ಛಗƄƢƄ߄ ႀ ޯࢵ ܻ‫ࡩݘࡺ࡯ޢࠦࠧެܻހ‬Ƅƣ ࣠ࡓ ‫ކ‬ƥ‫ۑ‬Ƈ ܻ‫ࡩީޢܻރ‬Ƅે ઻ၮơ ࣠ࢧ ॶॱ५໠ ໄ‫ީ ࠒ࣠މ‬ଐଌܻ‫ ྲހ‬႖ܻީଐଌƄơ ܻࢧ ‫ކ‬ଘଓƄ‫ࠁޢ ݟ‬ Ƥ߃ ࠧߎ ‫ࡩޢ‬ᇴᇍ ‫ކ‬ƥ‫ࡩ ࣃ ࠑ ࢧޢࠦۑ‬Ƅቿቷ‫ߥܻހ‬ଐଌƄࡼ ࡷଘଓਔ ৹ ፥ ፎ ܻࡺޯࠅ ଘଓƄࡣ ࢢ Ƅ࣠ޯࡩƄࡣ ‫ݶ‬ ܻࢵ ƥ ơ ܻࢧ ‫ކ‬Ƅቿቷ‫ࡁ࣠ ࢵޢ‬ ơ ܻࢧ ‫ކ‬Ƅቿቷ‫ ࢵޢ‬ƥƢ߄ ႀ ޯఱ ఠܻᐱ᏶ଛଓࡣ ߘƄƢ߄ ႀ ޯࣀ ଐଌƄ‫ࡩ࣠ߘ ݝ‬Ƅܻࢥ ‫ݹ‬ƄƥƆྲ ႖ܻީଐଌƄࠦߥ‫ރࠦހܻࡩ ࣐࢖ݹ‬Ǝܻߥଐଌܻ‫ߘ ࡣހ‬Ƅࠦߥ‫ࡩ ࣐࢖ݹ‬Ƅ ે ઻ၮƧ ࣠ޯ༨ဈ ཯དྷࡩƄࠦ‫ީܻރ‬ჴࢵ ଘଓƄơ ܻࢵ Ǝܾ‫ࡩީܻ ࠁ ݟހ‬ƄƢ࣠ߥ‫ݹ‬ቿቷࢵ Ƅ࢖ଐଌƄࠦߥ‫ ࣍࢖ݹ‬፥ ጿቼቴࢲ Ɓ࢓଍ଉƁࠣߢ‫࣊࢓ݶ‬ ‫ۑ‬Ɩ ྲ ႓Ƅ ‫ۑ‬iii ࠦ‫ޢ‬ዙኼ༟༎Ƅơ ܻࢧ ‫ކ‬ଘଓƄࡣ ‫ݶ‬ ܻࢵ ƥƢ߄ ႀ ޯఱ ఠܻ‫ܻࡩޢ‬ၰ ƥƢܻ‫ࡩݘ‬Ƅ ਔ ৲౰౎ࡩƄࢇ ‫ࡩީ ࠒ ݟݑ ࡼ ࣍࢖ݹ‬ƄƖ ࣠ࢠ࣠೘ೃࠦߥ࡯‫ޯݘ‬ଐଌƄቿቷ‫ ࢵޟ‬ଘଓƄ ‫ۑ‬ơ ܻ‫ߙݘ‬ᇯ ᇍ ఴ చบ ค౰౔‫ރ‬ଗଓᇴᆝ ࠦ‫ܻހ‬ၴ ଘଓƄቿቱ࣠ᐳ႓ƥܻᐱᐢ࣍ ܻఱ చƥƣ ܻ‫ީޢ‬ჴࢵ ଘଓƄ፥ ጿቿቷ‫ ࢵޟ‬ଘଓƄƊ ࣠ࢠƥࡼ ࡷࠦ‫ܻࡩޢ‬ၮࠦၴ Ǝƌ‫୓ ࣐࡯ ݪ‬ขቿቷࢵ Ƅ࢖ଐଌƄࠦߥ‫࣍࢖ݹ‬ ‫ݹ ࢥܻ ࠁޢࠦۑ‬ƄƇ ܻࡺ‫୓ݘ࡯ޢ‬กܻࠩࠁ ܻࢥ ‫ݹ‬Ƅ‫୓ ࣐࡯ ݪ‬෺ࠦީࡁ ࢖ଐଌƄቿቷ‫ ࢵޟ‬ଘଓƄƥܻᐱᐕ࡯ၮ Ƣ߄ ၮƥ ، ൎ൉ࡺޯࡓ ࡷƣ ܻ‫ީޢ‬ჴࢵ ଘଓƄƆƌܻ‫ ࢵޯݘ‬Ƅ፥ ጿ݊ ‫ޢ‬ସବଣ ƄበሼƎܻࣃ ࡷƣ ܻያዧ Ƣ߄ ၮቿቷࢵ Ƅ࢖ଐଌƄࠦߥ‫ ࣍࢖ݹ‬ƅ ࣠‫ࡷ ࡼݑޢ‬ ࠦߥ‫ࡩ ࣐࢖ݹ‬ƄƤ߃ ࠧߎ ‫ࡩޢ‬ᇴᇍ ‫ކ‬ƥ‫ܻۑ‬ಛಗሯሡሓ ޯࢵ ƄƢ߄ ‫ࠦࡩݘ‬ႅ ‫ޢ‬ჴ჈Ƈ ܻ‫ީܻރ‬ჴࢵ ଘଓƄ࢛ ‫ހݘ‬Ɣ ࣠ߥࢵ ƥ ،Ɩ ࣠ࢠ࣠ଐଌܻ‫ހ‬ƢܻࡺޯࠧଘଓƄ ‫ܻކ ߎۑ‬ಛಗƄƢƄ߄ ႀ ޯࢵ ܻ‫ހ‬ቿቷ‫ߥܻހ‬ଐଌƄࠦߥ‫ ࣍࢖ݹ‬ƥƆྲ ႖ܻީଐଌƄࠦ‫ࡩ ࣃ ࠑ ࢧޢ‬Ƅቿቷ‫ߥܻހ‬ଐଌƄࠦߥ‫ࡷ ࡼ ࣍࢖ݹ‬౰ే޲ࡓ ‫ߘ ࡣހ‬Ƅ ࡣ ၴ Ƅࡼ ࡷܻᐱᐢ࣏ ࠁ ൎ൉ࡷƥƆ࢖ܻ࣏ࠧ ࡩƄࠦީߘƄ࢖೘ೃ‫ࡩީܻ ࠁ ݟ‬ƄƢ࣠ߥ‫ࡩ ࣐࢖ݹߥࠦݹ‬ƄƤ߃ ࠧ፥ ጿ ࠣ‫ࡦ ࣀ ࠎ ࢤޟ‬Ɓƃ߁ ࠤܸࣀ ଍ଉƁ .IV Ɔ౰౗ࡓ ‫ࡩ ࣐࢖ݹߥࠦހ‬ƄƤ߃ ࠧƢ߄ ႀ ޯఱ ఠƥ‫ ࢵࠦۑ‬ƄƎኆ ቱƄƖ ࣠ࢠ࣠ࡷƆ࢖ܻ࣏ࠧ ࡩƄࡼ ‫ݑ‬Ƈ ܻࡷॅळ‫ݘ‬ଐଌƄƥƇ ܻ‫ࡩީޢܻރ‬Ƅે ઻ၮơ ࣠ࢧ ஊै Ƅ ௪ ௎ƥਔ ৿ƥ Ɖ ܻ୓คଘଓƄ ౰ేࢵ ࠦީߘƄ୹ ୪ ‫୓ ࣐࡯ ݪ‬ข ཯དྷࡩƄƥ ࠦ‫ࡩެ࣠ݑޢ‬Ƅ Ɔ࢖࣐ ‫ࡩ ࣀޢ‬Ƅƥ ܻᐱᏹƌ࣠ࠅ ƥ Ɖ ܻ୓คଘଓƄ Ƈ ܻ‫࡯ޢ‬ዙኸ ፥ ጿ Ǝಒ౷ޯࡷᇯ ᇍ ఴ చ࢛ ‫࢖ݹ‬ଐଌƄƇ ଘଓܻၰ ࠦީߘƄ࢖೘ೃ‫ ݪ‬ყީ࣐ ࡩƄƢ࣠ߥ‫ࡩ ࣐ ݪ ࠁޢ ݟݹ‬Ƅơ ܻ༬ீ ፥ ጿᑑ ᐽቦሼƥ‫ۑ‬Ƈ ܻ࣏ ࠁ ൎ൉ଐଌƄ ‫ࡩ ࢧ ࠒ ࣯ۑ‬Ƅࠢ‫ࢠݘ‬ƥƥࠢ‫ކ‬ଘଓܻၰ ơ ࣠ࠁ Ƈ ܻ࣏ ࠁ ൎ൉ଐଌƄࡣ ‫ ࠖޢ‬ఱ ఠƥƧ Ǝƥƌƥ [Questionnaire ] Ơ ܸ‫ަޟ‬ჱࢲ କଐƁ ࠣୃ෴ܸࢤ ߾ କଐƁƃƋܸ෩ේᚿ କଐƁ‫ێ ێ ێ‬ ƆƎܻ෬ෝᛂ ൎᑉᐽ߄ ‫ޢ‬எै Ƅযওࡺࢧ ޯࡩƄે ઻ၮ߄ ࡺޯ‫ݹݘ‬Ƈ ܻ‫ࡩީޢܻރ‬Ƅ݊ ዙዋ ፥ ጿƣ ܻ‫ީޢ‬ჴࢵ ଘଓƄࠦߥ‫ ࣍࢖ݹ‬ƢƄ߄ ႀ ޯࢵ ƄƊ ܻ༬စ ƥࠦ‫ߙݘܻࡩޢ‬ƣ ƀ ࠦ‫ީܻރ‬ჴࢵ ଘଓƄ ࠦ୆෷ܻࢧ ࠁ ଘଓƄƆƎܻ෬ෝᛂ ଘଓƄ፥ ጿࠦ୆งଘଓƄƔ ƥྲ ႡࡩƄ࢖ߙ࣠୓ขƣ Ƅ‫ ݪ‬༬࿻ ƥ‫ࠦۑ‬୆෷ܻࢧ ࠁ ଘଓƄ ܻࢥ ‫ݹ‬Ƅƥƣ ܻ‫ީޢ‬ჴࢵ ଘଓƄࡼ ࡷƓ ࢖‫ݓ‬ ࡩƄƥ‫ࡩީܻ ࠁ ݟ‬Ƅࠦ‫ࠧ࣠ݹ‬໠ ໄ୍งࠦߥߙ୹ ୪ቿቱܻࢵ Ǝࠦ‫ީܻރ‬ჴࢵ ଘଓƄƧ ࣠ޯ༨ဈ ƣ Ƅ‫ ݪ‬༬࿻ ‫ۑ‬I ፥ ፕ ƥቻቷၴ ଘଓ‫ࡺ ݝ‬ያዩ Ƨ ኅ ቱƄƓ ࢖‫ݓ‬ ࡩƄƄ߄ ၮܻࡷࠦࠩࠅ Ƨ ଛଓƇ ܻࡷॅळ‫ݘ‬ଐଌƄžܻࣃ ߙƄƢ߄ ၮƥܻᐱᐢ᠃ྲ ႓ࠦၴ Ǝƌ ‫ ࣯ߙߥ ࣊ۑ‬൞൙‫ࡩݘ‬Ƅ‫ࡩީ ࠒ ݟ‬Ƅ ƌ߄ഷവറƄƌ߄‫ࡩݘ‬Ƅ‫ۑ‬፟ ጕƌଘଓƄ߄ ஊै Ƅ፥ ጿƣ ࣠ࡓ ‫ކ‬ƣ Ƅ‫ ݪ‬༬࿻ ࠦ୆෷ܻࢧ ࠁ ଘଓƄƆƎܻ෬ෝᛂ ଘଓƄ፥ ጿቿቷ‫ ࢵޟ‬ଘଓƄƌ߄ ၮ ‫ۑ‬II ܻࠩࡺ‫ ݪކߥ ࣀޢ‬༬࿻ ቿቷ‫ ࢵޟ‬ଘଓƄ݊ ࢵ ࣠‫ކ‬ቿቱܻၰ ፥ ጿƥ ،‫ࡩީ ࠒ ݟ‬Ƅƞ Ƅ߄ ࠧƄƥƖ ࣠ࢠ࣠ࡷે ઻ၮ߄ ࡺޯ‫ީܻރࠦݹݘ‬ჴࢵ ଘଓƄቿቷ‫ ࢵޟ‬ଛଓ ‫ۑ‬Ǝƥܻ༨ீ ፟ ፑƄ 17 ܸୃกܸ଍ଉŽ–ໝ ແࡦफउ –ࠣ‫ݶ‬ Ɖܸࢤ ެߕକଐƁƢࠣ‫ࡦ ࢽޟܸ ࢲޟ‬ƁƢࠣ‫ޟ‬පඓ ൷Ɓ౭౑ࢲ କଐƁƄ ܸࢲ ƁƋኃ ᒔᐺ። ጏ࢓‫ݕ‬ ࡦƁ࣬ ࣊ Ɓ࢓ߢೕಹኃ ቮƁ೾೥࢓଍ଉƁƄ ƁƋƁ߁ ࢟ ſ ƛ ܸߢࢽ ࡦƁ߁ ዖዄ Ž૊ સၫƋࣝެ࠾ኃ ቮƁƊܸެࢲ ଘଐƁ ። ፎକଐ߁ ެࢲ କଐƁƢ। ्࢟ ࣝࡦƁŻܸࢤ ߾ ଔଐƁ ࠢ‫ࡩޢ‬ƄƆƌ࣠‫ࡩݘ‬ƄᎨ᎜ܻ୆෷Ƅ࣠ࣃ ‫ݑ‬༇໩ Ƅ፟ ፑƄƎ࣠༨ீ ࡼ ࡷ࢔ߦߥࡩƄƢ߄ ၮ،ܻ‫ܻࡷެ ࣐ߥޢ‬ᐱᐜყ჋मऒ‫ ݪ‬༬࿻ ƆƎܻ෬ෝᛂ ଘଓƄቿቷ‫ ࢵޟ‬Ƅ ‫ۑ‬III ‫ۑ‬Ʀ ༇໩ ƄƆ୹ ୪ ܻࢧ ࢧ ࠑ ޯࡷƈ ࣠ࠒ ީଐଌƄƣ ᇴᆝ ƄƍƄଘଓƄ‫ۑ‬ƈ ࣠ࠒ ީ൞ᒑᐽࡴ ࠩߦࡩƄቿቷᐳ႓ƥƆ౰ేࢧ ߘƣ ࣠ࡓ ‫ކ‬ƣ Ƅ‫ ݪ‬༬࿻ ቿቷ‫ ࢵޟ‬ଘଓƄ‫ۑ‬IV ‫ܻۑ‬ᐱᐜެ༬ဈ ‫ ݪ‬༬࿻ ࠦ‫ࡩޯ ࠑ ࢧ ࢧޢ‬Ƅƥࠦ‫ࡩߦެޢ‬ƄƇ ܻႁ ࡯࣐ ࢧ ଐଌƄƣ ܾߙᑑ ᐽኅ ቱܻߙൎ൉ၯ ،‫ࡩީ ࠒ ݟ‬ƄƖ ࣠ࢠ࣠ࡷ፥ ጿ ‫ܻ ࣍ޟࠦۑ‬ၯ Ƈ ܻ‫ܻހ‬ၴ Ƅ፟ ፑƄƧ ƌ࣓‫ܻކ‬ᐱ᏶ଛଓࠦ୆෷ܻࢧ ࠁ ଘଓƄƆƎܻ෬ෝᛂ ଘଓƄ፥ ጿ߃ ީ༨ဈ ଘଓࠦࢥ ࡷܻ‫ݓ‬ ࡩƄቿቷ‫ ࢵޟ‬ଘଓƄ .V Żܸࢤ ߾ କଐƁࣰࣴ ၫߋ ‫ ࣉܸݳ‬Ƣ‫ێ ێ‬ ፥ ጘଘଓƄ፥ ጿࠢࢧ ‫ ࡓ ࡼކ࡯ ࠑޢ‬೘಼žܻࢧ ࠁ ଘଓƄࣷࣳၮߎ ‫ݶ‬ ܻ࣌ ƥࡴ ࠧƀƣ ƀ ࠦߥ‫ܻހ ࣐࢖ݹ‬ᐱᐟ‫ݑ‬౰ేީ‫ࡩޯݘ‬ƄƥūܻᐳႩ ୹ ‫ࡷ ࡼݑ‬ƨଘଓਔ ৹ࠦߥ‫ߘޢ‬ƌƥࠦ༨༤ Ƅƥࠦ‫ݹ‬ƌ߄ ၮƆƎ࣠ࢧ ‫ݶ ࢇހ‬ ܻߥஊै Ƅࡼ ‫ݑ‬౰ేީ‫ࡩޯݘ‬Ƅ ‫ۑ‬I ‫ࠦۑ‬୆෷ܻఴ ఝƂ ܻࠩࡺࠩߙࡣ ‫ޢ‬ᐳ႞ჴࡩᑑ ᐽƍƥūࠦ୆คܻ୆กƇ ܻࡷ࣠ࢵ Ǝƥƀơ ƥƄ߄ ၴ ፥ ጿܻᐳႩ ࢖‫ࠦހݘ‬୆෷ܻࢧ ࠁ ଗଓƄƇ ܻ‫ࡩީޢܻރ‬Ƅ࣊ ‫ ࣀޢ‬ხ჋ ‫ۑ‬II ‫࡯ޢ࡯ܻࠩۑ‬༨ဈ ƥ ،ࠦߦ࡯ޯശവറƄ࢖ࠧƄ࣏࣠ ࡩƄ໠ ໄ‫ރࠦހ‬ƎܻߥଐଌƄࠦ‫࡯ޢ‬ዙኸ ࡣ ᐳྜཾ ‫ۑ‬III ‫ۑ‬Ƈ ܻ‫ࡩߦ࢖ࢠޢ‬ƄƎܻީޯࠀ Ƅƥࠦၩܻ‫ ࢢޢ‬፥ ጿ߄ ၮܻఱ ఓ ‫ۑ‬IV ‫ࡷ ࣀޯߥީ࡯ޢࠦۑ‬Ƈ ܻ‫ޢܻރ‬ხჅ࣓ީჱޯࡩƄࠦ‫࡯ޢ‬ዙኸ ፥ ጿ߄ ၮܻఱ ఓ ‫ۑ‬V ࠦߘኆ ቱƄࡼ ࡷࠦ‫ܻࡩޢ‬ၮࠦၴ Ǝਔ ৹ࠦީࢵ ܻެଐଌƄƇ ƄƎƄ࢖ߥࡩƄƌܻ༧ဈ Ƅƥၖ࿏ ܻޯެࡩƄƒ ൎ൉ႀ ޯࢵ Ƅ ‫ۑ‬VI ‫ެܻ ࢵީࠦۑ‬ଐଌƄƇ ܻࢵ ܻ‫ࡩ ࣀޢ‬Ƅࠦၩܻ‫ ࢢޢ‬፥ ጿ߄ ၮܻఱ ఓ ‫ۑ‬VII Żܸࢤ ߾ କଐƁࣰࣴ ၫƄ କଐܸ༩஽ ‫ێ ێ‬ ࡼ ࡷࣷࣳၮ߄ ၴ ࣠‫ݹ‬ଘଓ‫ݑެࠢۑ‬žܻެ‫ݓ‬ ޯࢵ ଘଓƄࡼ ࡓ ೘಼ଘଓƢॅळ‫ࡩݘ‬Ƅᇯ ᇍ ࡩࡣ ‫࡯ޢ‬༨ဈ ƆƄƌᇷᆝ ࿃ ႓ܻࢵ Ƅࣷࣳၮžܻࢧ ࠁ ଘଓ‫ا‬ ࠦ‫ࡩ ࣐ީޢݘޢ‬ƄƢॅळ‫ࡩݘ‬Ƅƥƀࠦ‫ ࠅޯࡺܻݑޢ‬ଘଓƄƢॅळ‫ࡩݘ‬ƄƇ ଘଓܻ༬ீ ፥ ጿบ คᇴᆝ ƀ ،ࠦ୆෷ܻࢧ ࠁ ଘଓƄƟ ࢖࣐ ࡩƄƢ߄ႀ ޯఱ ఓଘଓƢॅळ‫ࡩݘ‬Ƅ žܻ‫ ࡓޢࡺޢ‬ᚻࡩƄƥžܻ‫ݹ‬౮ేߦࡩƄƥᑻ ᑧߦࡩƄࣷࣳၮ ،ƥƄƇ ܻީެࡩƄƥƀžܻ‫ ࠁޢ‬ଘଓƄƥƄƌܻࢧ ޯߘଘଓƄƥƆƎܻႅ ޯࡩƄƥࠦၮܻެࢧ ࡩᇴᆝ ،ࠦߦ࡯ޯശവറƄ ‫ۑ‬ƌܻࢢ ƎଘଓƄࣷࣳၮƥࢪ ߦެࡩƄࣷࣳၮƥƖ ܻࡺޯࠅ ଘଓƄࣷࣳၮƥ‫ࡩ ࣐ ݪ‬Ƅƥࠦ୆ก౰౔ࡩƄƥ ƞ ܸዖኵ କଐƁƢŻܸࢤ ߾ କଐƁ‫ۑ ۑ ێ‬ ࡴ ߘܻ࣐ ࡩƄ،ࠦ‫ܻࡩޢ‬ଐଌƄƥƇ ܻ‫ݹ‬౰౔ࣃ ଐଌƄƥƇ ܻ‫ީޢݘ‬ଐଌƄƥƉ ܻ୓คଘଓƄ፥ ጿܻࠧྲ Ⴐ ࠁ ࡼ ࡓ ೘಼ơ ܻዙኸ ଘଓƄơ ܻ༬ீ ፥ ጿƇ ܻ࣍ ܻࣃ ჱࡩƄ ،ࠦ୆෷ܻࢧ ࠁ ଘଓƄ‫ ࢵܻࡩޢ ݪ‬ଘଓƄƆ߄ ၮܻࣀ ೘ೃ‫ۑ‬Ɔƌ࣠எै ƄࠦީߘƄ୹ ୪ƥƉ ܻ୓คଘଓƄƈ ࣠༨ဂ ƥࢇ ‫ ࣀ࣠ݹ‬ჴࡩƄƥࠦީࢵ ܻഷവറƄƥ ،१ ॐ‫ࡩ࣠ ࣌ޢ‬Ƅ ᑑ ᐽƍƥƆ౰౗ࡓ ‫ހ‬Ƣ߄ ႀ ޯఱ ఠࠦ୆෷ܻࢧ ࠁ ଘଓƄ‫ ࢵܻࡩޢ ݪ‬ଘଓƄƣ ܾߙ ،ഃ೨ኅ ቱƄࠦߦࡩܻࣀ ࡩƄơ ܻዙኸ ଘଓƄƇ ଘଓܻੋ੅ਰள ஭‫୓ݘ‬෺ܻࡺ‫ߙޢ‬ ࠦၩܻ‫ݶ ࡣࡩ ࢧޢ‬ ܻಛಗƄƢƄ߄ ႀ ޯࢵ ଘଓƄƇ Ƅƍ፥ ፕ ƥܻᐱᐕ࡯ၮơ ࣠ࢧ ஊै Ƅࡼ ࡓ ೘಼཯དྷࡩƄƥࠦ‫ޢ‬ዙኒ ࡩƄƇ ܻࡷॅळ‫ݘ‬ଐଌƄƆ࢖ߙ࣠ࡩ मऒ࣠‫ ࡓ࣠ࡷީޢ‬ᚻࡩƄƆ࢔ࠩࠅ Ƅ፥ ጿࠦ࣌ ࣠ߦ༨ீ ƥƄमअƎܻߥ‫ކ‬ƆƎ࣠ࢧ ‫ހ‬ƣ ࣠ࡓ ‫ކ‬ƣ Ƅࡼ ࡓ ೘಼Ƈ ܻࡷॅळ‫ݘ‬ଐଌƄƥ‫ެܻ ࢵީࠦۑ‬ଐଌƄ Ƈ ܻࢵ ܻ‫ࡩ ࣀޢ‬Ƅ 18 ܸୃกܸ଍ଉŽ–ໝ ແࡦफउ –ࠣ‫ݶ‬ Ɖܸࢤ ެߕକଐƁƢࠣ‫ࡦ ࢽޟܸ ࢲޟ‬ƁƢࠣ‫ޟ‬පඓ ൷Ɓ౭౑ࢲ କଐƁƄ ܸࢲ ƁƋኃ ᒔᐺ። ጏ࢓‫ݕ‬ ࡦƁ࣬ ࣊ Ɓ࢓ߢೕಹኃ ቮƁ೾೥࢓଍ଉƁƄ ƁƋƁ߁ ࢟ ſ ƛ ܸߢࢽ ࡦƁ߁ ዖዄ Ž૊ સၫƋࣝެ࠾ኃ ቮƁƊܸެࢲ ଘଐƁ ። ፎକଐ߁ ެࢲ କଐƁƢ। ्࢟ ࣝࡦƁŻܸࢤ ߾ ଔଐƁ ࡼ ࡷƖ ࣠ࢠ࣠ଐଌܻ‫ߘࠦހ‬ൎ൉‫ࡩݘ‬ƄƇ ƄƍƇ ܻࡷॅळ‫ݘ‬ଐଌƄƒ ൎ൉ႀ ޯࢵ Ƅ፥ ጿે ઻භඣ ൺमअ߄ ଐଌƄƆƎ߄ ߘƥ‫ۑ‬Ƈ ܻߦ࡯ࡷƥƀƇ ൎ൉༬೛ ƥƀ Ɔ౰ేީࡓ ᚻࡩƄƇ ܿࣃ ჱଐଌƄ፥ ጿࢇ ‫ ࣀ࣠ݹ‬ჴࡩƄƈ ࣠༨ဂ ƣ ܾߙơ ܻ޲ଐଌƄࡣ ‫ޢ‬ხࢵ ે ઻ၮ‫ۑ‬ƎƄ࢖ߥࡩƄƍܻ༧ဈ Ƅ፥ ጿܻᐱᚑƄ߄ ႀ ޯࢵ ƄƥƇ ܻ‫ࡩީޢܻރ‬Ƅ Ɖ ܻ୓คଘଓƄमअ߄ ࡷ‫ۑ‬Ɔਔ ৲߄ ၴ Ƈ ܻႅ ޯެࡷमअ࣐࣠ ‫ކ‬ƥƉ ܻ୓คଘଓƆ߄ ၮܻࣀ ൞ᒑᐽᑻ ᑧᐱᐢࣀ ଐଌƄƠ ॅळఱ చࠦߥ࡯‫ࡷޯݘ‬Ƈ ܻ‫ܻރ‬୆กƢ߄ ႀ ޯఱ ఠ ơ ƥƄ߄ எै Ƅ‫ޢ ݊ۑ‬ჱࢧ ޯࡩƄƥƉ ܻ୓คଘଓƄƇ ܻ‫࡯ޢ‬ዙኸ ፥ ጿࡣ ‫ހܻࡩޯݘ ߄ݹ‬Ƣ࣠ߥ‫ݹ‬ཀྵདྷࡷƎ࢖ߥ‫ࡩޢ‬Ɔƌ࣠எै ƄࠦީߘƄ୹ ୪Ƈ ܻ‫ܻރ‬୆ก፟ ፑƄ࢖࣏ ୐෺ ࡣ ޲ଐଌܻ‫ހ‬ƥ‫ޢࠦۑ‬ዙኼ༟༎Ƅࢇ ‫ݶ‬ ܻߥஊै Ƅ౰ేߙ࣠ޯࡩƇ ܻ‫ީޢݘ‬ଐଌƄžƄƎ߄ࡷࡣ ީߘࡼ ࡷƎƄ࢖෬ෝᛂ ܻ‫ހ‬Ƣ߄ ႀ ޯఱ ఠࠦ୆෷ܻࢧ ࠁ ଘଓƄ࣊ ‫ݶ‬ Ƅ࢖உै Ƅƥ ࡣ ‫࡯ޢ‬༨ဈ ‫ ࢵܻࡩޢ ݪ‬ƀ‫ۑ‬ᑑ ᐽƍ፥ ጿƦ ౰౎ࡁࠦ‫ޢ‬ዙ዗Ƅࡣ ޯ༨࿻ žܻࢧ ࠁ ଘଓƄƥ ،ܻᐱᐢყީჷ჆ࡼ ࡓ ೘಼݊ ࡯ࣀ ᑃᐽࠦީࣀ ჱࡩܻ‫ހ‬Ǝܻ‫ ࢵݘ‬ଘଓƄ ౰ే‫ܻࡩޢࠦކܾމ‬ၮࠦߘਔ ৹ƥࠦ‫ޢ‬༨࿻ Ǝܾ‫ީ࣓ހ‬ჱޯࡩƄ፥ ጿơ ܻዙኸ ଘଓƄơ ܻၴ Ǝ߄ ၮܻఱ ఓơ ܻዙኸ ଘଓƄ፥ ጿƇ Ƅ࣓ީჱޯࡩƄƥࠦ‫ࡷެޢ‬༝༎Ƅࡣ ࢵ ൎ൉ࣀ ࡩƄ ‫ۑ‬Ƈ Ƅ౰ే‫ݓ‬ ޯଐଌƄࡼ ࡷࡣ ‫ݶ‬ ܻಛಗƄఅ௷ ᚻࡩƄ Ɖܸࢤ ެߕକଐƁƢŻܸࢤ ߾ କଐƁ‫ێ ێ ێ‬ Ƈ ܻ‫ࡩީޢܻރ‬Ƅ‫ ࡣۑ‬ၯ ኆ ቱƄƥƎܻࡺ޲ჴࢵ ଘଓƄƥƎܻၯ ƌଘଓƄƥƠ ൎ൉ᐱᐢࢵ ଘଓƄᑑ ᐽቦሼƥƆƥ౰౗ࡩƄ݊ ‫ݹ‬Ə࣠‫ކ‬ƥƉ ܻ୓คଗଓܻ‫ހ‬ƌܻࢧ ޯߘଘଓƄযঠᐱᏬ Ƈ ܻࢵ ܻ‫ࡩ ࣀޢ‬Ƅࠦၩܻ‫ ࢢޢ‬ƥࠦ‫ݹ‬ƌܻࢧ ޯߘଘଓƄᇯ ᆝ ܻࣃ ଐଌƄࡴ ࠩߙ፥ ጿƦ ౰౎ࡁࠦ‫ޢ‬ዙ዗ƀࡣ ޯ༨ဈ ࠦ୆෷ܻࢧ ࠁ ଘଓƄƟ ࢖࣐ ࡩƄƥࠦ୆෷ܻࢧ ࠁ ଘଓƄ ܻᐱ᏶ܺߙ ،ࠦީࢵ ܻެଐଌƄƇ ܻࢵ ܻ‫ࡩ ࣀޢ‬Ƅࠦၩܻ‫ ࢢޢ‬፥ ጿ߄ ၮܻఱ ఠܻᐱ᏶Ƅว คܻၴ ፟ ፑƄ ،ࠦ୆෷ܻࢧ ࠁ ଘଓƄƟ ࢖࣐ ࡩƄ‫ݹࠦۑ‬ƌܻࢧ ޯߘଘଓƄ ൎ൉‫ݶ‬ ܻࠧሯሡሁ Ɖ ܻޯ༨ဈ ቿቷ‫ ࢵޟ‬ଘଓƄƤ߃ ࠧߴ ޯჱ჈ࡼ ଐଌƥߎ ‫ ࡁޢ‬ƥߴ ޯჱ჈Ƅƍܻࡷơ ܻ޲ଐଌƄࡣ ‫ޢ‬ხࢵ ે ઻ၮ‫ܻۑ‬ᐱᏹƄ౰ే‫ކܾމ‬যওߥ‫ݹ ࢥܻކ‬ƀ ‫ۑ‬Ɠ ࢖‫ݘ‬ᑃᐽࡣ ‫ߥܻހ‬ଐଌƄ‫ࡩ ࣐࡯ ݪ‬Ƅࡣ ‫ކݘ ߄ݹ‬፥ ጿܻ‫ ߄ރ‬ၮܻఱ ఠƉ ܻ୓คଘଓƄƇ ܻ୆෷ܻࢧ ࠁ Ƃ‫ࠦۑ‬୆෷ܻࢧ ࠁ ଘଓƄƇ ܻ‫ࡩީޢܻރ‬Ƅࡼ ࡷ ƣ ᇴᇍ ࣀ ࡩƄƇ ܻߥީ࣍ ߎ ࡯ޯ༧ீ ܻಛಗൎ൉ၯ ࡼ ࡷࡼ ࡓ ೘಼཯དྷࡩƄࠦߥ‫ࡩ ࣐࢖ݹ‬Ƅƌܻ༬࿻ Ƅࡼ ࡷܻެެࡓ ೘೉Ơ ൎ൉ᐱᐢࢵ ଘଓƄƇ ܻ୆෷ܻࢧ ࠁ Ƅƥ ‫ ࠀޢ࡯ࠩ ࡴۑ‬Ƅ߄ ࡷƟ ܻߦ‫ރ‬Ƅࡼ ࡷ ‫ߘޯ ࢧܻد‬ଘଓƄࣷࣳၮ፥ ጿܻ࣌ ࣠ࠒ ࡯ࡷƄ౰ే‫ކܾމ‬Ƈ मकƄƥžܻࢧ ࠁ ଗଓܻ‫ހ‬Ɔ߄ ‫ ࣍ޢ‬ƥࠦߘൎ൉ၮܻಛಗ཯དྷࡩƄܻ‫ޢ‬ხఱ ఝࠦୖ෺߄ ஊै ƄƢॅळ‫ࡩݘ‬Ƅࡼ ࡷƥ ፥ ጿࠦ୆෷ܻࢧ ࠁ ଘଓƄƟ ࢖࣐ ࡩƄࢇ ‫୓ ࢥࡺ ࡼކ ࣐ީޢ‬෺Ƨ ኅ ቱƄ> Econometrics ]࿃ ႓ܻ‫ࡩߥޢ‬Ƅƌܻࢧ ޯߘଘଓƄࣷࣳၮ࣠ࠧ ‫ݹࠦۑ‬ƌܻࢧ ޯߘଘଓƄƈ ࣠ࠒ ީࡩƄ፥ ጿ݊ ࢵ Ƅƥᇯ ᇍ ఴ చƢ߄ႀ ޯఱ ఓƥࠦ‫ݹ‬ƌܻࢧ ޯߘଘଓƄࠦ‫ࡩެ ࣏࢖ݹ‬Ƅ Ƅ ܸ‫ݶܸ ࢝ޟ‬༜་ƁƢŻܸࢤ ߾ କଐƁ‫ێ ێ ێ‬ ࠦ୆෷ܻࢧ ࠁ ଘଓƄƟ ࢖࣐ ࡩƄ፥ ጿƕ ࣠ࠒ ࡯ଐଌƄƎ࣐࣠ ޯࡩƄ‫ۑ‬Ƈ ܻ‫ݹܻࢠޢ‬༟༎Ƅƥžܻࢧ ࠁ ଘଓƄ໠ ໄ‫ ࠅ࣠ࠧ࢖ݹࠦހ‬Ɣ ୍ܻงƎƄࠦߘൎ൉ၮ߄ ၴ ࣠‫ކ‬ ࠦ‫ ރ ࣏࢖ݹ‬ƣ ƀ‫ࡷ ࡣ ۑ‬ᇴᇍ ޯࡩƄƥ ࡣ ࢠܻߦޯࡩƄƥ ౰౎எै ᇴᆝ ، Ƈ ܻ‫ݹܻࢠޢ‬༟༎Ƅ Ɩ ƥ࢖ߙ ߎ ࡯ޯ༧ீ ƢƄ߄ ႀ ޯࢵ ଘଓ ࠦႅ ‫ޢ‬ჴ჈ ࣠ࠧ ƣ ƥƌžܻࢧ ࠁ ଘଓƄࠦࣃ ߘܻެࡷࡼ ࡓ ೘಼ଘଓƥ፥ ገܻࢧ ࠁ ଘଓƄࡣ ‫࡯ޢ‬ႁ ޯࡩܻ‫୍ܻ ܻ࣍ ࠅ࣠ࠧ࢖ݹܻހ‬งƎƄ࣊ ୍งमऒƇ ଘଓܻࡺޯࠁ ଘଓƄ ‫ۑ‬Ƈ ଘଓܻࡺޯࠁ ଘଓƄࠦ‫ޢ ߄ࡩެ ࣏࢖ݹ‬எै Ƅƅ ܻ‫ޢݘ‬ჴࢵ ଘଓƄ ࠣ‫ݕޟ‬ ‫ࡦ ࣍ަޟ‬ƁƢࠣୃ෴ܸ‫ݶ‬౫ౄߣࡦƁƟूर‫ݕ‬ ࡦƁƢŻܸࢤ ߾ କଐƁ‫ێ ێ ێ‬ 19 ܸୃกܸ଍ଉŽ–ໝ ແࡦफउ –ࠣ‫ݶ‬ Ɖܸࢤ ެߕକଐƁƢࠣ‫ࡦ ࢽޟܸ ࢲޟ‬ƁƢࠣ‫ޟ‬පඓ ൷Ɓ౭౑ࢲ କଐƁƄ ܸࢲ ƁƋኃ ᒔᐺ። ጏ࢓‫ݕ‬ ࡦƁ࣬ ࣊ Ɓ࢓ߢೕಹኃ ቮƁ೾೥࢓଍ଉƁƄ ƁƋƁ߁ ࢟ ſ ƛ ܸߢࢽ ࡦƁ߁ ዖዄ Ž૊ સၫƋࣝެ࠾ኃ ቮƁƊܸެࢲ ଘଐƁ ። ፎକଐ߁ ެࢲ କଐƁƢ। ्࢟ ࣝࡦƁŻܸࢤ ߾ ଔଐƁ Ɵ ࢖࣐ ࡩƄ‫ ߄ࡷ ݝ‬ႀ ޯࢵ Ƅ཯ དྷࡩƄƇ ଘଓ഻ܻവറƄࡼ ࡷ፥ ፕ ،žܻ‫ݹ‬౮ేߦࡩƄƥ ،ܻ‫ޢ࣠ࡩ࣠ ࠅޢ‬எै Ƅ ،ᑻ ᑧߦࡩƄࣷࣳ‫ ࡁݘ‬،ࠦ୆෷ܻ‫ݹ‬౮ేߦࡩƄƢॅळ‫ࡩݘ‬Ƅ ࠦ୆෷ܻࢧ ࠁ ଘଓƄ‫ ࢵܻࡩޢ ݪ‬ଘଓƄƢ߄ႀ ޯఱ ఠƥ‫ۑ‬Ƣॅळ‫ࡩݘ‬ƄƤ߃ ࠧमअ࣐࣠ ‫ކ‬፥ ጿ‫ࡺ ݝ‬ᐳ႓ƄƥƆಒ౷ީࡷƇ Ƅ౰౔ߙ፥ ጿࠦ୆෷ܻࢧ ࠁ ଘଓƄ ‫ۑ‬ƌܻࢢ ƎଘଓƄࣷࣳၮƥ ،ࠦࢵ ߄ ެಛಗƄƥžܻ‫ ࡓޢࡺޢ‬ᚻࡩƄ፥ ጿࡼ ࠧƄ༟༎Ƅ‫ࡩ࣠ߘ ݝ‬Ƅ፥ ጿࠦߙܻຫ຀‫ހ‬ ƣ Ƅ࣠‫ޢ‬ஊै Ƅƥ‫ࡩ ࣐ ݪ‬Ƅƥžܻ‫ ࠁޢ‬ଘଓƄƢॅळ‫ࡩ ࣐ީޢݘޢࠦۑߙݘ‬ƄƢॅळ‫ࡩݘ‬ƄƇ ଘଓܻ༬ீ ፥ ጿ‫ ߄ࡷ ݝ‬ႀ ޯࢵ Ƅࠦ୆෷ܻࢧ ࠁ ଘଓƄ‫ ࢵܻࡩޢ ݪ‬ଘଓƄ ፥ ጿൎ൉޲ዙኽ‫ࠦۑ‬୆෷ܻ‫ࡩߦ࢔ݹ‬ƄƢॅळ‫ރࠦހܻࡩݘ‬Ǝܻߥࡷߎ ޲ࡓ ࡷᇯ ᇍ ఴ చࠦߦ࡯ޯശവറƄࠦ୆෷ܻࢧ ࠁ ଘଓƄƟ ࢖࣐ ࡩƄƢ߄ ႀ ޯఱ ఠƇ ܻީެࡩƄƥ ࠦߥ࡯‫ޯݘ‬ଐଌƄƥ࢛ ‫࢖ݹ‬ଐଌƄቿቱܻၰ ࡼ ‫ݑ‬Ƈ ܻ‫ܻރ‬୆กࠢ‫ݹ‬ኆ ቱ࢖ߙ࣠୓ขƣ Ƅ‫ ݪ‬༬࿻ Ɠ ࢖ଐଌƄ‫ ݪ‬ყީ࣐ ࡩƄ࢝ ࠑ ఴ ఓഭ ഑ ᚻࡩ‫ࡩ ࣐ ݪ‬Ƅ ࠦஎै ܻ‫ݘ‬ଐଌ໠ ໄ‫ࡷݘ‬Ǝܻߥ‫ߙݘܻࡩޢࠦݑ‬Ʀ ߄ ࡷે ઻ၮఅ௷ ॶॱ५ܻࢥ ‫ݹ‬Ƅƥ‫ۑ‬ƆƎƄ࢖ஊै Ƅࠦၴ Ǝƌƥ‫ࡩߥ࡯ ݪ‬ƄƇ ܻߘƌƥƢኆ ቱƄ࣊ ‫ݓ‬ ࢥ‫ހ‬ ࠦߙ࢖‫୓ ࣐࡯ ݪࡷݘ‬෺Ƅ߃ ࠧƥܻࠩ࡯ࣃ ߙƥƄࠦ‫ࡩޯ ࠖ࢖ހ‬ƄƊ ܻ༬စ ે ઻ၮƞ ࢖‫ޯݘ‬ᑃᐽƦ ࢖༬ဈ ƣ Ƅ‫ ݪ‬༬࿻ ƅ Ǝܻႅ ޯࡩƄƣ ܻߙ࢛ ‫࢖ݹ‬ଐଌƄ Ƈ ܻ୆෷ܻࢧ ࠁ ଘଓƄે ઻ၮ߄ ࡺޯ‫ݹݘ‬ƣ Ƅ‫ ݪ‬༬࿻ žܻ‫ ࠁޢ‬ଘଓƄࣷࣳၮ፥ ጿ‫ۑ‬Ǝܻߥ‫ݘ‬ᑃᐽࡴ ಛಗƥܻ୐ง߄ ‫ހݘ‬Ƅ࣠எै ܻ‫ކݘ‬௪ ௎ኅ ቱƄƒ ܻ༧ೞ ଘଓƄƌ߄ ၮ ‫ހࠦۑ‬౰౔ࡩƄƥƆƎƄ࢖ஊै ƄƇ ܻၴ Ǝਔ ৹Ƈ ܻީެࡩƄमकܾ‫ކ‬Ʀ ߄ ࡷે ઻ၮƞ ࢖‫ޯݘ‬ᑃᐽᑑ ᐽƍƥƅ Ǝܻႅ ޯࡩƄžƄ༇໮ ଘଓ ࣬ ൛ൖ‫ݕ‬ ࡦƁ‫ࡦަ ࠏݜ‬ƁƢŻܸࢤ ߾ କଐƁ‫ێ ێ ێ‬ Ɖ ܻ୓ค፥ ፕ ࠦߙ࢖‫ݘ‬ଐଌƄ፥ ጿࡣ ‫ݶ‬ ܻಛಗƄƎ࣐࣠ ޯࡩƄƥ‫ޢࠦۑ‬൞൙‫ࡩݘ‬Ƅƈ ࣠ࠒ ީࡩƄ፥ ጿࠢެ‫ݑ‬ཀྵཏ‫ ݌‬ଘଓƥ࿃ ႓ܻࢵ Ƅžܻࢧ ࠁ ଘଓƄࣷࣳၮ Ɖ ܻ୓คƄơ ࣠ࠁ บ ෺༇໮ Ƅ཯དྷࡩƄƅ Ǝܻႅ ޯࡩƄൎ൉޲ዙኽ‫ࠦۑ‬୆෷ܻࢧ ࠁ ଘଓƄ‫ ࢵܻࡩޢ ݪ‬ଘଓƄƢƄ߄ ႀ ޯࢵ ܻ‫ހ‬บ ෺༇໮ Ƅ཯དྷࡩƄƅ Ǝܻႅ ޯᑃᐽ Ɩ Ƅ࣠‫ރ‬Ƅƞ ൎ൉ޯࠀ ଘଓܻߥީ࣍ Ƈ ܻ‫ރ‬Ƅ࣠‫ޢ‬ஊै Ƅࠦ୆กमऒƥƀࠦ‫ހ‬౰౔ࡩƄƥƆ߄ ያዜ ଘଓƄࡼ ࡷࠦߦ࡯ޯശവറƄƖ Ƅ࣠‫ރ‬ଘଓƄƥࠦ‫ݑޢ‬ƄƎ༝༎Ƅࡣ ‫ܻ ࢢޢ‬ഷവറƄ ‫ࠦۑ‬୆෷ܻࢧ ࠁ ଘଓƄƟ ࢖࣐ ࡩƄƢƄ߄ ႀ ޯࢵ ܻ‫ހ‬ᑑ ᐽƍƥࡣ ‫࡯ޢ‬ႁ ޯࡩƄžƄ༇໮ Ƅƥܻࠩࡺ‫ކ ࢧࡺޢ‬যঠ‫ܻࡷܻݹ‬೘ಹƄƌ ،ࠦ‫ޟ‬ყީࡩƄƥࠦ‫ݓ ߃ݹ‬ ޯࡩƄ ࠦႁ ࢧ ࡩƄƥ‫ࡩ ࣐ ݪ‬Ƅ፥ ጿܻ‫࡯ޢ‬ၴ Ƅ౰ే‫ކܾމ‬Ƈ मकƄƥࠦ‫ࡩ ࣐ީޢ‬ƄƢॅळ‫ࡩݘ‬Ƅ፥ ጿƆ߄ ఴ చƢ߄ ႀ ޯఱ ఠࠦ୆෷ܻࢧ ࠁ ଘଓƄ‫ ࢵܻࡩޢ ݪ‬ଘଓƄƥ ‫ࡩݘܻࡷࠦۑ‬Ƅ Ɩ ƥ࢖ߙƏƥमऌƦ ƌƀࠦ‫ ࠅޯࡺܻݑޢ‬ଘଓƄƥࠦ‫ࡩ ࣐ީޢݘޢ‬ƄƢॅळ‫ࡩݘ‬Ƅ፥ ጿࠦ୆෷ܻࢧ ࠁ ଘଓƄ‫ ࢵܻࡩޢ ݪ‬൑ᑉᐽߎ ޲ࡓ ᅚଌƄƢƄ߄ ႀ ޯࢵ ଘଓƄƣ ƀ Ɛ Ǝਔ ৲Ƨ ኅ ቱ‫ ا‬࿃ ႓ܻ‫ࡩߥޢ‬Ƅƌܻࢧ ޯߘଘଓƄࣷࣳၮว คܻၴ ፟ ፑƄ‫ۑ‬ቿቷࢧ ߦެࡷࠦ‫ߙ࢖ݑޢ‬Ƈ ଘଓܻ༬ீ ᇯ ᇍ ఴ ఠࠦ୆෷ܻࢧ ࠁ ଘଓƄƢॅळ‫ݘ‬ᑃᐽ Ʀ ༇໩ Ƅࠦ‫ߙ࢖ݑޢ‬ƢॅळၮƠ ܻެࠧƣ ܻߙܻᐱᏹ୍ܻชƄ፥ ጿܻࢥ ‫ݹ‬ƄƢ߄ ႀ ޯఱ ఓƥܻ‫ޢ‬ዙኤ ܻᐱᏹܻߥ‫ކ ࣐ީޢ‬ƥࠦ‫ݹ‬ƌܻࢧ ޯߘଘଓƄƇ ܻ‫ࡩެ ࣏࢖ݹ‬Ƅ žܻࢧ ࠁ ଘଓƄ ، Ƨ ࣠‫ހ‬౰౔ࡩƄ žܻࢧ ࠁ ଘଓƄ ، Ƨ ƌܻࢧ ޯߘଘଓƄ žܻࢧ ࠁ ଘଓƄƥ ، ࿃ Ⴉ ܻ‫ݹ‬༟༎Ƅ žܻࢧ ࠁ ଗଓᇴᆝ Ƈ Əमऌ Ɩ ƥ࢖ߦࡩƄƤ߃ ࠧᇯ ᆝ ‫ۑ‬፥ ገܻެஎै Ƅžܻࢧ ࠁ ଘଓƄƥ፥ ጕᇴᇍ ࣀ ࡩƄžܻࢧ ࠁ ଘଓƄƥƧ ࣠‫ޢ‬ஊै Ƅžܻࢧ ࠁ ଘଓƄƥ ،፥ ፎ ܻࡺޯࠅ ଘଓƄ ࡼ ࡷơ ܻ༬ீ ᇯ ᆝ ፥ ጿܻᐱᐢࢵ ƄƎƌযঠ‫཯ݹ‬དྷࡩƄ݊ ‫ࢠޢ‬Ƅ࣠ଐଌܻ‫ހ‬Ɣ ୍ܻงƎଘଓܻ‫ހ‬ᑑ ᐽƍƥࠦ୆෷ܻࢧ ࠁ ଘଓƄ‫ ࢵܻࡩޢ ݪ‬ଘଓƄƢ߄ ႀ ޯఱ ఠ ‫ ࠅޯࡺܻݑޢࠦۑ‬ଘଓƄƥࠦ‫ࡩ ࣐ީޢݘޢ‬ƄƢॅळ‫ࡩݘ‬ƄƇ ଘଓܻ༬ீ 20 ܸୃกܸ଍ଉŽ–ໝ ແࡦफउ –ࠣ‫ݶ‬ Ɖܸࢤ ެߕକଐƁƢࠣ‫ࡦ ࢽޟܸ ࢲޟ‬ƁƢࠣ‫ޟ‬පඓ ൷Ɓ౭౑ࢲ କଐƁƄ ܸࢲ ƁƋኃ ᒔᐺ። ጏ࢓‫ݕ‬ ࡦƁ࣬ ࣊ Ɓ࢓ߢೕಹኃ ቮƁ೾೥࢓଍ଉƁƄ ƁƋƁ߁ ࢟ ſ ƛ ܸߢࢽ ࡦƁ߁ ዖዄ Ž૊ સၫƋࣝެ࠾ኃ ቮƁƊܸެࢲ ଘଐƁ ። ፎକଐ߁ ެࢲ କଐƁƢ। ्࢟ ࣝࡦƁŻܸࢤ ߾ ଔଐƁ Ļ ļû ʆ ļ Ļ ĻĻ ļ ʫŅźȶ őǀ Ż ŷŉĿɩŅřǔ ſ ńņ ŅɩŅŉĿŉľŮńţ ȏĿŉřǓ 21 ܸୃกܸ଍ଉŽˍ ໝ ແࡦफउˍ ࠣ‫ݶ‬ Ɖܸࢤ ެߕକଐƁƢࠣ‫ࡦ ࢽޟܸ ࢲޟ‬ƁƢࠣ‫ޟ‬පඓ ൷Ɓ౭౑ࢲ କଐƁƄ ܸࢲ ƁƋኃ ᒔᐺ። ጏ࢓‫ݕ‬ ࡦƁ࣬ ࣊ Ɓ࢓ߢೕಹኃ ቮƁ೾೥࢓଍ଉƁƄ ƁƋƁ߁ ࢟ ſ ƛ ܸߢࢽ ࡦƁ߁ ዖዄ Ž૊ સၫƋࣝެ࠾ኃ ቮƁƊܸެࢲ ଘଐƁ ። ፎକଐ߁ ެࢲ କଐƁƢ। ्࢟ ࣝࡦƁŻܸࢤ ߾ ଔଐƁ ࠢ‫ހܾރ‬໠ ໄ୆෷ܻࢧ ࠁ ଘଓƄࡼ ࡷ౰ే޲ࡓ ᚻࡩƄƇ ܻ୆กƌƀ፥ ጿƌƎƥሯሡሁ ،žܻࢧ ࠁ ଘଓƄࣷࣳၮߎ ‫ކݘ࢖ݹ‬Ꭸ᎙ơ ƥଘଓƄࡣ ࢧ ߦࡩƄ፥ ጿ ܻࠧ౰ేࣀ ߦ‫ކ‬ƥܻࠩ࡯‫࡯ޢ‬༨ဈ ƥࠦ୆෷ܻࢧ ࠁ ଘଓƄƇ ܻ‫ࡩީޢܻރ‬ƄƓ ୹ ‫ݑ‬ƥ،ว ෺୍࣠งƥߎ ‫ޢ‬ჱࢧ ‫ ކ‬،݊ ዙዋ ݊ ࡷࡣ ࡷܻ‫୓ݘ‬෺Ƨ ኅ ቱƄࣷࣳ‫ࡩݘ‬Ƅ যঠ‫ ࢵޢ‬፥ ጕܻ޲ࡩƄࡣ ࢧ ߦࡩƄ፥ ጿƥ‫ܻࠧۑ‬Ǝƌܻࢧ ࡷƥࠦ୆෷ܻࢧ ࠁ ଘଓƄƇ ܻ‫ࡩީޢܻރ‬Ƅ݊ ዙዋ Ɵ ࢖࣍ ߰ ‫ݹ ࢥܻކ࣠ࢠޢ‬ƄᎨ᎙ƥ‫ۑ‬ .ࠦ୆෷ܻࢧ ࠀ ଘଓƄƇ ܻ‫ࡩީޢܻރ‬ƄƓ ୹ ‫ݑ‬ƥว ෺୍࣠งƥߎ ‫ޢ‬ჱࢧ ‫ކ‬Ɵ ࢖࣍ Ɠ Ƅ࢖‫ ࢵޯݘ‬Ƅ Ƈ ܻ‫ࡩީޢܻރ‬Ƅߎ ‫ޢ‬ჱࢧ ‫ ކ‬.2.1 ཯དྷࡩƄƥūࠦ‫ ࠅݘ‬Ƅ࢖ଐଌƄቿቷၰ ୹ ୪፥ ጘܾ‫ކ‬ཀྵདྷࠁ ūƇ ܻ‫ࡩީޢܻރ‬Ƅ݊ ዙዋ ࠦ‫࡯ޢ‬ዙኸ ཯᝵ᐢჱ჋ƣ ƀܻࡷ Ƈ ܻ‫ࡩީޢܻރ‬Ƅࠦ‫ ࠅݘ‬Ƅ୹ ୪ : ‫ۑ ۑ‬ ࠦႁ ‫ࡩ ࢧ ࠒޢ‬Ƅࠦ‫ܻހ‬ၴ ଗଓƄƇ ƄƍƇ ƄƎܻ෬ෝᛂ ଘଓƄॡ ॏީჴఱ ఠƣ ƀે ઻ၮࢇ ߘਔ ৿ƥūࠦ୆෷ܻࢧ ࠁ ଗଓƄƇ ƄƎܻ෬ෝᛂ ଘଓƄܻᐱᐕߙ࢝ ࠒ ߦ‫ކ‬ ‫ࠦۑ‬ႁ ‫ࡩ ࢧ ࠒޢ‬Ƅ౰ేၩƇ ܻ‫ܻހ‬ၴ ଗଓƄƇ ƄƍƇ ƄƎܻ෬ෝᛂ ଘଓƄƥƀūࠦࢧ ߘܻެࡩƄƇ ƄƎܻ෬ෝᛂ ଘଓƄ߄ ‫ީݘ‬ჴఱ ఠƥūቿቷࡷᇴᇍ ࡩƄ Ʀ ߄ ࡷࠦߙ࢖‫ࡷݘ‬ƥƇ ܻ‫ࡩީޢܻރ‬Ƅࠦ‫ ࠅݘ‬Ƅ୹ ୪౰౎ၮࡼ ࡷ༝༎Ƅƥ߄ ࠩஎै Ƅ౰ేߙ࣠‫ ࡣކ‬ၴ Ƅࡼ ࡷƥ Ƈ ܻ‫ࡩީޢܻރ‬Ƅቿቱƥ߄ ၴ ‫ۑ ۑ‬ ౮Ʈేၰ ࢖‫ݓ‬ ࢢ ܾ‫ࢠݘܻࠩހ‬ƥƧ ƀ‫ܻۑ‬ᐱᐢࡩƥ߄ ၴ ࠦ‫࡯ޢ‬ዙኸ ܻᐱᐟࡷᎨ᎜ ،‫ࡩީ ࠒ ݟ‬Ƅƞ Ƅ߄ ࠧƀ݊ ࡷܻࠩߥߙƄ࣠‫ކ‬ƥū‫ީܻ ࠁ ݟ‬ᑃᐽܻᐱᏹਔ ৯ܻߙ ơ ƥƄ߄ எै Ƅƥ‫ۑ‬ơ ƥ߄ எै Ƅ ᚻ‫ࡺ ࣩހ‬ఱ ఓ ،ܻࠩࡺ‫୐ ࣏ޢ‬งƥܻࠧƏ࢖ߙ߄ ‫ހݘ‬Ƈ ܻ‫ࡩީޢܻރ‬Ƅࠢ‫ކ࣠ࢠ ݊ߙޢ‬Ƨ ኅ ቱƄ‫ ݪ‬ყ჋౰౔ࡩƄ ƥūࡼ ࡓ ያዪ ߰ ࡯ࢧ ‫ܻݹ‬ᐱᐟࡷᇯ ᆝƥ‫ࡩ޲ܻރ࣠ݹࠦۑ‬Ƅܻᐱᐟࡷƥūࠦ‫ࡩޢ‬ƥଛଓƄơ ƥƄ߄ எै ƄܻᐱᐟࡷƍƂūࠦၮ࣠ެޯࡷƥࠦߦ࡯ޯ༧ீ ơ ᇴᇍ ࢸ ƀે ઻ၮƣ ࣠ࡓ ‫ࠧެܻކ‬ ߎ ޲ࡓ ࡷࡴ ᄽ஛ ፥ ጿܻࠩࠒ ‫ކ࣠ࢠޢ‬ƥƇ ܻ‫ࡩީޢܻރ‬ƄƏƄमऌƂ፟ ፑƂƞ ߄ ᐱᏹƨܻ‫ޢݘ‬ዙዋ ܻᐱ᏶ƀƮଘଓƂࠦެ‫ࡷݘޢ‬Ƈ ଘଓܻၰ ፥ ጿƢƄ߄ ႀ ޯࢵ ൎᑉᐽ .࢖‫ݓ‬ ࢧ ࡷƥ ߄ ‫ߙީݘ‬ว ෺࣠ީޯࡩƄࠦ‫࡯ޢ‬ዙኸ ፥ ጿࠦࡺࠩଐଌƄƇ Ƅ࣐࣠ உै Ƅࡼ ࡷƇ ܻ‫ࡩީޢܻރ‬Ƅߎ ‫ޢ‬ჱࢧ ‫ ݹݘ ߄ކ‬Ƈ ܻ‫ࡩީޢܻރ‬Ƅߎ ‫ޢ‬ჱࢧ ‫ކ‬2.1.3 ‫ࡩީޢܻރܻت‬ƄƏ࢖ߙࠦ‫࡯ޢ‬ዙኸ ፥ ጘܾ‫ܻކ‬ᐱᐢ‫ ࠅݘ‬Ƅ୹ ୪ƥࠦ‫ॅހ‬ळ࣐ ଐଌƄƇ ܻ‫ݘެޢࠦހܻࡩީޢܻރ‬ଐଌƄƇ ƄƎܻ෬ෝᛂ ଘଓƄ݊ ዙዋ ࠦ‫࡯ޢ‬ዙኸ ࡼ ࡷžܻᐱᐢ᠍ଘଓƄ Ƈ ܻߦࢧ ࡩƄࡼ ࡷ࢛ ‫ހݘ‬፥ ጿࠦၮ࣠ዙಣಝᇯ ᆝ Ơ ౰౔ఴ ఠƣ ᇷᆝࠦެ‫ࡷݘޢ‬Ɔ߄ ၮܻߘܻࠧ߄ ၰ ࣠‫ކ‬ūƆ౰ే‫ݓ‬ ࢢ ƞ ܻެࢢ ƀƥ݊ ‫ܻࡷޢ‬༬ீ ፟ ፑƂ Ƈ ܻީ࡯࣐ ޯࡷ‫ ࣀ ݪ‬༨ဂ ᅵै Ƅ‫ ࡣۑۑۑ‬ၯ ኆ ቱƄࠦ‫ߙޟ‬ūࠦ‫ ྲݹ‬႓ଛଓƄቿቱܻஊै Ƅūƣ Ə࣠ࡩƄū࢖ࡺ‫ࡩݘ‬Ƅūơ ࣐࣠ ࡩƄ ࢝ ‫ݶ‬ ܻࢧ உै Ƅƥƀ ‫ࡩީ ࠒ ݟۑ‬Ƅ ว ෺࣠ީჴჅ߄ ࢧ ߥ‫ݹ‬ƥƇ ܻ‫ࡩީޢܻރ‬Ƅว ෺୍࣠งࠦ‫ހݘࡺ࡯ޢ‬ƀ߄ ୍คƇ ܻ‫ࡩީޢܻރ‬Ƅߎ ‫ޢ‬ჱࢧ ‫ކ‬Ƣܻ೘೉Ƅ߄ ‫ ހݘ‬Ƅ ܸ‫ࡦަޟܸހ‬Ɓฤ ෷ࣝ୊ค ‫ێ ێ‬ ܻࠩߙൎ൉ޯࠀ Ƅ ፥ ጘܾ‫ݹ‬ūࠦߦ࡯ޯ༧ீ ‫ ࢵܻࡩޢ ݪ‬ƀว ෺࣠ީޯᑃᐽƥ‫ۑ‬ơ ƥƄ߄ ၴ ፥ ጿƇ ܻ‫ࡩީޢܻރ‬Ƅ݅ ‫ކߦ࢖ݹ‬ƥߎ ‫ޢ‬ჱࢧ ‫࡯ޢࠦކ‬ዙኸ ūƇ ܻ‫ࡩީޢܻރ‬Ƅ ،ܻᐱᐜ᠃୍࣠ง߄ ‫ހݘ‬Ƈ ܻ‫ࡩީޢܻރ‬ƄܻᐱᏳƢ ߄ႀ ޯఱ ఠƞ ࣠ࢵ ཯དྷࡩƄࠦ‫ ࡓޢߦޢ‬ᚻࡩƄƄቦሼƥūܻᐱᐜ᠃୍࣠งƌ Ƅ࢖ଐଌƄƇ ܻ‫ࡩީޢܻރ‬Ƅࠦ‫ ࣀ ݪ ࣍ީޢݘ‬༨ဂ ว ෺࣠ީޯࡩƄƖ Ƅ࣠‫ރ‬ƀࡼ ࡷƥ ᇯ ᆝ ƣ Ƅ Ɛ ܻࢵ Ƅે ઻ၮơ ƥƄ߄ ၴ ፥ ጿܻᐱᐜყ჋मऒƥࠦߦެࢧ ଐଌƄƇ ܻ‫ࡩީޢܻރ‬Ƅ݊ ‫ޢ‬ዙ၅ေࡼ ‫ݑ‬ƆƎܻީ‫ݑ‬። ጼƁ࢓‫ݐ‬ ஋ॅ Ɓฤ ෷ࣝަެࡦƁ  .໠ ໄ‫ࡷݘ‬Ƨ ƎƄƌƄযওࣀ ߥ‫ކ‬ƥƄࠦެ‫ߙޢࠦࡷݘޢ‬Ƅ࢖‫ݓ‬ ࠅ Ɔ߄ ၰ ࣠‫ހ‬ƒ ܻၯ ܻᐱᐟࡷ݊ ዙዋ ᇯ ᆝ ƣ ƄƐ ܻࢵ Ƅે ઻ၮơ ƥƄ߄ ၴ ፥ ጿܻᐱᐜყ჋मऒƥࠦߦެࢧ ଐଌƄƇ ܻ‫ࡩީޢܻރ‬Ƅ݊ ‫ޢ‬ዙ၅ေࡼ ‫ݑ‬ƆƎܻީ‫ཬݑ‬ཌࡴ༚་Ɓฤ ෷ࣝަެࡦƁ  .ࠦެࣀ ࡩƄƥƄ࢖ᐳႡࡩƄƥƄƖ ࣠ީࢵ ଘଓƄƥƄƢ࣠‫ࡩޢ‬ᇴᆝࠦެ‫ࡷެޢࠦࡷݘޢ‬ƏƆ߄ ၰ ƥ፟ ፑƄƌ࣠‫ܻݹݘ‬ᐱᐟࡷ݊ ዙዋ 22 ܸୃกܸ଍ଉŽ–ໝ ແࡦफउ –ࠣ‫ݶ‬ Ɖܸࢤ ެߕକଐƁƢࠣ‫ࡦ ࢽޟܸ ࢲޟ‬ƁƢࠣ‫ޟ‬පඓ ൷Ɓ౭౑ࢲ କଐƁƄ ܸࢲ ƁƋኃ ᒔᐺ። ጏ࢓‫ݕ‬ ࡦƁ࣬ ࣊ Ɓ࢓ߢೕಹኃ ቮƁ೾೥࢓଍ଉƁƄ ƁƋƁ߁ ࢟ ſ ። ፎକଐ߁ ެࢲ କଐƁƢ। ्࢟ ࣝࡦƁŻܸࢤ ߾ ଔଐƁ ƛ ܸߢࢽ ࡦƁ߁ ዖዄ Ž૊ સၫƋࣝެ࠾ኃ ቮƁƊܸެࢲ ଘଐƁ ‫ ان‬Ɛ ܻࢵ Ƅે ઻ၮࠦࢢ ܻၯ ơ ƥƄ߄ ၴ ፥ ጿܻᐱᐜყ჋मऒƥƇ ܻެყީࡩƄ݊ ‫ޢ‬ዙ၅ေࠦ‫࡯ޢ‬ዙኸ ࡼ ‫ݑ‬ƆƎܻީ‫ ݑ‬። ፋ ࣝީࡦƁฤ ෷ࣝަެࡦƁ  ‫ࡩ࣠ ࣌ޢߦࠦۑ‬Ƅ ،ࠦ‫ࡩޯݘ࡯ޢࡺޢ‬Ƅቿቱܻஊै Ƅūࠦ‫ ࠅޯࡺܻݑޢ‬ଘଓƄቿቱܻஊै Ƅūࢪ ჱஎै ᇴᆝࠦެ‫ހ ࢧߦࠦࡷݘޢ‬Ơ ౰౔ఴ ఓܻᐱᐟࡷ݊ ዙዋ ᇯ ᆝ Ɔ࢖ܻࠧ࣌ ࡼ ‫ݑ‬౰Ʈ౎‫ࠧެܻݹݘ‬Ɩ ࣠ެࡩƄƥū Ɩ ࣠ެࡩƄࠦߦࢢ ‫ ࣀ ݪ‬༨ဂ ว ෺࣠ީޯࡩƄࡼ ࡷƖ ࣠ެࡩƄƄ߃ ࠧ፥ ጿƇ ܻ‫ࡩީޢܻރ‬Ƅơ ƥ߄ ༬ဈ . Ɛ ܻ‫ߥޢ‬ᑃᐽቿቷ‫ߘܻހ‬౰ేၩܻᐱ᏶ܾ‫ހ‬౮ే๻๬ใܻެࠧࠦࢵ ƄƎኆ ቱƄࡣ ༨ீ ࠦߥߦࢧ ࡩƄ‫و‬ ࡣ ޲೘೉Ƈ ܻ‫ࡩީޢܻރ‬Ƅƥ ،‫ࡷެܻ ࢵ ݪ‬Ƨ ƎƄಒ౷‫ކ‬ơ ƥ߄ ၴ ፥ ጿࠦ୆งଘଓƄ ࠦ‫ࡩ࣠ ࢢߦޢ‬Ƅ ࠦ‫ࡩެ࣠ݑޢ‬ƄƇ ܻ‫ࡩީޢܻރ‬Ƅ݊ ࢠ ‫ ۑ‬ơ ܻ޲ࡷ ࿃ ႓ƄƎኆ ቱƄƢܻ‫ݘ‬ᑃᐽࠦࡷܻ‫ࡩݘ‬Ƅࠦ‫ࡩ޲ܻރ࣠ݹ‬Ƅƅ ൎ൉࣐ ࡩƇ ܻ‫ݹܻࢠޢ‬༟༎ƄƟ ܻࣀ ࡷ፥ ጿܻީࡩܻ࣍ 50Ƈ ܻၴ Ǝƌࡼ ࡷࠦެ‫ݑޢ‬ ߄ ‫ࠅޢ‬ Ƅ߄ ၴ ơ ࣠ީߥࡷ Əܻޯያዪ ߄ ‫ࠅޢ‬ Ƅ߄ ၴ ߄ ‫ࠅޢ‬ Ƅ߄ ၴ ‫ ࢵ ݪ‬ƄƎ ‫ ࢵ ݪ‬ƄƎ ߄‫ࠅޢ‬ ߄‫ࠅޢ‬ ‫ ࢵ ݪ‬ƄƎ ‫ ࢵ ݪ‬ƄƎ ߄‫ࠅޢ‬ ơ ࣠ީߥࡷ ߄‫ࠅޢ‬ ߄‫ࠅޢ‬ Əܻޯያዪ ߄‫ࠅޢ‬ ߄‫ࠅޢ‬ ߄‫ࠅޢ‬ ‫ ࢵ ݪ‬ƄƎ ߄‫ࠅޢ‬ ơ ࣠ީߥࡷ ߄‫ࠅޢ‬ ߄‫ࠅޢ‬ ‫ ࢵ ݪ‬ƄƎ Əܻޯያዪ ߄ ‫ࠅޢ‬ Ƅ߄ ၴ ߄‫ࠅޢ‬ ߄ ‫ࠅޢ‬ Ƅ߄ ၴ ‫ ࢵ ݪ‬ƄƎ ߄ ‫ࠅޢ‬ Ƅ߄ ၴ ߄ ‫ࠅޢ‬ Ƅ߄ ၴ ‫ۑ‬2001/2000 ߄‫ࠅޢ‬ ߄‫ࠅޢ‬ ơ ࣠ީߥࡷ Əܻޯያዪ Əܻޯያዪ ‫ ࢵ ݪ‬ƄƎ Əܻޯያዪ ߄‫ࠅޢ‬ ơ ࣠ީߥࡷ ߄‫ࠅޢ‬ ߄‫ࠅޢ‬ ߄ ‫ࠅޢ‬ Ƅ߄ ၴ ߄‫ࠅޢ‬ ߄ ‫ࠅޢ‬ Ƅ߄ ၴ ơ ࣠ީߥࡷ ‫ ࢵ ݪ‬ƄƎ ơ ࣠ީߥࡷ ơ ࣠ީߥࡷ ࠦ‫ࡩެ࣠ݑޢ‬Ƅ)ƥࠦ‫ޢ‬ዙኒ ࡩƄƇ ܻ‫ީޢܻރ‬ᑃᐽว ෺୍࣠งࠦ‫࡯ޢ‬ዙኸ ࣠ࠧ፥ ገܻࢧ ࠁ ଘଓƄơ ƥ߄ எै Ƅƣ ܾ‫ ހ‬،ࢇ ީࢵ ܻࡺ‫ ྲރܻߙޢ‬႖Ƅሯሡሁ ፥ ገܻࢧ ࠁ ଘଓƄơ ƥ߄ எै ƄƌƄ߄ ၮƀ߄ ެ‫ݑ‬ƥ‫ܻۑ‬ᐱᐟࡷƆƌܻߦޯࢵ ଘଓƄƥܻᐱᐢࢵ ƄƎƌࡣ ‫ޢ‬ᐳྜྋ߄ ࢧ ߥ‫ߙߥޢܻހ‬Ƅƥܻ‫ݹ‬ƌ࣠ዙኸ ܻᐳႰ ‫ކ࡯ ࠑޢ‬ƥ :ࠦ୆งଘଓƄ߄ ၮƄ࣠ߥࡩƄƆܻၮƄ୹ ୪ࡼ ࡷਔ ৹ଘଓ ، ࠢ‫ޯ࣠ݹܻކ‬༨ீ ࢪ ࡓ ‫ܻݹݘ‬༨༤ Ƅƥܻ‫ރ‬Ƅ࣠ެ‫ݑ‬ơ ƥ߄ ॺॱ५ƣ ࣠ࡓ ‫ݹ‬ƣ ƀ ܻࠩࡺ‫ߘޢ‬मऒƥƟ ࣠ߦࢧ ࡩƄƥƆ߄ ዙኸ ଘଓƄ‫ ݪ‬ყ჋मऒࡼ ࡷਔ ৹ଘଓ Ƥഃ೨ƍ‫ ݪ‬༬࿻ ໠ ໄ‫ࡷݘ‬Ǝ߄ࢧ ࡷ‫ ࠁ ࣀ ݪ‬ƣ ࣠ࡓ ‫ކ‬ơ ƥ߄ எै Ƅ፥ ጿƆƌƎƄ࣠ࡩƄƇ ܻࡷॅळ‫ݘ‬ଐଌƄ 23 I. II. III. ܸୃกܸ଍ଉŽ–ໝ ແࡦफउ –ࠣ‫ݶ‬ Ɖܸࢤ ެߕକଐƁƢࠣ‫ࡦ ࢽޟܸ ࢲޟ‬ƁƢࠣ‫ޟ‬පඓ ൷Ɓ౭౑ࢲ କଐƁƄ ܸࢲ ƁƋኃ ᒔᐺ። ጏ࢓‫ݕ‬ ࡦƁ࣬ ࣊ Ɓ࢓ߢೕಹኃ ቮƁ೾೥࢓଍ଉƁƄ ƁƋƁ߁ ࢟ ſ ። ፎକଐ߁ ެࢲ କଐƁƢ। ्࢟ ࣝࡦƁŻܸࢤ ߾ ଔଐƁ ƛ ܸߢࢽ ࡦƁ߁ ዖዄ Ž૊ સၫƋࣝެ࠾ኃ ቮƁƊܸެࢲ ଘଐƁ ࣯ ൞൙‫ࡩݘ‬Ƅࡣ ‫ࡩޯ ࠒ ࢧޢ‬Ƅࡣ ޲೘೉཯དྷࡩƄƥ ࠦ‫ ࢢߦޢࠦ ރ࣠ݑޢ‬ƥ፥ ፕ Ƈ ܻ‫ࡩީޢܻރ‬Ƅƣ ܻߙ ، ( ‫ߘ ࡴ ۑ‬Ǝơ ܻ޲ଐଌƄ፟ ፑƄƆƌ࣠‫ހܻࡩݘ‬ƥ ‫ݹ ࢧݘ ݪ‬،Ƣܻၯ Ƈ ܻ‫ܻރ‬୆ก፥ ፕ ơ ܻ޲ଐଌƄ፥ ጿƆƌƎƄ࣠ࡩƄƇ ܻ‫ࡩީޢܻރ‬Ƅƣ ƀࣉ ࠁ ൎ൉‫ރ‬ƥ‫ۑ‬Ƈ ܻ‫ݹܻࢠޢ‬༟༎ƄƟ ܻࣀ ࡷ፥ ጿƅ ൎ൉࣐ ᑃᐽ ࠦ‫ࡩ࣠ ࢢߦޢ‬ƄƇ ܻ‫ީޢܻރ‬ᑃᐽƧ ƎƄಒ౷ޯࡩƄ݊ ‫ݹ‬Ə࣠ޯࡩƄơ ƥ߄ ၴ ࣩ ࡺఱ ఓơ ƥ߄ ၴ ፥ ጿܻࠩ‫ࢠݘ‬ƥࡼ ࡷਔ ৹ଘଓࠢ‫࡯ޢ‬ၮƥܻᐱᐟࡷƆƌܻߦޯࢵ ଘଓƄ ፥ ጘଘଓᇴᆝ ơ ƥ߄ எै Ƅ௪ ௎࣠ࡓ ‫ކ‬যঠ‫ݹ‬ƥ ࠦ‫ࡩެ࣠ݑޢ‬Ƅƥƀ ࠣߣ࢟ ࣝࡦƁƄ ܸ‫ަޟܸހ‬ᑀᐺƤ ƋƁಏ౴ެࡦƁ݇ ‫ݶ‬ƌࣝެࡦƁƞ Ƣ߁ ၱ ‫ ێ‬ƞ Ƣ߁ ၱ Ƈ ƄƎƄಒ౷ޯࡩƄ ƅ ൎ൉࣐ ࡩƄƌ߄ ၮ 6 9 18 8 9 50 ݅ ‫ࡩޯߦ࢖ݹ‬ƄƇ ܻࡷൎ൉ၮ / //// //// //// /// //// //// //// - Ƈ ܻߦࢧ ࡩƄ ƅ ൎ൉࣐ ࡩƄƇ Ƅमअ߄ ߥ‫ކ‬ Əܻޯያዪ Ƅ߄ ၴ ߄ ‫ࠅޢ‬ ߄‫ࠅޢ‬ ơ ࣠ީߥࡷ ‫ ࢵ ݪ‬ƄƎ Ɩ ࣠ࡺ഻വറƄ ࠣ‫ޟ‬ዖ኏ ࡦƁƄ ܸ‫ࡦަޟܸހ‬Ɓฤ ෷ࣝ୊ค  Ƈ ܻߦࢧ ᑃᐽܻߥީ࣍ ܻᐱᐜ᠃୍࣠งযঠ‫ݹ‬ƥ‫ܻۑ‬ᐳ႓ܻ‫ ࡓ ࡼߘޢ‬೘಼཯དྷࡩƄƇ ܻ‫ࡩީޢܻރ‬Ƅᑻ ᑧ‫ކ‬፟ ፑƄ౰ేఴ ఓࠦ‫ޢ‬ዙኒ ࡩƄƇ ܻ‫ࡩީޢܻރ‬Ƅว ෺୍࣠ง ᅵै Ƅ‫ۑۑ‬Ɖ ܻ୓คଘଓƄ ،Ɗ ܻ‫ހ‬ƎଘଓƄ ،Ƈ ܻ‫ީޢݘ‬ଐଌƄ ،ࡣ ၯ ኆ ቱƄ ،ơ ࣐࣠ ࡩƄƥƣ Ə࣠ࡩƄࡣ ޲ࡷ . ፥ ፕ ƥơ ƥƄ߄ ၴ Ɔ߄ ၮ፥ ጿܻᐳႩ ୹ ‫ݑ‬যঠ‫ޢࠦݹ‬ዙኒ ࡩƄƇ ܻ‫ࡩީޢܻރ‬Ƅ ࣊ ‫ ࣀޢ‬ხࡩƄƧ ƎƄಒ౷ޯࡩƄơ ƥ߄ எै Ƅ Ƈ ܻ‫ߙޟ‬ƥƍƧ ƎƄಒ౷ޯࡩƄơ ƥ߄ எै Ƅ ߄ ၮܻࢢ ݊ ࡺࠖ ޯࡷƧ ƎƄಒ౷‫ކ‬ơ ƥ߄ ၴ ࣊ ‫ࡷޯ ࠖࡺ ܻ݊ࠧހ‬Ƨ ƎƄಒ౷‫ކ‬ơ ƥ߄ ၴ .i .ii .iii .iv ࡼ ࡷƣ ܻ‫ީޢ‬ჴࢵ ଘଓƄƆƎܻ෬ෝᛂ Ƅ‫ ࠁ ࣀ ݪ‬Ɔྲ ႓ƀ໠ ໄࣀ ዙ዆ ơ ܻߦ࣍ ƀƌ߄ ၮ ،ࡣ ޲೘೉ࠦ୆งଘଓƄƇ ܻ‫ࡩީޢܻރ‬Ƅ ‫ ۑ‬ơ ܻ޲ࡷ ‫ࡷެܻ ࢵ ݪ‬Ƨ ƎƄಒ౷‫ކ‬ơ ƥ߄ ၴ ፥ ጿƇ ܻ‫ࡩީޢܻރ‬ƄƤ߃ ࠧƓ ୹ ‫ݑ‬ƅ ॅळ࣐ ଐଌƄ‫ࡷܻۑ‬ኆ ቷީࡩƢܻ‫ࡩݘ‬ƄƌƄ߄ ‫ࡩޯݘ‬Ƅ݊ ߘƄƥ 24 ܸୃกܸ଍ଉŽ–ໝ ແࡦफउ –ࠣ‫ݶ‬ Ɖܸࢤ ެߕକଐƁƢࠣ‫ࡦ ࢽޟܸ ࢲޟ‬ƁƢࠣ‫ޟ‬පඓ ൷Ɓ౭౑ࢲ କଐƁƄ ܸࢲ ƁƋኃ ᒔᐺ። ጏ࢓‫ݕ‬ ࡦƁ࣬ ࣊ Ɓ࢓ߢೕಹኃ ቮƁ೾೥࢓଍ଉƁƄ ƁƋƁ߁ ࢟ ſ ƛ ܸߢࢽ ࡦƁ߁ ዖዄ Ž૊ સၫƋࣝެ࠾ኃ ቮƁƊܸެࢲ ଘଐƁ ። ፎକଐ߁ ެࢲ କଐƁƢ। ्࢟ ࣝࡦƁŻܸࢤ ߾ ଔଐƁ 4 3 1 2 4 5 1 2 2 3 5 5 3 4 1 2 3 0 3 2 1 5 4 1 6 3 2 0 2 3 1 1 3 3 0 2 1 4 2 5 2 0 6 1 3 3 4 3 4 5 Ƥܻ‫ރ‬ƌƀƧ ƎƄಒ౷ޯࡩƄơ ƥ߄ எै Ƅ‫ ࠁ ࣀ ݪ‬Ƥൎ൉ၮƄƢܻஉै ƄƇ ܻ‫ࡩީޢܻރ‬ƄƗ ࢖ߦ‫ ࡣރ‬ஊै Ƅ ƃྯ ႐Ž50Ƅ ܸ‫ࡦަޟܸހ‬Ƥ ƋƁಏ౴ެࡦƁƞ Ƣ߁ ஋ॅ Ɓ ‫ ێ‬ƞ Ƣ߁ ၱ ཯ཌࣀ ჱࡩƄƎƄಒ౷ޯࡩƄ ƎƄಒ౷ޯࡩƄ ݅ ‫ࡩޯߦ࢖ݹ‬ƄƇ ܻࡷൎ൉ၮ Ɔྲ ႓ଘଓƄƌƄ࢖ߙƄƌ߄ ၮ 0.18 9 //// //// 1 12 // //// //// 6 / //// 0.08 4 0.20 10 0.14 7 0.24 0.12 0.04 1.00 2 50 //// 0 //// //// 2 3 // //// - // 4 5 Ɩ ࣠ࡺ഻വറƄ 6 ‫و‬،ࠦࢵ ƄƎኆ ቱƄࡣ ༨ீ Ɩ ࣠ࢠ࣠൞ᒑᐽūᇯ ᇍ ࡩƄࡼ ࡷ໠ ໄ‫ࡷݘ‬ž༅໮ ࠦީఱ ఝࠦߙ࢖‫ ࠁޢܻރܻࡷݘ‬ƀ‫୓ ࣐࡯ ݪ‬෺ ཬཉࢽ ხࡦƁƋƁಏ౴ެࡦƁ  ‫ࡩݘޢެࠦۑ‬Ƅࡴ ᄽ஛ ે ઻ၮƇ ƄƎƄಒ౷ޯࡩƄࠦࡺࣀ ߥ‫཯ހ‬ཌࣀ ჱࡩƄƎƄಒ౷ޯࡩƄ‫ ࣀ ݪ‬༨࿻ ƥ‫࡯ޢ ࡣۑ‬ႁ ޯࡩƄࠦ‫࡯ޢ‬ዙኸ ࡣ ᐳྜཾƄ߃ ࠧ ࠦެ‫ࡩݘޢ‬Ƅࡴ ᄽ஛ ƎƄಒ౷ޯࡩƄ ཯ཌࣀ ჱࡩƄƎƄಒ౷ޯࡩƄ ƣ ܻ‫ީޢ‬ჴࢵ ଘଓƄ፥ ጿơ ܻߦ࣍ ଘଓƄƌ߄ ‫ࡩݘ‬ơ ƥ߄ எै Ƅ፥ ጿ཯ཌࣀ ჱࡩƄƎƄಒ౷ޯࡩƄƅ ܻࣀ ࠁ যঠᚻ 2.2ࡴ ߘƎơ ܻ޲ଐଌƄ፥ ጿƥ Ɔྲ ႓ƀ 50ࡼ ࡷƣ ࣠ࡓ ᅚଌƄ Ƅ ܸ‫ߖޜ‬ƢƊƤ ƋƁಏ౴ެࡦƁƞ Ƣ߁ ஋ॅ Ɓ  Ɵ ܻࣀ ࡷƣ ܻႁ ޯࡷƄ፥ ጿܻ‫ܻࡷݘޢ‬ၴ ܻީࡩܻ࣍ ໠ ໄࣀ ସଧଣ ࠦ୆෷ܻᐱᐟࡩƄƇ ܻၴ Ǝኆ ቱƄࠦ‫ࡩޯܻࡩޢ‬ƄযওߥࡩƄ໠ ໄ୍ง ‫ ۑ‬ơ ܻ޲ࡷ ‫ࡷެܻ ࢵ ݪۑ‬Ƨ ƎƄಒ౷‫ކ‬ơ ƥ߄ၴ ፥ ጿƇ ܻၴ Ǝኆ ቱƄƤ߃ ࠧ݊ ‫ݹ‬Ə࣠‫ކ‬ƅ ॅळ࣐ ଐଌƄ ‫ۑ‬ơ ƥଘଓƄƦ ࣠ޯࣀ ଐଌƄƅ ൎ൉࣐ ࡩžܻࢧ ࠁ ଘଓƄ 25 ܸୃกܸ଍ଉŽ–ໝ ແࡦफउ –ࠣ‫ݶ‬ Ɖܸࢤ ެߕକଐƁƢࠣ‫ࡦ ࢽޟܸ ࢲޟ‬ƁƢࠣ‫ޟ‬පඓ ൷Ɓ౭౑ࢲ କଐƁƄ ܸࢲ ƁƋኃ ᒔᐺ። ጏ࢓‫ݕ‬ ࡦƁ࣬ ࣊ Ɓ࢓ߢೕಹኃ ቮƁ೾೥࢓଍ଉƁƄ ƁƋƁ߁ ࢟ ſ ƛ ܸߢࢽ ࡦƁ߁ ዖዄ Ž૊ સၫƋࣝެ࠾ኃ ቮƁƊܸެࢲ ଘଐƁ ። ፎକଐ߁ ެࢲ କଐƁƢ। ्࢟ ࣝࡦƁŻܸࢤ ߾ ଔଐƁ 74 50 76 62 78 88 57 73 80 65 75 95 60 79 83 71 61 89 78 96 79 57 74 67 68 55 65 74 87 78 69 95 76 77 78 65 94 75 85 72 63 71 82 98 75 62 78 66 72 61 ࡣ ஊै Ƅ Ƣ߄ ႀ ޯఱ ఝܻ୐คܾߙ‫ࡷެܻ ࢵ ݪ‬Ƨ ƎƄಒ౷‫ކ‬ơ ƥ߄ၴ ፥ ጿ (2.3 ࡴ ߘƎơ ܻ޲ଐଌƄ፥ ጿƇ ܻ‫ࡩީޢܻރ‬Ƅ݅ ‫ ࡣކߦ࢖ݹ‬ၴ ƀࡼ ࡷ Ƈ ܻ‫ߙޟ‬ƥƍƧ ƎƄಒ౷ޯࡩƄơ ƥ߄ எै Ƅ :ࠦ୆งଘଓƄƇ Ƅ࣐࣠ ॵॱ५ܻߥީ࣍ ơ ƥ߄ எै ƄƄ߃ ࠧযওࡺࢧ ‫ކ‬যঠ‫ݹ‬ƥ ે઻ၮƀࡼ ࡷƇ ܻ‫ࡩީޢܻރ‬Ƅ፥ ጿࠦࡺ‫ߘޢ‬፟ ጕƌƀƊ ࢖࣐ ‫ހ‬ᑑ ᐽƍƥƢܻஉै ƄƇ ܻ‫ީޢܻރ‬ᑃᐽ Range) Ʀ ߄ ଐଌƄƌܻ༬࿻ ƀ ࠣࡷ‫ߕޟ‬፜ ጒƉŽ–ࠣࡷ‫ ૄߕޟ‬સၫŽ ƣ ߁ ଍ଉƁ .I .ࠦࡺ‫ߘޢ‬ Ƈ ܻ‫ࡩߦޟ‬Ƅƌ߄ ၮે ઻ၮƦ ߄ ଐଌƄࠦࡺࣀ ߥ‫ހ‬ᑑ ᐽƍƥࠦ‫ࡩߦޟ‬Ƅơ ࣠࣍ ߄ ၴ ࣠‫ ރ‬. II Ƅ ܸ‫ࡦߣޜ‬ƁƉ߁ ၫૄ સၫƣ ߁ ଍ଉƁ ࠣ‫ࡦߣޜ‬Ɓƞ ࣝ࣊ ܻࢵ ܻࢵ ƀ߄ ࡺޯ‫ݹݘ‬Ƈ ܻ‫ࡩߦޟ‬Ƅƌ߄ ၮਔ ৲߄ ༨ဈ ƥ‫ۑ‬Ƈ ܻ‫ࡩߦޟ‬Ƅƌ߄ ၮଘଓƥƀƌ߄ ༨စ ƣ Ƅਔ ৹ଘଓūࠦ‫ࡩߦޟ‬Ƅơ ࣠࣍ ƌܻ༬࿻ ଗଓƥ ‫ أي‬، 20 ‫ و‬5໠ ໄ‫ހ‬Ɗ ƥƄ౰౔‫ޢ ݟކ‬༨ဂ Ƈ ܻ‫ࡩߦޟ‬ƄƢ߄ ႀ ޯఱ ఠƆƌܻၮƥ‫ۑ‬Ƈ Ƅƌ࢖ߦଐଌƄƌ߄ၮƥ‫ࡩީܻ ࠁ ݟ‬ƄƆ౰౎ၯ ે ઻ၮ ‫ۑ‬20ࡼ ‫ݑ‬ਔ ৲बऒଘଓƥ 5ࡼ ‫ކߥ ࡣݑ‬ଘଓ Struges)ࢪ ࠅ ౰౔ࢵ Ɔ߄ ၮܻߘ፥ ፕ ƥūƇ ܻ‫ࡩߦޟ‬Ƅƌ߄ၮƌܻ༬࿻ ଗଓܻᐱᚑƄ߄ႀ ޯࢵ Ƅࡼ ࡓ ೘಼Ɔ߄ ၮܻߘƠ ܻެࠧƥ K = 1 + 3.3 log N ( Rule :Ơ Ž‫߾ޟݜ‬ Ƈ ܻ‫ࡩߦޟ‬Ƅƌ߄ ၮ= k Ƈ Ƅƌ࢖ߦଐଌƄƌ߄ၮ N ƌ߄ ‫ࡩݘ‬Ƅযঠ‫ݹ‬Ǝܻၩ࣠ࡩ Log 26 ܸୃกܸ଍ଉŽ–ໝ ແࡦफउ –ࠣ‫ݶ‬ Ɖܸࢤ ެߕକଐƁƢࠣ‫ࡦ ࢽޟܸ ࢲޟ‬ƁƢࠣ‫ޟ‬පඓ ൷Ɓ౭౑ࢲ କଐƁƄ ܸࢲ ƁƋኃ ᒔᐺ። ጏ࢓‫ݕ‬ ࡦƁ࣬ ࣊ Ɓ࢓ߢೕಹኃ ቮƁ೾೥࢓଍ଉƁƄ ƁƋƁ߁ ࢟ ſ ƛ ܸߢࢽ ࡦƁ߁ ዖዄ Ž૊ સၫƋࣝެ࠾ኃ ቮƁƊܸެࢲ ଘଐƁ ። ፎକଐ߁ ެࢲ କଐƁƢ। ्࢟ ࣝࡦƁŻܸࢤ ߾ ଔଐƁ Sūࠦ‫ߦޟ‬ᑃᐽે ઻ၮଘଓƄ߄ ஊै Ƅ L ƣ ƀ‫ ࠁޢ ݟ‬Ƥܻ‫ރ‬ƌƄƣ ࣠‫ߥܻރ‬ᑃᐽܻߥީ࣍ I ࠦ‫ࡩߦޟ‬Ƅơ ࣠࣍ ߄ ၴ ࣠‫ރ‬Ꭸ᎜ࡼ ࡷ ‫ۑ‬Ʀ ߄ ଐଌƄ I=L–S/k R ūࠦ‫ߦޟ‬ᑃᐽ፟ ጕƌଛଓƄ߄ ஊै Ƅ R=L–S I = R /k I = R/ 1+3.322 log N ፟ ጕƌƀ፟ ፑƄࠦ‫ࡩߦޟ‬Ƅơ ࣠࣍ ࠦߙܻࢠܺᄣᑑ ᐽƍƥ፟ ፑƥଘଓƄࠦ‫ࡩߦޟ‬Ƅ߄ ၴ ࣠‫୐ܻރ‬คܾߙࠦ‫ࡩߦޟ‬Ƅơ ࣠࣍ ƅ ܻࣀ ࠁ ߄ ‫ۑހݘ‬III Ƈ ܻ‫ࡩީޢܻރ‬Ƅ፥ ጿࠦࡺ‫ߘޢ‬ ‫ۑ‬Ƣܻஉै ƄƇ ܻ‫ࡩީޢܻރ‬Ƅ፥ ጿࠦࡺ‫઻ ેߘޢ‬ၮƀ፟ ፑƄࡣ ࢧ ‫ރ‬ƣ ƀ፟ ፑƄഭ ୨‫୓ܻހ‬ขᇯ ᇍ ఴ చƦ ༇໩ ଘଓƄƇ ܻ‫ࡩߦޟ‬Ƅ߄ ၴ ࣠‫ރ‬ : (3 ࡴ ߘƎơ ܻ޲ଐଌƄ፟ ፑƄƆƌ࣠‫ހܻࡩݘ‬ ( R ) Ʀ ߄ ଐଌƄ߄ ၴ ࣠‫ۑރ‬I R=L–S = 98- 50 = 48 ࠦߘൎ൉‫ࡩݘ‬Ƅࡼ ࡷࠦ‫ࡩߦޟ‬Ƅơ ࣠࣍ ߄ ၴ ࣠‫ۑރ‬Ii I = R /k K = 1 + 3.322 log N K = 1 + 3.322 log (50) K = 1+ 3.322 ( 1.6989 ) K = 1+ 5.6437 = 6.6437 ~ 7 I = /k = 44/7 = 6.2 = 6 5= ( K Ƈ ܻ‫ࡩߦޟ‬Ƅƌ߄ ၮƣ ƄƓ ౰౔ߦ‫ރ‬Ƅ߃ ܻࠧެࡩܻ޲ࡷ፥ ጿࡼ ࡓ ᚻࡩ I = 48/5 = 9.6 ( k = 5) I = 10 50 ࠦ‫ߦޟ‬ᑃᐽ፟ ጕƌଛଓƄ߄ ஊै Ƅ‫ۑ‬Iii 10 ࠦ‫ࡩߦޟ‬Ƅơ ࣠࣍ 60 = 10 + 50 ፟ ፑƥଘଓƄࠦ‫ࡩߦޟ‬Ƅƣ ƍƀ ( 60-50 ፟ ፑƥଘଓƄࠦ‫ࡩߦޟ‬Ƅે ઻ၮࡣ ࢧ ༨စ ƥ 27 ܸୃกܸ଍ଉŽ–ໝ ແࡦफउ –ࠣ‫ݶ‬ Ɖܸࢤ ެߕକଐƁƢࠣ‫ࡦ ࢽޟܸ ࢲޟ‬ƁƢࠣ‫ޟ‬පඓ ൷Ɓ౭౑ࢲ କଐƁƄ ܸࢲ ƁƋኃ ᒔᐺ። ጏ࢓‫ݕ‬ ࡦƁ࣬ ࣊ Ɓ࢓ߢೕಹኃ ቮƁ೾೥࢓଍ଉƁƄ ƁƋƁ߁ ࢟ ſ ƛ ܸߢࢽ ࡦƁ߁ ዖዄ Ž૊ સၫƋࣝެ࠾ኃ ቮƁƊܸެࢲ ଘଐƁ ። ፎକଐ߁ ެࢲ କଐƁƢ। ्࢟ ࣝࡦƁŻܸࢤ ߾ ଔଐƁ ࠦ୆คܻ޲ࡩƄࠦ‫ࡩߦޟ‬Ƅે ઻ၮࡣ ࢧ ༨စ ƥ፟ ፑƥଘଓƄࠦ‫ߦޟ‬ᑃᐽે ઻ၮଘଓƄ߄ ၰ ଘଓƄ፟ ፑƄࠦ‫ࡩߦޟ‬Ƅơ ࣠࣍ ߎ ‫ރ ࢥޢ‬ƥ 70 = 10+ 60 = ( 70-60 ࠦ୆คܻ޲ࡩƄࠦ‫ࡩߦޟ‬Ƅƣ ƍƀ ࠦ‫ࡩޯܻࡩޢ‬Ƅࠦ‫ݓ‬ ‫ހܻࡩ ࢧޢ‬Ƈ ܻ‫ࡩߦޟ‬Ƅ࢖࣏ࠩ ‫ކ‬ƥ Ƅቦ቎ࠧƥ‫ۑۑۑۑۑۑ‬ū 90- 80 ، 80- 70 ، 70- 60 ، 60-50 ፥ ጘଘଓƄᇯ ᇍ ࣃ ࡩܻ‫ހ‬Ƈ ܻ‫ࡩߦޟ‬ƄƧ ƍƧ ƎƄಒ౷ޯࡩƄ݊ ‫ݹ‬Ə࣠ޯࡩƄơ ƥ߄ ၴ ᇯ ᇍ ࢸ ࢖࣏ࠩ ‫ݹ‬ƥ Żܸࢤ ߾ କଐƁƜ ܸࢽ ࡴ። ጼܸަࡦܸ࣊ 50Ƅ ܸၱ Ƌኃ ቮƄ ܸ‫ࡦߣޜ‬ƁƤ ƊƤ ƋƁಏ౴ެࡦƁ݇ ‫ݶ‬ƌࣝެࡦƁƞ Ƣ߁ ၱ ‫ ێ‬ƞ Ƣ߁ ၱ ƎƄಒ౷ޯࡩƄ 4 13 ݅ ‫ࡩޯߦ࢖ݹ‬ƄƇ ܻࡷൎ൉ၮ //// /// //// //// 21 / //// //// //// //// 5 //// 7 50 // //// Ƈ ܻ‫ࡩߦޟ‬Ƅ 60 – 50 70 - 60 80- 70 90 - 80 - 90 100 Ɩ ࣠ࡺ഻വറƄ Cumulative ࣯ ዙኤ Ƅ౰౔ࡩƄƎƄಒ౷ޯࡩƄƥ [Relative Frequency] ཬཉࢽ ხࡦƁƋƁಏ౴ެࡦƁ  [ frequency ] .Ƈ ƄƎƄಒ౷ޯࡩƄƖ ࣠ዙಣಝે ઻ၮƨܻࡷ࣠ࣀ ߥࡷƤƎƄಒ౷‫ކ‬Ƨ ƥܻఱ ఓमअ߄ ߥޯࡩ཯ཌࣀ ჱࡩƄƎƄಒ౷ޯࡩƄ  ౰౗ࡁ ଛଓƄે ઻ၮमअ߄ ߥޯࡩƄᑑ ᐽƍࡴ ಛಗ௪ ௎ኅ ቱƄࠦެ‫ࡩݘޢ‬Ƅྲ Ⴋ ܻެ‫ݑ‬ƌ߄ ၮƧ ƥܻఱ ఓमअ߄ ߥޯࡩ࣯ ዙኤ Ƅ౰౔ࡩƄƎƄಒ౷ޯࡩƄ  Ƈ ƄƎƄಒ౷ޯࡩƄƖ ࣠ዙಣಝે ઻ၮƨܻࡷ࣠ࣀ ߥࡷ࣯ ዙኤ Ƅ౰౔ࡩƄƤƎƄಒ౷‫ކ‬Ƨ ƥܻఱ ఓमअ߄ߥޯࡩ཯ཌࣀ ჱࡩƄ࣯ ዙኤ Ƅ౰౔ࡩƄƎƄಒ౷ޯࡩƄ  ፥ ጘଘଓᇴᆝࠦ‫ޢ‬ዙኤ Ƅ౰౔ࡩƄƥࠦ‫ޢ‬ხࣀ ჱࡩƄƇ ƄƎƄಒ౷ޯࡩƄƅ ܻࣀ ჴၰ Ƅࡼ ࡓ ೘಼ 3 ࡴ ߘƎơ ܻ޲ଐଌƄ፟ ፑƄƆƌ࣠‫ހܻࡩݘ‬ƥ ࠣ‫ޟ‬ዖኡ Ɓ౭౑ࡦƁƢࠣ‫ޟ‬ძࢽ ხࡦƁƄ ƁƋƁಏ౴ެࡦƁ ‫ ێ‬ƞ Ƣ߁ ၱ 28 ܸୃกܸ଍ଉŽ–ໝ ແࡦफउ –ࠣ‫ݶ‬ Ɖܸࢤ ެߕକଐƁƢࠣ‫ࡦ ࢽޟܸ ࢲޟ‬ƁƢࠣ‫ޟ‬පඓ ൷Ɓ౭౑ࢲ କଐƁƄ ܸࢲ ƁƋኃ ᒔᐺ። ጏ࢓‫ݕ‬ ࡦƁ࣬ ࣊ Ɓ࢓ߢೕಹኃ ቮƁ೾೥࢓଍ଉƁƄ ƁƋƁ߁ ࢟ ſ ። ፎକଐ߁ ެࢲ କଐƁƢ। ्࢟ ࣝࡦƁŻܸࢤ ߾ ଔଐƁ ƛ ܸߢࢽ ࡦƁ߁ ዖዄ Ž૊ સၫƋࣝެ࠾ኃ ቮƁƊܸެࢲ ଘଐƁ ࣯ ዙኤ Ƅ౰౔ࡩƄƎƄಒ౷ޯࡩƄ ཯ཌࣀ ჱࡩƄ ཯ཌࣀ ჱࡩƄƎಒ౷ޯࡩƄ Ƨ ࣠‫ޟ‬ଐଌƄ 8 ƎƄಒ౷ޯࡩƄ 8 34 26 90 14 76 ཯ཌࣀ ჱࡩƄ ‫ ۑ‬ 42 100 4 Ƈ ܻ‫ࡩߦޟ‬Ƅ 60 – 50 13 70 – 60 ‫ ۑ‬ 7 90 – 80 ‫ ۑ‬ 100 (f) ‫ ۑ‬ ‫ ۑ‬ 10 - ƎƄಒ౷ޯࡩƄ ‫ ۑ‬ 21 5 50 80- 70 100 – 90 Ɩ ࣠ࡺ഻വറƄ ࣇ‫ݽ‬ ܸಘಔƁƢ߁ ၫܸࢤ ࡦƁ݇ ࡷࠓ ެ଍ଉƁƋƁಏ౴ެࡦƁ  ࠦ‫ߙޟ‬ᇯ ᆝ ࡣ ‫ݑ ࡼߘޢࡺࠦࡷݘޢެࠦࡷߥܻހ‬ਔ ৲बऒƥƄࡣ ߥ‫཯ކ‬དྷࡩƄযওߥࡩƄਔ ৲߄ ඳඣ ൴Ƈ ܻ‫ݹݘ‬Ə࣠ޯࡩƄࡼ ࡷƖ ࣠ެࡩƄƄ߃ ࠧযঠᐱᏬ ܻዙ዗໠ ໄၮ࣠‫ޯ ࠖࡺݘࠦރ‬ଐଌƄࠦ‫ݹ‬ƎƄಒ౷ޯࡩƄƇ ܻ‫ݹݘ‬Ə࣠ޯࡩƄƣ ࣠ࡓ ‫ކ‬ƥ݊ ‫ݹ‬Ə࣠ޯࡩƄƇ ܻ‫ࡷ ࡼߙޟ‬ ߁ၫܸࢤ ࡦƁ݇ ࡷࠓ ެ଍ଉƁƋƁಏ౴ެࡦƁ  ્ ઻ࢢ ଘଓƄơ ƥ߄ எै ƄƇ ƄƎƄಒ౷‫ކ‬Ꭸ፫ ƄमऒƥƄ݊ ‫ޢ‬ዙ၅ေơ ൎ൉ၯ ࡼ ࡷ߄ ၮܻࢧ ࡩƄ݊ ࡺࠖ ޯଐଌƄƎƄಒ౷ޯࡩƄે ઻ၮơ ࣠ࢧ ஊै Ƅࡼ ࡓ ೘಼ ƎƄಒ౷ޯࡁ Ƈ ƄƎƄಒ౷ޯࡩƄƖ ࣠ዙಣಝે઻ၮࡣ ࢧ ༨စ ƣ Ƅ፟ ፑƄࠢެࡷƆ౰ేၯ ଘଓƄࠦ‫ࡩߦޟ‬ƄƎƄಒ౷୓กžܻᐱᐢ᠍Ƅƥ፟ ፑƥଘଓƄࠦ‫ࡩߦޟ‬ƄƎƄಒ౷୓กžਔ ৹ ߄ ၮܻࢢ ݊ ࡺࠖ ޯࡷ ࣇ ‫ݽ‬ ܸಘಔƁ݇ ࡷࠓ ެ଍ଉƁƋƁಏ౴ެࡦƁ  ࡼ ࡷ્ ઻ࢢ ଘଓƄơ ƥ߄ எै ƄƇ ƄƎƄಒ౷‫ކ‬Ɗ ࢖࣍ ơ ൎ൉ၯ ࡼ ࡷơ ƏܻެࡩƄ݊ ࡺࠖ ޯଐଌƄƎƄಒ౷ޯࡩƄે ઻ၮơ ࣠ࢧ ஊै Ƅࡼ ࡓ ೘಼ ࡣ ࢧ ༨စ ƣ Ƅ፟ ፑƄࠢެࡷƆ౰ేၯ ଘଓƄࠦ‫ࡩߦޟ‬ƄƎƄಒ౷୓กžܻᐱᐢ᠍Ƅƥ፟ ፑƥଘଓƄࠦ‫ࡩߦޟ‬ƄƎƄಒ౷୓กžਔ ৹፥ ፑƄ࣠ޯࡩƄે ઻ၮƇ ƄƎƄಒ౷ޯࡩƄƖ ࣠ዙಣಝ ‫ߦޟࠦۑ‬ᑃᐽơ Əܻ‫ࡷޯ ࠖࡺ ݊ރ‬ƎƄಒ౷ޯࡁ ౰ేၯ ଘଓƄƎƄಒ౷ޯࡩƄે ઻ၮ ࣠ࠒ ެࡩƄે ઻ၮ࣊ ‫ܻހ‬ಛಗƄƥ߄ ၮܻࢧ ࡩƄ݊ ࡺޯ഻വറƄƎƄಒ౷ޯࡩƄƅ ܻࣀ ჴၰ Ƅࡼ ࡓ ೘಼ ‫ߘ ࡴ ۑ‬Ǝơ ƥ߄ எै Ƅ፟ ፑƄƆƌ࣠‫ހܻࡩݘ‬ƥ ‫ۑ‬፥ ጘଘଓƄ ࣊ ‫ܻހ‬ಛಗƄ݊ ࡺࠖ ޯଐଌƄƎƄಒ౷ޯࡩƄ 29 Ƈ ܻ‫ࡩߦޟ‬ƄƥƍƧ ƎƄಒ౷ޯࡩƄ݊ ‫ݹ‬Ə࣠ޯࡩƄ( 2.5 ơ ƥ߄ ၴ ߄ၮܻࢧ ࡩƄ݊ ࡺࠖ ޯଐଌƄƎƄಒ౷ޯࡩƄ (f ƎƄಒ౷ޯࡩƄ Ƈ ܻ‫ࡩߦޟ‬Ƅ ܸୃกܸ଍ଉŽ–ໝ ແࡦफउ –ࠣ‫ݶ‬ Ɖܸࢤ ެߕକଐƁƢࠣ‫ࡦ ࢽޟܸ ࢲޟ‬ƁƢࠣ‫ޟ‬පඓ ൷Ɓ౭౑ࢲ କଐƁƄ ܸࢲ ƁƋኃ ᒔᐺ። ጏ࢓‫ݕ‬ ࡦƁ࣬ ࣊ Ɓ࢓ߢೕಹኃ ቮƁ೾೥࢓଍ଉƁƄ ƁƋƁ߁ ࢟ ſ ƛ ܸߢࢽ ࡦƁ߁ ዖዄ Ž૊ સၫƋࣝެ࠾ኃ ቮƁƊܸެࢲ ଘଐƁ 50 ። ፎକଐ߁ ެࢲ କଐƁƢ। ्࢟ ࣝࡦƁŻܸࢤ ߾ ଔଐƁ 4 4 60 – 50 21 80- 70 46 17 13 12 45 7 33 5 - 38 50 - 5 50 70 - 60 90 - 80 100 - 90 Ɩ ࣠ࡺ഻വറƄ ࠣୃ෴ܸࢤ ߾ କଐƁƄ ܸ‫ࡦަޟܸހ‬ƁƐ ୶ ‫ ݎ‬.2 ܻ‫ࡩޢ‬ƥ߄ ၴ ܻᐳႩ ୹ ‫ݑ‬ƥ ࠦ୆෷ܻࢧ ࠁ ଘଓƄ Ƈ ܻ‫ࡩީޢܻރ‬Ƅ ࢝ ‫ ކ࡯ ࠑޢ‬ƥ যও࣏ ୐ง Ɵ ࢖࣍ Ɠ Ƅ࢖‫ ࢵޯݘ‬Ƅ Ꭸ᎙ Ƣ߄ ߥ‫ߙޢࡺܻ ކ‬ ࠦ‫ݹ‬ƎƄಒ౷ޯࡩƄܻᐱᏹܻ‫ݹݘ‬Ə࣠‫ކ‬ƥƇ ܻ‫ࡩީޢܻރ‬Ƅࡼ ‫ࠦݑ‬༨༤ Ƅƥƥቿቷࡷܻࢸ ƆƎ࣠ࢢ ࣯ ࣐ ‫཯ކݘ‬དྷࡩƄࠦ‫ݹ‬ƎƄಒ౷ޯࡩƄơ ƥƄ߄ எै ƄƢƄ߄ ႀ ޯࢵ ܻ‫ހ‬ ࡼ ‫ ࣊ݑ‬ఱ చƄƥၒল ƥƀƆƎ࣠ࢢ ࣯ ࣐ ‫ܻݹݘ‬୆คܻ୆กࠦ‫ݹ‬ƎƄಒ౷ޯࡩƄơ ƥƄ߄ எै Ƅࢇ ‫ݑ ࡼ ࣍࢖ݹ‬Ƈ ܻ‫ࡩީޢܻރ‬ƄƓ ୹‫ ࡓ ࡼݑ‬ᚻࡩ‫ۑ‬ ࠦ‫ܻݹ‬ၮኆ ቱƄ ፥ ጿƆ౰౗ࡓ ‫ހ‬Ƣ߄ ႀ ޯఱ ఠƥࠦࡷܻ‫ݘ‬ᑃᐽࠦ୆กƍܻၴ ౰౗ࡁ ƀ [Graphs] [ ࠦ୆คܻ‫ࡩީޢ‬ƄƢ࣠ࢵ ༟༎Ƅƥ‫ ࢵࠦۑ‬ƥƎ߄ ଐଌƄƆ࢖ܻ࣏ࠧ ࡩƄ ࢖࣏ ‫ ࠅࠩࠦރ‬ƥࡼ ࡷƥūƇ ܻ‫ީޢܻރ‬ᑃᐽࠦީࣀ ჱࡩܻ‫ހ‬ཀྵཏ‫ݘ‬ଐଌƄ፟ ፑƄܻ‫ޟ‬ყࢸ ߎ ‫ކ ࢥޢ‬ଘଓࠦ୆คܻ‫ࡩީޢ‬ƄƢ࣠ࢵ ༟༎Ƅƥ‫ۑ‬ƣ ൎ൉ၮଘଓƄƥ ࡼ ࡷƄ౰ే޲ࡁ Ƣ߄ ႀ ޯఱ ఠܻᐱ᏶ƀࡼ ࡷࡴ ‫ ݌‬༟༎ܻ‫ۑހ‬፥ ገܻࢧ ࠁ ଘଓƄࡣ ‫࡯ޢ‬ႁ ޯࡩƄ፥ ጿƆ߄ ‫ ࣀ ݝࡷߦޢ‬ყࡩ໠ ໄ޲ࠁ ܻީࡩƄƥ໠ ໄ୆෷ܻࢧ ࠁ ଘଓƄ .໠ ໄ޲ࠁ ܻީࡩƄࡣ ީߘ ࠣୃกܸ‫ࡦަޟ‬ƁƟࣝࢲ ༜་ƁŻܸఱ చଘଐࠣ‫ ࢲܸ ࢲޟ‬କଐƁ߁ ၫƁࣝߢࡦƁ  ᄎ ႓༟༎ƄƓ ୹ ‫ ݌‬ƥࠦ‫ޢ‬ዙ዗ƀࢪ ࡓ ‫ݹݘ‬Ƅྲ Ⴐ ޯ༧ீ ܻ‫ރ‬Ƅ࣠ެ‫ݑ‬፥ ጕܻ‫ࡩީޢ‬Ƅᄎ ႓༟༎ƄƧ ࣠ޯ༨࿻ ƣ ƀ‫ ݪ‬༬࿻ ƣ Ƅ࣠ެ‫ࡩݘ‬Ƅ ƥƄ፥ ገܻ୐ชࢇ ‫ߘޢ‬ƌƥ‫ࡷެܻ ࢵ ݪ‬ᄎ ႓ƎƐ ܻ‫ߥޢ‬೘ೃ፥ ጕܻ‫ࡩީޢ‬Ƅᇯ ᇍ ࣃ ࡩƄᄎ ႓मअƣ ƀ‫ ݪ‬༬࿻ ᄎ ႓༟༎ƄƐ ܻ‫ࡷߥޢ‬ .i .ii (30 ، 20 ، 10 ‫ أو‬15 ، 10 ، 5ൎ൉޲ࡷ Ƨ ྲ Ⴁ‫ݑ‬ƥƀ࿃ ႓ܻዙ዆ ࠦ‫ݹ ࢥܻ ࠁޢ‬ƄƇ ܻ࣏ ࠁ ൎ൉ࡷƞ ܻࢥ ‫ކ‬ƣ ƀ‫ ݪ‬༬࿻ ࠦ‫ܻ ࢸޢ‬ஊै Ƅ፥ ጿ ፥ ጕܻ‫ࡩީޢ‬Ƅᄎ ႓༟༎Ƅࡣ ߦࢵ ƀࠦ‫ܻ ࢸޢ‬ஊै Ƅ Ɛ ܻ‫ࡩߥޢ‬ƄƇ Ƅ߄ ၰ ࣠‫ ࢢߦࠢހ‬ƥ‫ ݪ‬༬࿻ १ ॏߙଘଓƄƥ࿃ ႓Ƅ༟༎Ƅƌ࣠ࡺ‫ࡩݘ‬Ƅ१ ॏߙଘଓƄƥ࿃ ႓Ƅ༟༎Ƅ፥ ጛƄ߄ ၰ ଘଓƄ ࠣୃกܸ‫ࡦަޟ‬ƁƟࣝࢲ ༜་ƁƓ Ɓࣝ‫ހ‬Ž. 2.2.1 .iii .iv [ Bars] ƃ߁ ዖኵ କଐƁ  ౰ేၩƇ Ƅ౰ే‫ݓ‬ ޯ൞ᒑᐽƇ ܻ‫ࡩީޢܻރ‬ƄƓ ࢖‫ࡩݘ‬Ɔ߄ ዙኸ ଘଓƄƢ߄ ႀ ޯఱ ఠƥ‫ܻࡩީܻۑ‬ၩࠦ୆คܻ‫ࡩީޢ‬ƄƢ࣠ࢵ ༟༎Ƅ፥ ጿƆ߄ ዙኸ ଘଓƄƢ߄ ႀ ޯఱ ఠ ፥ ጘଘଓƄƆܻၮƄ୹ ୪‫ ݪ‬༬࿻ ፥ ጕܻ‫ࡩީޢ‬ƄƓ ࢖‫ݘ‬ᑃᐽƆ߄ ዙኸ ଘଓƄƢƄ߄ ႀ ޯࢵ Ƅ߄ ެ‫ݑ‬ƥ‫ࡩެ࣠ݑޢࠦۑ‬Ƅƥƀቿቷࢧ ޯଐଌƄ ܻ୓กܻ‫މ‬ƣ ࣠ࡓ ‫ݹ‬ƣ Ƅ‫ ݪ‬༬࿻ Ɔ߄ ዙኸ ଘଓƄƓ ୹ ‫ ݑ‬.i Ɔ߄ ၰ ࣠ࡷƥࠦ୓กܻ‫މ‬ƣ ࣠ࡓ ‫ކ‬ƣ Ƅ‫ ݪ‬༬࿻ Ɔ߄ ዙኸ ଘଓƄ໠ ໄ‫ ࣀܻߙࠦހ‬ଐଌƄ ࠦ‫ ࢵޢ‬ƄƎƥƀࠦ‫ߙߥޢ‬Ƅƣ ࣠ࡓ ‫ކ‬ƣ Ƅࡼ ࡓ ೘಼Ɔ߄ ዙኸ ଘଓƄ 30 .ii .iii ܸୃกܸ଍ଉŽ–ໝ ແࡦफउ –ࠣ‫ݶ‬ Ɖܸࢤ ެߕକଐƁƢࠣ‫ࡦ ࢽޟܸ ࢲޟ‬ƁƢࠣ‫ޟ‬පඓ ൷Ɓ౭౑ࢲ କଐƁƄ ܸࢲ ƁƋኃ ᒔᐺ። ጏ࢓‫ݕ‬ ࡦƁ࣬ ࣊ Ɓ࢓ߢೕಹኃ ቮƁ೾೥࢓଍ଉƁƄ ƁƋƁ߁ ࢟ ſ ። ፎକଐ߁ ެࢲ କଐƁƢ। ्࢟ ࣝࡦƁŻܸࢤ ߾ ଔଐƁ ƛ ܸߢࢽ ࡦƁ߁ ዖዄ Ž૊ સၫƋࣝެ࠾ኃ ቮƁƊܸެࢲ ଘଐƁ ( ƄƎܻ‫ ࠀޯޢ‬Ƅ ƌ࣠ࡺ‫ࡩݘ‬ƄƐ ƀ༟༎ࡣ ‫ߥܻހ‬ଐଌƄࡴ ߘ༟༎Ƅࠦ‫ૻܻހ‬૓ࡼ ࡓ ೘಼Ɔ߄ ዙኸ ଘଓƄžܻఴ ఝƀ߄ ެ‫ ݑ‬.iv Ɔ߄ ዙኸ ଘଓƄࡼ ࡷࠦ‫ࡩޯܻࡩޢ‬ƄƖ Ƅ࣠‫ރ‬ଘଓƄ߄ ၴ ࣠‫ކ‬ƥ ࣐ࠦ ‫ ࣀޢ‬ხࡩƄƆ߄ ዙኸ ଘଓƄ  ࠦߥࢢ ൎ൉ޯଐଌƄƆ߄ ዙኸ ଘଓƄ  Ɔƀ࢔഻വറƄƆ߄ ዙኸ ଘଓƄ  [Histogram] ƎƄಒ౷ޯࡩƄƉ Ǝ߄ ଐଌƄ  [ Polygon ] Ƨ ƎƄಒ౷ޯࡩƄ݊ ࡯ࢥ ଐଌƄ  [ frequency curve ] Ƨ ƎƄಒ౷ޯࡩƄཀྵཏࠒ ެଐଌƄ  ࣊ ‫ܻހ‬ಛಗƄƥ߄ ၮܻࢧ ࡩƄƧ ƎƄಒ౷ޯࡩƄ݊ ࡺࠖ ޯଐଌƄཀྵཏࠒ ެଐଌƄ  ࣐ࠦ ‫ ࣀޢ‬ხࡩƄƆ߄ ዙኸ ଘଓƄƢƄ߄ ႀ ޯࢵ ܻ‫ܻހ‬୆คܻ୆กࠦ‫ࡩޯܻࡩޢ‬ƄƇ ܻ‫ࡩީޢܻރ‬ƄƓ ୹ ‫ݑ‬ƀ ‫ ێ‬ƞ ܸޯࡴ E D 800 c 600 B 450 A 350 Ƈ ᇴᆝྲ ႡࡩƄ Ɔ࢔ࠩࠅ ƀƌ߄ ၮ 200 ࠦၮܻީଐଌƄƣ ࣠‫ࡩޯ࡯ߦ࢔ݹ‬Ƅ ࠣၫܸަ଍ଉƁƠ ࣝ‫ࡦެ࡬ߣ࢑ݶ‬Ɓƃ࢑ࠦࠂ Ž ‫ ێ‬ᇬ ᇊ ࢵ ‫ﻋدد اﺟﮭزة اﻟﺗﻠﻔزﯾون اﻟﻣﺑﺎﻋﮫ‬ 800 ‫ﻋ‬ ‫د‬ ‫د‬ ‫ا‬ ‫ﺟ‬ ‫ﮭ‬ ‫ز‬ ‫ة‬ ‫ا‬ ‫ﻟ‬ ‫ﻧ‬ ‫ﻠ‬ ‫ف‬ 800 700 600 600 450 500 350 400 300 200 200 100 0 A B C D E ‫اﻟﺷرﻛﺎت‬ 31 ܸୃกܸ଍ଉŽˍ ໝ ແࡦफउˍ ࠣ‫ݶ‬ Ɖܸࢤ ެߕକଐƁƢࠣ‫ࡦ ࢽޟܸ ࢲޟ‬ƁƢࠣ‫ޟ‬පඓ ൷Ɓ౭౑ࢲ କଐƁƄ ܸࢲ ƁƋኃ ᒔᐺ። ጏ࢓‫ݕ‬ ࡦƁ࣬ ࣊ Ɓ࢓ߢೕಹኃ ቮƁ೾೥࢓଍ଉƁƄ ƁƋƁ߁ ࢟ ſ ። ፎକଐ߁ ެࢲ କଐƁƢ। ्࢟ ࣝࡦƁŻܸࢤ ߾ ଔଐƁ ƛ ܸߢࢽ ࡦƁ߁ ዖዄ Ž૊ સၫƋࣝެ࠾ኃ ቮƁƊܸެࢲ ଘଐƁ -2000ƟƁࣝ‫ ݎ‬ൎᑆᐺƄ ܸߣ࣌ ީ଍ଉƁቝሹྯ ႞ࡦƇ ܸ‫ݽ‬ ƋକଐƁƢƄ ܸ‫ݕ‬ ‫ަޟ‬଍ଉƁໝ ແ୊෷። ፎܸެࡦƁƞ Ƣ߁ ஋ॅ Ɓ ‫ ێ‬ƞ ܸޯࡴ ( ƞ ܸ‫ݶ‬Ƌߋ ࡦƁ Ƅ ܸ‫ࡦަޟܸހ‬ƁƐ ࢓‫ݕ‬ ࡦࠣߢ࢟ ോെެ଍ଉƁƃ߁ ዖኵ କଐƁƟ߁ ၽ ެࢲ Ž‫ێ‬2002 ፥ ጿܻࢧ ࡩƄၒ࿙ ༟༎Ƅ 15 ્ ᇣᇍ ࡩƄၒ࿙ ༟༎Ƅ Ƈ ܻ‫ީޢݘ‬ଐଌƄ ƢƄ࣠‫ݑ‬ଘଓƄ 50 130 2001 40 20 30 70 100 2000 160 2002 Ƅ ܸߣ࣌ ީ଍ଉƁቝሹྯ ႓Ƈ ܸ‫ݽ‬ ƋƁƢƄ ܸ‫ݕ‬ ‫ ێ ࡴަޟ‬ᇬ ᇊ ࢵ ‫ﻣﺑﯾﻌﺎت وارﺑﺎح ﺷرﻛﺔ اﻟﻣﻧظﻔﺎت‬ 180 160 140 120 100 80 60 40 20 0 ‫اﻟﻣﺑﯾﻌﺎت‬ ‫اﻟرﺑﺢ اﻟﻛﻠﻲ‬ ‫اﻟرﺑﺢ اﻟﺻﺎﻓﻲ‬ 2000 2001 2002 ‫اﻟﺳﻧوات‬ ፥ ጿܻީࡩܻ࣍ 15Ƈ ܻၴ Ǝƌၒল ࣠‫཯ކ‬དྷࡩƄࠦ‫ࡩޯܻࡩޢ‬ƄƇ ܻ‫ީޢܻރ‬ᑃᐽƆƀ࢔഻വറƄƆ߄ ዙኸ ଘଓƄƢ߄ ႀ ޯࢵ ƀ ‫ ۑ‬ơ ܻ޲ࡷ žܻ‫ݹ‬౮ేߙ 60 52 70 Ƨ ࣠‫މܻࡩ ݟމܻރ‬ƥ Ƨ ࣠‫މܻރ‬፥ ጕܻ‫މ‬ƥƧ ࣠‫މܻރ‬ơ ƥଘଓƄߎ ࢧ ᑃᐽࠦ୆෷ܻᐱᐟࡩƄƇ ܻ‫ܻރ‬ႁ ޯࡷଘଓƄ žܻ‫ݹࠦ ࡁޢࡺޢ‬౮ే࡯༬စ ଘଓƄࠦ‫ݓ‬ ᑃᐽƄ 57 75 76 70 63 65 Ƈ ܻ‫ݹܻࢠޢ‬༟༎Ƅ ࠦ୆ก࢖‫ࡩݘ‬Ƅࠦ‫ݓ‬ ᑃᐽƄ ߎ ࢧ ࡩƄ 90 65 Ƨ ࣠‫މܻރ‬ơ ƥƀ 95 62 Ƨ ࣠‫މܻࡩ ݟމܻރ‬ 84 70 Ƨ ࣠‫މܻރ‬፥ ጕܻ‫މ‬ ። ጒܸ‫ކ‬ƢƤ ࣝ‫ކܸހ‬ƞ ƢକଐƁߋ ࢤ ᑀᐺࠣୃ෴ܸᐮᐜࡦƁƄ ܸ‫ܸހ‬ၾ ެࡴକଐƁ። ጼܸަࡦܸ࣊ 15Ƅ ܸၱ ƋƉ၏য ࣝ‫ ރ‬: ‫ ێ‬ᇬ ᇊ ࢵ Ƥ ࣝ‫ކܸࡦݜކܸހ‬ƢƤ ࣝ‫ކܸހ‬ 32 ܸୃกܸ଍ଉŽ–ໝ ແࡦफउ –ࠣ‫ݶ‬ Ɖܸࢤ ެߕକଐƁƢࠣ‫ࡦ ࢽޟܸ ࢲޟ‬ƁƢࠣ‫ޟ‬පඓ ൷Ɓ౭౑ࢲ କଐƁƄ ܸࢲ ƁƋኃ ᒔᐺ። ጏ࢓‫ݕ‬ ࡦƁ࣬ ࣊ Ɓ࢓ߢೕಹኃ ቮƁ೾೥࢓଍ଉƁƄ ƁƋƁ߁ ࢟ ſ ። ፎକଐ߁ ެࢲ କଐƁƢ। ्࢟ ࣝࡦƁŻܸࢤ ߾ ଔଐƁ ƛ ܸߢࢽ ࡦƁ߁ ዖዄ Ž૊ સၫƋࣝެ࠾ኃ ቮƁƊܸެࢲ ଘଐƁ 400 350 300 ‫ﻓﯾزﯾﺎء‬ 250 ‫ﻛﯾﻣﯾﺎء‬ 200 ‫اﻟﻠﻐﺔ اﻻﻧﺟﻠﯾزﯾﺔ‬ 150 ‫اﻟرﯾﺎﺿﯾﺎت‬ 100 ‫اﻟﻠﻐﺔ اﻟﻌرﺑﯾﺔ‬ 50 0 ‫أول ﺛﺄﻧوي‬ ‫ﺛﺎﻧﻲ ﺛﺎﻧوي‬ ‫ﺛﺎﻟث ﺛﺎﻧوي‬ ƢƄ࣠‫ݑ‬൑ᑉᐽ ƎଘଓƥƌƎܻ‫ ࡷ࡯ޢ‬ƣ Ƅኆ ቷީࡩƄ߄ ၰ ଘଓƇ ܻၮ࣠ߙ߄ ଐଌƄƣ Ƅ౮ేࡷ໠ ໄ୍෺፥ ፑܻޯࡩƄơ ƥ߄ எै Ƅ ‫ ۑ‬ơ ܻ޲ࡷ ௪ ௎߄ ࡷ Ɔ߄ ዙኸ ଘଓƄ Ƈ ܻ‫ࡩީޢܻރ‬ƄƓ ࢖‫ެܻ ࢵ ݪࡩݘ‬ଐଌƄᇯ ᇍ ࣃ ࡩƄƢ߄ ႀ ޯࢵ ƀ‫ ۑ‬ 23 ௪ ோƄƌ߄ ‫ ࢢޢ‬Ǝ ƢƄ࣠‫ݑ‬ଘଓƄ 46 2006 16 26 51 2005 47 2007 ܸࡴኃ ቴަࡦƄ ܸၫࣝߖ߁ ଍ଉƁƠ Ɓ౫ౄࡴ ‫ ێ‬ᇬ ᇊ ࢵ 60 50 40 30 20 10 0 ‫رﺻﯾد داﺋن‬ ‫ﻣدﯾن‬ 2005 2006 2007 Titre de l'axe 33 ܸୃกܸ଍ଉŽˍ ໝ ແࡦफउˍ ࠣ‫ݶ‬ Ɖܸࢤ ެߕକଐƁƢࠣ‫ࡦ ࢽޟܸ ࢲޟ‬ƁƢࠣ‫ޟ‬පඓ ൷Ɓ౭౑ࢲ କଐƁƄ ܸࢲ ƁƋኃ ᒔᐺ። ጏ࢓‫ݕ‬ ࡦƁ࣬ ࣊ Ɓ࢓ߢೕಹኃ ቮƁ೾೥࢓଍ଉƁƄ ƁƋƁ߁ ࢟ ſ ። ፎକଐ߁ ެࢲ କଐƁƢ। ्࢟ ࣝࡦƁŻܸࢤ ߾ ଔଐƁ ƛ ܸߢࢽ ࡦƁ߁ ዖዄ Ž૊ સၫƋࣝެ࠾ኃ ቮƁƊܸެࢲ ଘଐƁ Ɔྲ ႓ଘଓƄࠦ୆คƄ౮ేଐଌࠦ‫ࡩޯܻࡩޢ‬ƄƇ ܻ‫ࡩީޢܻރ‬ƄƓ ࢖‫ ࡩݘ‬pie diagram मंƄƥኆ ቱƄƢ߄ႀ ޯࢵ ƀ ‫ ۑ‬ơ ܻ޲ࡷ Bྲ ႓ଘଓƄ AƆྲ ႓ଘଓƄ 420 250 ᄅ ྅ൎ൉ଐଌƄ 60 যও࡯‫ࡩޯݘ‬Ƅ 500 320 200 160 1600 ࠦ୆คƄ౮ేଐଌƄƌ࣠୐ก 400 žƄ߃ ‫ݓ‬ ࡩƄ 150 Ǝܻ༬࿻ ଘଓƄ 140 1000 Ǝܻၯ ƌଘଓƄ݊ ࡷ Ʀ ༇໩ ƄƇ ܻߥߦ‫ރ‬ ፥ ፑܻዙዋ ଘଓƄ ( A ƃྯ ႐କଐƁࠣୃกƁ౫ౄࡴ ‫ ێ‬ᇬ ᇊ ࢵ A ‫اﻻﺳرة‬ ‫اﻟﻐذاء‬ ‫اﻟﻣﻼﺑس‬ ‫اﻻﯾﺟﺎر‬ ‫اﻟﺗﻌﻠﯾم‬ (‫ﻧﻔﻘﺎﺗﺎﺧرى )ﻣﻌﺎﻻدﺧﺎر‬ (B ƃྯ ႐କଐƁࠣୃกƁ౫ౄࡴ ‫ ێ‬ᇬ ᇊ ࢵ B ‫اﻻﺳرة‬ ‫اﻟﻐذاء‬ ‫اﻟﻣﻼﺑس‬ ‫اﻻﯾدﺟﺎر‬ ‫اﻟﺗﻌﻠﯾم‬ ‫ﻧﻔﻘﺎت اﺧرى‬ 34 ܸୃกܸ଍ଉŽˍ ໝ ແࡦफउˍ ࠣ‫ݶ‬ Ɖܸࢤ ެߕକଐƁƢࠣ‫ࡦ ࢽޟܸ ࢲޟ‬ƁƢࠣ‫ޟ‬පඓ ൷Ɓ౭౑ࢲ କଐƁƄ ܸࢲ ƁƋኃ ᒔᐺ። ጏ࢓‫ݕ‬ ࡦƁ࣬ ࣊ Ɓ࢓ߢೕಹኃ ቮƁ೾೥࢓଍ଉƁƄ ƁƋƁ߁ ࢟ ſ ። ፎକଐ߁ ެࢲ କଐƁƢ। ्࢟ ࣝࡦƁŻܸࢤ ߾ ଔଐƁ ƛ ܸߢࢽ ࡦƁ߁ ዖዄ Ž૊ સၫƋࣝެ࠾ኃ ቮƁƊܸެࢲ ଘଐƁ ። ጒܸ‫ࡦަޟ‬Ɓࣇ ஆॅ Ɓ ࠦ୆งଘଓƄƇ ܻ‫ࡩީޢܻރ‬ƄƓ ࢖‫ ࡩݘ‬Line Graphs ࠦ୆คܻ‫ࡩީޢ‬ƄƔ ࣐࣠ உै ƄƢ߄ ႀ ޯࢵ ƀ ‫ ۑ‬ơ ܻ޲ࡷ ƣ ࣠‫ ࡷ࡯ޢ‬2009 - 2005ƢƄ࣠‫ݑ‬൑ᑉᐽࠦ୆෷Ƅ߃ ‫ݓ‬ ࡩƄƇ ܻၮܻެࢧ ᑃᐽበሼྲ ႖Ƈ ƄƌƄमअƄ໠ ໄ୍෺፥ ፑܻޯࡩƄơ ƥ߄ எै Ƅ‫ۑ‬i 2009 2008 ‫ۑ‬ 2007 ‫ۑ‬ 2006 ‫ۑ‬ 2005 ‫ۑ‬ Ƈ Ƅ࣠ެࣀ ࡩƄ ‫ۑ‬ Ǝଘଓƥƌ Ƈ ƄƌƄमअଗଓƄ - 2005ƟƁࣝ‫ ݎ‬ൎᑆᐺࠣୃ෴Ɓ߀ ‫ݐ‬ ࡦƁƄ ܸၫܸީࢤ ᑀᐺቝሹྯ ႓Ƅ ƁƉƁफंƁ 2.7)ᇬ ᇊ ࢵ ƋକଐƢƉƠ ࣝ‫ ࡴ࡬ޟ‬2009 ‫اﻻﯾردات‬ 14 12.8 12 10.3 10 8.5 8 5.3 6 ‫اﻻﯾردات‬ 3.8 4 2 0 2005 2006 2007 2008 2009 ‫اﻟﺳﻧوات‬ ƢƄ࣠‫ݑ‬൑ᑉᐽ ƆƎܻ‫ࡩ ߎ ࢵޢ‬Ƅ Ƈ ƄƎܻ‫ࡩ ࣀޢ‬ƄƉ ܻ୓คଗଓࠦ୆งଘଓƄƇ ܻ‫ࡩީޢܻރ‬ƄƓ ࢖‫ࡩݘ‬፥ ጕܻ‫ࡩީޢ‬Ƅ࣊ உै ƄƢ߄ ႀ ޯࢵ ƀ ii) 200 200 200 200 200 200 60 55 48 40 35 30 7 35 6 5 4 3 2 2008-2002 Ƅ࣠ެࣀ ࡩƄ ‫ت‬ Ɖ ܻ୓คଘଓƄ ܸୃกܸ଍ଉŽ–ໝ ແࡦफउ –ࠣ‫ݶ‬ Ɖܸࢤ ެߕକଐƁƢࠣ‫ࡦ ࢽޟܸ ࢲޟ‬ƁƢࠣ‫ޟ‬පඓ ൷Ɓ౭౑ࢲ କଐƁƄ ܸࢲ ƁƋኃ ᒔᐺ። ጏ࢓‫ݕ‬ ࡦƁ࣬ ࣊ Ɓ࢓ߢೕಹኃ ቮƁ೾೥࢓଍ଉƁƄ ƁƋƁ߁ ࢟ ſ ። ፎକଐ߁ ެࢲ କଐƁƢ। ्࢟ ࣝࡦƁŻܸࢤ ߾ ଔଐƁ ƛ ܸߢࢽ ࡦƁ߁ ዖዄ Ž૊ સၫƋࣝެ࠾ኃ ቮƁƊܸެࢲ ଘଐƁ 2008-2002ƟƁࣝ‫ ݎ‬ൎᑆᐺ ƃƋܸ‫ࡦ ߋ ࢲޟ‬Ɓ Ƅ ƁƋܸ‫ࡦ ࢽޟ‬ƁƆ ܸ୐กŽ ‫ ێ‬ᇬ ᇊ ࢵ Ɔ ܸ୐กକଐƁ ‫اﻻﻧﺗﺎج‬ 70 60 55 60 48 50 40 35 40 30 30 ‫اﻻﻧﺗﺎج‬ 20 10 0 2002 2003 2004 2005 2006 2007 ‫اﻟﺳﻧوات‬ (Histogram Ƨ ƎƄಒ౷ޯࡩƄƉ Ǝ߄ ଐଌƄƢƄ߄ ႀ ޯࢵ ܻ‫ࠦހ‬୆งଘଓƄƇ ܻ‫ࡩީޢܻރ‬ƄƓ ୹ ‫ݑ‬ƀ ‫ ۑ‬ơ ܻ޲ࡷ -70 -60 -50 -40 -30 -20 -10 1 5 7 14 22 17 8 80 36 70 60 50 40 30 20 10-0 3 ࢖ࡺ‫ࡩݘ‬ƄƇ ܻ‫ߙޟ‬ ƌ߄ ၮ ƒ ܻ༧ೞ ଘଓƄ ࢓ࡷ‫ݕ‬ ࡦƁƄ ܸ‫ ߾ ࢽ ݧߖޜ‬Ə ܸ༤೛ କଐƁƉ߁ ၫ ‫ ێ‬ᇬ ᇊ ࢵ ܸୃกܸ଍ଉŽ–ໝ ແࡦफउ –ࠣ‫ݶ‬ Ɖܸࢤ ެߕକଐƁƢࠣ‫ࡦ ࢽޟܸ ࢲޟ‬ƁƢࠣ‫ޟ‬පඓ ൷Ɓ౭౑ࢲ କଐƁƄ ܸࢲ ƁƋኃ ᒔᐺ። ጏ࢓‫ݕ‬ ࡦƁ࣬ ࣊ Ɓ࢓ߢೕಹኃ ቮƁ೾೥࢓଍ଉƁƄ ƁƋƁ߁ ࢟ ſ ። ፎକଐ߁ ެࢲ କଐƁƢ। ्࢟ ࣝࡦƁŻܸࢤ ߾ ଔଐƁ ƛ ܸߢࢽ ࡦƁ߁ ዖዄ Ž૊ સၫƋࣝެ࠾ኃ ቮƁƊܸެࢲ ଘଐƁ Ə ܸ༤೛ କଐƁ Ƈ ܻ‫ߙޟ‬ ࢖ࡺ‫ࡩݘ‬Ƅ - Ɩ ࣠ࡺ഻വറƄ 60 Ƨ ƎƄಒ౷ޯࡩƄ݊ ࡯ࢥ ଐଌƄƢƄ߄ ႀ ޯࢵ ܻ‫ࠦހ‬୆งଘଓƄƇ ܻ‫ࡩީޢܻރ‬ƄƓ ୹ ‫ݑ‬ƀ ‫ ۑ‬ơ ܻ޲ࡷ 65 1 - 55 13 - 45 6 - 35 11 - 25 11 - 15 13 - 5 5 Ƈ ܻ‫ߙޟ‬ ࢖ࡺ‫ࡩݘ‬Ƅ ƎƄಒ౷ޯࡩƄ ƋƁಏ౴ެࡦƁƋƁಏ౴ެࡦƁ݇ ࡬ࢢ ଍ଉƁ ‫ ێ‬ᇬ ᇊ ࢵ ȏ °¦ʙؒ ʯƃ¦ŴƄʷ ʸƃ¦ 15 10 5 0 0 37 -5 -15 -25 -35 -45 -55 -65 -75 ࢖ࡺ‫ࡩݘ‬ƄƇ ܻ‫ߙޟ‬ ܸୃกܸ଍ଉŽˍ ໝ ແࡦफउˍ ࠣ‫ݶ‬ Ɖܸࢤ ެߕକଐƁƢࠣ‫ࡦ ࢽޟܸ ࢲޟ‬ƁƢࠣ‫ޟ‬පඓ ൷Ɓ౭౑ࢲ କଐƁƄ ܸࢲ ƁƋኃ ᒔᐺ። ጏ࢓‫ݕ‬ ࡦƁ࣬ ࣊ Ɓ࢓ߢೕಹኃ ቮƁ೾೥࢓଍ଉƁƄ ƁƋƁ߁ ࢟ ſ ። ፎକଐ߁ ެࢲ କଐƁƢ। ्࢟ ࣝࡦƁŻܸࢤ ߾ ଔଐƁ ƛ ܸߢࢽ ࡦƁ߁ ዖዄ Ž૊ સၫƋࣝެ࠾ኃ ቮƁƊܸެࢲ ଘଐƁ ። ጒܸޯࡦƁࡠ ࢤ ߣࡦƁ ௧ ோƋܸೕೆ ơ ƥଘଓƄƦ ࣠ޯࣀ ଐଌƄƅ ൎ൉࣐ ࡩࡣ ࢧ ߦࡩƄơ ܻዙኸ ƀ፥ ጿܻީࡩܻ࣍ 52Ƈ ܻၴ Ǝƌࡣ ޲೘಼Ƥܻ‫ރ‬ƌƀơ ƥ߄ எै Ƅ .1 .ࠦ‫ݹ‬ƎƄƌଘଓƄƢॅळ‫ࡩݘ‬Ƅࠦ‫ޢ‬ᆿᆝ ū 50ࡼ ࡷࠦၴ Ǝኆ ቱƄ ࠦީࢵ ܻഷവറƄƃ ƌܻީࡷƟ ܻࣀ ࡷ፥ ጿࠦީࢵ ܻ༨ீ ‫ࡷެܻ ࢵ ݪ‬Ƨ ƎƄಒ౷‫ކ‬ơ ƥ߄ ၴ ፥ ጿƇ ܻ‫ࡩީޢܻރ‬Ƅƅ ࣠‫ ހ‬ 4 3 1 2 4 5 5 3 4 1 1 1 5 6 4 3 2 3 0 1 2 1 3 3 0 Ƨ ࣠‫ޟ‬ଐଌƄ཯ཌࣀ ჱࡩƄƎƄಒ౷ޯࡩƄƥ཯ཌࣀ ჱࡩƄƎƄಒ౷ޯࡩƄ߄ ၴ ƥƀ  5 1 2 2 3 2 3 0 3 2 2 4 2 1 2 5 0 6 1 3 3 3 4 4 5 Ɔ౰౔ߦࡩƄơ ൎ൉ၯ Ƈ ƄƎܻ‫ࡩ ࣀޢ‬ƄƉ ܻ୓คଗଓ݊ ‫ ࢧܻރ‬ଐଌƄ߄ ၰ ƀ፥ ጿࠦၮܻީଐଌƄƇ ƄƎܻ‫ࡩ ࣀޢ‬Ƅƌ߄ ၮ໠ ໄ୍෺Ƥܻ‫ރ‬ƌƀơ ƥ߄ எै Ƅ‫ۑ‬2 ፥ ጕܻ‫ࡩީޢ‬Ƅ࣊ உै Ƅ ƅ 2004 1200 2003 Ɔƌ࢖ߦެଐଌƄƆ߄ ዙኸ ଘଓƄ ƀ ƢƄ߄ ႀ ޯࢵ ܻ‫ހ‬Ƈ ܻ‫ࡩީޢܻރ‬ƄƓ ୹ ‫ݑ‬ƀū 2005-2000 2002 1700 1900 2001 2800 2000 2100 ƢƄ࣠‫ݑ‬ଘଓƄ Ƈ ƄƎܻ‫ࡩ ࣀޢ‬Ƅ ƌ߄ ၮ ࠦၮܻީଐଌƄ ࠦ‫ࡩޯܻࡩޢ‬ƄƇ ܻ‫ࡩީޢܻރ‬ƄƓ ࢖‫ ࡩݘ‬pie diagramƆमंƄኆ ቱƄƢ߄ ႀ ޯࢵ ƀ‫ۑ‬3 ࠦީࣀ ჱࡩƄ 20 ࡣ ၯ ኆ ቱƄ žƄ߃ ‫ݓ‬ ࡩƄ 12 Ƈ ൎ൉ࢢ Ƅ࣠ଐଌƄ 25 যও࡯‫ࡩޯݘ‬Ƅ 15 10 18 100 38 ࡼ ࡷࠦީࣀ ჱࡁ Ƨ ࢖ᐳႡࡩƄƟ ܻߦ‫ރ‬ଘଓƄƌ࣠୐ก Ǝܻ༬࿻ ଘଓƄ ࠦႁ ࢧ ࡩƄ Ʀ ༇໩ ƀƇ ܻߥߦ‫ރ‬ ፥ ፑܻዙዋ ଘଓƄ ܸୃกܸ଍ଉŽ–ໝ ແࡦफउ –ࠣ‫ݶ‬ Ɖܸࢤ ެߕକଐƁƢࠣ‫ࡦ ࢽޟܸ ࢲޟ‬ƁƢࠣ‫ޟ‬පඓ ൷Ɓ౭౑ࢲ କଐƁƄ ܸࢲ ƁƋኃ ᒔᐺ። ጏ࢓‫ݕ‬ ࡦƁ࣬ ࣊ Ɓ࢓ߢೕಹኃ ቮƁ೾೥࢓଍ଉƁƄ ƁƋƁ߁ ࢟ ſ ƛ ܸߢࢽ ࡦƁ߁ ዖዄ Ž૊ સၫƋࣝެ࠾ኃ ቮƁƊܸެࢲ ଘଐƁ ። ፎକଐ߁ ެࢲ କଐƁƢ। ्࢟ ࣝࡦƁŻܸࢤ ߾ ଔଐƁ ᄎ ႓ƎƀƤܻ‫ރ‬ƌƀƧ ƎƄಒ౷ޯࡩƄơ ƥ߄ எै Ƅࡼ ࡷ‫ۑ‬4 ( Histogram )Ƨ ƎƄಒ౷ޯࡩƄƉ Ǝ߄ ଐଌƄ  (Polygon)Ƨ ƎƄಒ౷ޯࡩƄ݊ ࡯ࢥ ଐଌƄ  ƎƄಒ౷ޯࡩƄ ƅ ൎ൉࣐ ࡩƄƌ߄ ၮ Ƈ ܻ‫ࡩߦޟ‬Ƅ 5 10 - 0 20 30-20 16 28 20-10 40-30 15 50–40 5 70-60 10 2 39 (Frequency curve Ƨ ƎƄಒ౷ޯࡩƄཀྵཏࠒ ެଐଌƄ  60–50 80-70 ܸୃกܸ଍ଉŽ–ໝ ແࡦफउ –ࠣ‫ݶ‬ Ɖܸࢤ ެߕକଐƁƢࠣ‫ࡦ ࢽޟܸ ࢲޟ‬ƁƢࠣ‫ޟ‬පඓ ൷Ɓ౭౑ࢲ କଐƁƄ ܸࢲ ƁƋኃ ᒔᐺ። ጏ࢓‫ݕ‬ ࡦƁ࣬ ࣊ Ɓ࢓ߢೕಹኃ ቮƁ೾೥࢓଍ଉƁƄ ƁƋƁ߁ ࢟ ſ ƛ ܸߢࢽ ࡦƁ߁ ዖዄ Ž૊ સၫƋࣝެ࠾ኃ ቮƁƊܸެࢲ ଘଐƁ 40 ። ፎକଐ߁ ެࢲ କଐƁƢ। ्࢟ ࣝࡦƁŻܸࢤ ߾ ଔଐƁ ܸୃกܸ଍ଉŽˍ ໝ ແࡦफउˍ ࠣ‫ݶ‬ Ɖܸࢤ ެߕକଐƁƢࠣ‫ࡦ ࢽޟܸ ࢲޟ‬ƁƢࠣ‫ޟ‬පඓ ൷Ɓ౭౑ࢲ କଐƁƄ ܸࢲ ƁƋኃ ᒔᐺ። ጏ࢓‫ݕ‬ ࡦƁ࣬ ࣊ Ɓ࢓ߢೕಹኃ ቮƁ೾೥࢓଍ଉƁƄ ƁƋƁ߁ ࢟ ſ ። ፎକଐ߁ ެࢲ କଐƁƢ। ्࢟ ࣝࡦƁŻܸࢤ ߾ ଔଐƁ ƛ ܸߢࢽ ࡦƁ߁ ዖዄ Ž૊ સၫƋࣝެ࠾ኃ ቮƁƊܸެࢲ ଘଐƁ ࢛ ‫઻ ેހݘ‬ၮƞ ࢖‫ޯݘ‬ᑃᐽࠦެ‫ ࢵ ࢪࡷݘޢ‬Ƅࢇ ߙƥƇ ܻ‫ࡩީޢܻރ‬Ƅࡼ ࡷቿቷࣀ ࡯ࢵ ࠦࢵ ƄƎƌ཯ཏ‫ݹࠦކݘ‬ഁ೨࢖ଐଌƄࠦၮ౮౑ࡩƄࢪ ყႾܻߥࡷ ࠦࣀ ዙ዆ Ơ ܻެࠧƥ‫ݹࠦۑ‬ഁ೨࢖ଐଌƄযওߥࡩƄᑻ ᑧ‫࣠ކ‬༨စ ܻᐱᐢ‫ݑ‬बएƥቿቷࣀ ࡯ࣀ ࡩƄƇ Ƅƌ࢖ߦࡷࡣ ‫ࡷޢ‬໠ ໄ୍ง཯དྷࡩƄࠦ‫ݹ‬ഁ೨࢖ଐଌƄযওߥࡩƄ ፥ ፕ ࢪ ყႾܻߥଐଌƄƤ߃ ࠧƥ‫ۑ‬໠ ໄ޲ࠁ ܻީࡩƄƥ໠ ໄ୆෷ܻࢧ ࠁ ଘଓƄࡣ ީߘࡼ ࡷ݊ ‫ݶ‬ ܻࣃ ࡩƄƢƄ߄ ႀ ޯࢵ ଘଓƄƇ Ƅƍࢪ ყႾܻߥࡷ ፥ ጘଘଓᇴᆝ (Arithmetic Mean )፥ ጒܻࣀ ஊै Ƅ࣊ ࢵ ࣠ࡩƄ 3.1 ܻࠧ౰౗ࡁ Ƅ [Central Tendency Measure] ࠦ‫ݹ‬ഁ೨࢖ଐଌƄࠦၮ౮౑ࡩƄࢪ ყႾܻߥࡷ߄ ၰ ƀ࣠ࠧ፥ ጒܻࣀ ஊै Ƅ࣊ ࢵ ࣠ࡩƄ .ܻ‫ݹܻࢠޢ‬ƎƤਔ ৲߄ ༨ဈ ƥࠢ‫ ࣀܻހ‬ჴၰ Ƅቿቱ࣠ᐳ႞ࡩƄ࢖࣏ ‫ࡷܻ ރ‬Ƅ߄ ႀ ޯࢵ Ƅ ࡣ ࢢ ܻၰ ࠢ‫ ހܾރ‬x3……xn ، x2 ،x1 যওߥࡩƄࡼ ࡷࠦၮ࣠ࡺ഻വറ> The Mean ]፥ ጒܻࣀ ஊै Ƅ࣊ ࢵ ࣠ࡩƄƞ ࢖‫ݹݘ‬ƥ ƣ ƀƧ ƀ‫ܻࠧۑ‬ƌ߄ ၮે ઻ၮযওߥࡩƄᑻ ᑧ‫ކ‬Ɩ ࣠ዙಣಝࠦࡺࣀ ߘ ‫ݔ‬ҧ = ௫భ ،௫మ ، ௫మ ، ...........، ௫೙ ‫ݔ‬ҧ = ෍ ‫ ݔ‬/݊ ௡ [ un grouped data] ࠣ‫ݽ‬ ࣝަ଍ଉƁ౭ౄၦƄ ܸ‫ަޟܸހ‬ᑀᐺ። ጏܸࢽ இॅ Ɓࣇ ࢲ ࣝࡦƁƂ ܸࢽ ߾ 1.1.1 ࠯ ެީࡩƄ፥ ጿ໠ ໄߦ࣌ ࣠ࡷƆྲ Ⴁ‫ࡩݘ‬Ƨ ࢖ᐳႡࡩƄว ง࢖ଐଌƄ໠ ໄ୍งࠦ୆งଘଓƄƇ ܻ‫ࡩީޢܻރ‬Ƅ ‫ ۑ‬ơ ܻ޲ࡷ ‫ۑ‬፥ ጒܻࣀ ஊै Ƅ࣊ ࢵ ࣠ࡩƄ‫ ࠁ ࣀ ݪ‬ƀ‫཯ ۑ‬ཏ೘಼ơ ܻ‫ݹ‬Ǝߎ ࡩƄ 150 ‫ݔ‬ҧ = 100 70 90 85 80 60 55 50 xଵ ،xଶ ، xଶ ، . . . . . . . . . . . ، x௡ ݊ 30 ࡠ இॅ Ɓ ‫ݔ‬ҧ = ෍ ‫ ݔ‬/݊ ‫ݔ‬ҧ = 30 + 50 + 55 + 60 + 80 + 85 + 90 + 70 + 100 + 150 = 770/10 = 77 41 10 ܸୃกܸ଍ଉŽ–ໝ ແࡦफउ –ࠣ‫ݶ‬ Ɖܸࢤ ެߕକଐƁƢࠣ‫ࡦ ࢽޟܸ ࢲޟ‬ƁƢࠣ‫ޟ‬පඓ ൷Ɓ౭౑ࢲ କଐƁƄ ܸࢲ ƁƋኃ ᒔᐺ። ጏ࢓‫ݕ‬ ࡦƁ࣬ ࣊ Ɓ࢓ߢೕಹኃ ቮƁ೾೥࢓଍ଉƁƄ ƁƋƁ߁ ࢟ ſ ። ፎକଐ߁ ެࢲ କଐƁƢ। ्࢟ ࣝࡦƁŻܸࢤ ߾ ଔଐƁ ƛ ܸߢࢽ ࡦƁ߁ ዖዄ Ž૊ સၫƋࣝެ࠾ኃ ቮƁƊܸެࢲ ଘଐƁ ơ ܻ‫ݹ‬Ǝ 77000 ࠯ ެީࡩƄ፥ ጿ໠ ໄߦ࣌ ࣠ଐଌƄƇ ୍ܻง࢖ଐଌ፥ ጒܻࣀ ஊै Ƅ࣊ ࢵ ࣠ࡩƄ [ Grouped Data ]ࠣ‫ݽ‬ ࣝަ଍ଉƁƄ ܸ‫ަޟܸހ‬ᑀᐺ። ጏܸࢽ இॅ Ɓࣇ ࢲ ࣝࡦƁ ‫ێ ێ‬ ፥ ጒܻࣀ ஊै Ƅ࣊ ࢵ ࣠ࡩƄƌܻ༬࿻ ଗଓ፥ ጘଘଓƄƣ ࣠‫ࡩߥܻރ‬ƄƢ߄ ႀ ޯఱ ఝܻ୐คܾߙࠦ‫ީ࣠ހ‬ଐଌƄƇ ܻ‫ࡩީޢܻރ‬Ƅቿቱܻၰ ፥ ጿ ‫ݔ‬ҧ= ∑ ݂x ∑݂ Ʀ ࣠ޯࣀ ଐଌƄƅ ൎ൉࣍ Ƈ ܻၴ Ǝኆ ቱࠦ୆งଘଓƄࠦ‫ީ࣠ހ‬ଐଌƄƇ ܻ‫ީޢܻރ‬ᑃᐽ፥ ጒܻࣀ ஊै Ƅ࣊ ࢵ ࣠ࡩƄ‫ ࠁ ࣀ ݪ‬ƀ : ( ‫ ێ‬ƞ ܸޯࡴ ‫ܻ ࢵީࠦۑ‬ഷവറƄƃ ƌܻީࡷƟ ܻࣀ ࡷ፥ ጿ،ࠦ‫ݹ‬ƎƄƌଘଓƄƢॅळ‫ࡩݘ‬Ƅࠦ‫ޢ‬ᆿᆝ ،ơ ƥଘଓƄ -90 -80 -70 -60 -50 -40 -30 -20 -10 -0 100 90 2 3 80 8 70 10 60 50 7 6 40 30 5 4 20 10 3 2 Ƈ ܻ‫ࡩߦޟ‬Ƅ Ƈ ܻၴ Ǝኆ ቱƄ ƎƄಒ౷ޯࡩƄ ፥ ጘଘଓƄ࣠ࠒ ެࡩƄે ઻ၮࡣ ஊै ƄƇ Ƅ࣐࣠ ࠀ ࢝ ࠑ ࡯‫ݹ‬ଘଓƥ߄ ၴ ƣ ࣠ࡓ ‫ ࡠރ‬இॅ Ɓ ƀ ƉܸަࡴƜ ܸࢽ ࡴ። ጼ،ࠣ‫ݶ‬ƋƁƉକଐƁƟूर‫ݕ‬ ࡦƁࠣ‫ޟ‬ᆼᆚ ،ƞ ƢକଐƁƣ ࣝެࢽ ଍ଉƁƂ ോെ࣊ Ƅ ܸၱ ƋƉ ‫ ێ‬ƞ Ƣ߁ ၱ Ƈ ܻၴ Ǝƌ Ƈ ܻ‫ࡩߦޟ‬Ƅ ƅ ൎ൉࣐ ࡩƄ 0 – 10 10-20 20-30 30-40 40-50 50-60 60-70 42 ƎƄಒ౷ޯࡩƄ ࠦ‫ࡩߦޟ‬Ƅഁ೨୹ ୪ 3 15 5 35 2 4 6 7 10 Fx 5 10 25 100 45 270 55 65 ‫ܸ ࢲަࠣێ‬ഴലമƁ 45 175 385 650 ܸୃกܸ଍ଉŽ–ໝ ແࡦफउ –ࠣ‫ݶ‬ Ɖܸࢤ ެߕକଐƁƢࠣ‫ࡦ ࢽޟܸ ࢲޟ‬ƁƢࠣ‫ޟ‬පඓ ൷Ɓ౭౑ࢲ କଐƁƄ ܸࢲ ƁƋኃ ᒔᐺ። ጏ࢓‫ݕ‬ ࡦƁ࣬ ࣊ Ɓ࢓ߢೕಹኃ ቮƁ೾೥࢓଍ଉƁƄ ƁƋƁ߁ ࢟ ſ ƛ ܸߢࢽ ࡦƁ߁ ዖዄ Ž૊ સၫƋࣝެ࠾ኃ ቮƁƊܸެࢲ ଘଐƁ ። ፎକଐ߁ ެࢲ କଐƁƢ। ्࢟ ࣝࡦƁŻܸࢤ ߾ ଔଐƁ 70-80 8 75 600 90-100 2 95 190 80-90 Ɩ ࣠ࡺ഻വറƄ 3 85 50 ‫ݔ‬ҧ = 2680/50 = 53.6 - ‫ݔ‬ҧ= 255 2680 ∑ ݂x ∑݂ [ Geometric Mean] ࿀ ႐߁ ީಘಔƁࣇ ࢲ ࣝࡦƁ ‫ێ‬ ቻቱ୷ ୪मअƥযওߥࡩƄƤ߃ ࠧƅ ྲ Ⴉ ࡣ ࢢ ܻஊै ፥ ጕ࣠ެࡩƄƎ߃ எै Ƅࠢ‫ހܾރ‬যওߥࡩƄࡼ ࡷࠦၮ࣠ࡺ഻വറ࿃ ႓߄ ެಛಗƄ࣊ ࢵ ࣠ࡩƄƞ ࢖‫ݹݘ‬ [ G ]୷ ୪༟༎ܻ‫ހ‬ x1 ، x2 ، x3 ، ………………..xn যওߥࡩƄܻ୐෺ኆ ቱƣ ᇴᆝ Ƅƍƀ ፥ ጘଘଓƄƣ ࣠‫ [ ݹݘ ࣐ ࣩހܻࡩߥܻރ‬G ]࿃ ႓߄ ެಛಗƄ࣊ ࢵ ࣠ࡩƄƣ ܾߙ G = ࢔ට ࢞૚ ، ࢞૛ ، ࢞૜ ، … … … . . ࢞࢔ ، ፥ ጘଘଓƄƣ ࣠‫ࡩߥܻރ‬ƄƢƄ߄ ႀ ޯࢵ ܻ‫ހ‬যঠ‫ݹ‬ƎܻၩॅळࡩƄƢƄ߄ ႀ ޯࢵ ܻ‫ހ‬࿃ ႓߄ ެಛಗƄ࣊ ࢵ ࣠ࡩƄƅ ܻࣀ ࠁ ܻࢥ ‫ݹ‬ƀࡼ ࡓ ೘಼ƥ Log G = 1/n [ log x1 + log x2 + log x3 + ….+ log xn ] Log G = 1/n [ ∑௡௜ୀ଴ ‫]݅ݔ‬ G = Anti log [ 1/n ∑௡௜ୀ଴ ‫] ݅ݔ‬ ‫ۑۑۑ‬،11 ، 8 ، 7 ، 3ࠦ୆งଘଓƄযওߥᑃᐽ࿃ ႓߄ ެಛಗƄ࣊ ࢵ ࣠ࡩƄ߄ ၴ ƥƀ ‫ ۑ‬ơ ܻ޲ࡷ G = ೙ට ‫ݔ‬ଵ ، ‫ݔ‬ଶ ، ‫ݔ‬ଷ ، … … … . . ‫ݔ‬௡ ، G = √3 x 7x 8 x 11 ర 43 ܸୃกܸ଍ଉŽ–ໝ ແࡦफउ –ࠣ‫ݶ‬ Ɖܸࢤ ެߕକଐƁƢࠣ‫ࡦ ࢽޟܸ ࢲޟ‬ƁƢࠣ‫ޟ‬පඓ ൷Ɓ౭౑ࢲ କଐƁƄ ܸࢲ ƁƋኃ ᒔᐺ። ጏ࢓‫ݕ‬ ࡦƁ࣬ ࣊ Ɓ࢓ߢೕಹኃ ቮƁ೾೥࢓଍ଉƁƄ ƁƋƁ߁ ࢟ ſ ። ፎକଐ߁ ެࢲ କଐƁƢ। ्࢟ ࣝࡦƁŻܸࢤ ߾ ଔଐƁ ƛ ܸߢࢽ ࡦƁ߁ ዖዄ Ž૊ સၫƋࣝެ࠾ኃ ቮƁƊܸެࢲ ଘଐƁ G = = √1848 ర G = 6.5 ࠦ୆งଘଓƄƇ ܻ‫ީޢܻރ‬ᑃᐽ࿃ ႓߄ ެಛಗƄ࣊ ࢵ ࣠ࡩƄ߄ ၴ ƥƀ ‫ ۑ‬ơ ܻ޲ࡷ Log 215 ، 35 ، 75 ، 14 ، 112 ،11 ، 169 ، 6 = 1/n [ ∑௡௜ୀ଴ ‫]݅ݔ‬ G = Antilog [ 1/n ∑௡௜ୀ଴ ‫] ݅ݔ‬ X বঝ‫ݶ‬ƋܸၦूरࡦƁࢄ ‫ ݎ ࡹ ࣊࢓ݶ‬࿀ ႐߁ ީಘಔƁࣇ ࢲ ࣝࡦƁƂ ܸࢽ ߾ ‫ ێ‬ƞ Ƣ߁ ၱ Log10 x 6 0.7781 169 2.2279 112 2.0492 11 1.0414 14 1.1461 75 1.8750 35 1.3979 215 2.3324 ∑ log10 x Log G 10.8066 = 1/n [ ∑௡௜ୀ଴ ‫]݅ݔ‬ Log G = 10.8066/8 = 1.3508 G = Antilog (1.3508) G = 22.43 [ Harmonic Mean ]। ौߖƁࣝެࡦƁࣇ ࢲ ࣝࡦƁ ‫ێ‬ ቻቱ୷ ୪मअƥযওߥࡩƄƅ ॅळߥଐଌ፥ ጒܻࣀ ஊै Ƅ࣊ ࢵ ࣠ࡩƄƅ ॅळߥࡷࠢ‫ހܾރ‬যওߥࡩƄࡼ ࡷࠦၮ࣠ࡺ഻വറ१ ॏߙƄ࣠ޯࡩƄ࣊ ࢵ ࣠ࡩƄƞ ࢖‫ݹݘ‬ H = ∑࢔ (࢔) ࢏స ૚/࢞࢏ 44 ( H ୷ ୪༟༎ܻ‫ހ‬ ܸୃกܸ଍ଉŽ–ໝ ແࡦफउ –ࠣ‫ݶ‬ Ɖܸࢤ ެߕକଐƁƢࠣ‫ࡦ ࢽޟܸ ࢲޟ‬ƁƢࠣ‫ޟ‬පඓ ൷Ɓ౭౑ࢲ କଐƁƄ ܸࢲ ƁƋኃ ᒔᐺ። ጏ࢓‫ݕ‬ ࡦƁ࣬ ࣊ Ɓ࢓ߢೕಹኃ ቮƁ೾೥࢓଍ଉƁƄ ƁƋƁ߁ ࢟ ſ ። ፎକଐ߁ ެࢲ କଐƁƢ। ्࢟ ࣝࡦƁŻܸࢤ ߾ ଔଐƁ ƛ ܸߢࢽ ࡦƁ߁ ዖዄ Ž૊ સၫƋࣝެ࠾ኃ ቮƁƊܸެࢲ ଘଐƁ ، 45 ، 30 ، 18 ، 50 ، 70 ، 20ࠦ୆งଘଓƄƇ ܻ‫ީޢܻރ‬ᑃᐽ१ ॏߙƄ࣠ޯࡩƄ࣊ ࢵ ࣠ࡩƄ߄ ၴ ƥƀ ‫ ۑ‬ơ ܻ޲ࡷ H = ∑࢔ H= H= H= (࢔) ࢏స ૚/࢞࢏ 66 ଻ భ భ భ భ భ భ భ ା ା ା ା ା ା మబ ళబ ఱబ భఴ యబ రఱ లల ଻ ଴.଴ହା଴.଴ଵସା଴.଴ଶା଴.଴ହହା଴.଴ଷଷା଴.଴ଶଶା଴.଴ଵହ ଻ = 33.49 ଴.ଶ଴ଽ [ Median]ࣇ ‫ࡦࣝ ࢲޟ‬Ɓ ‫ێ‬ ࡼ ࡷࠦၮ࣠ዙಣಝᄎ ႞ ߥ‫཯ކ‬དྷࡩƄƆƌ߄ ഷവറƄࠦࡺ‫ࡩߥޢ‬Ƅࠢ‫ހܾރ‬ƞ ࢖‫ݹݘ‬ƥ‫ݹࠦۑ‬ഁ೨࢖ଐଌƄࠦၮ౮౑ࡩƄࢪ ყႾܻߥࡷ߄ ၰ ƀ࣊ ‫ࡩ࣠ ࢵޢ‬Ƅ߄ ‫ݹݘ‬ ࢪ ࡓ ‫ࡩݘ‬Ƅƥ ،ࠢެࡷ౰౎ࡁ ƀࠢ‫཯ކ࡯ޢ‬དྷࡩƄƥ ،ࠢެࡷ࢖‫ݓ‬ ࢢ ƀቻቷީߘ཯དྷࡩƄযওߥࡩƄƣ ࣠ࡓ ‫ކ‬ƥ‫ۑ‬໠ ໄ‫ݹ‬ƥܻࣀ ჴࡷ໠ ໄߦࢧ ‫ރ‬፟ ፑƄযওߥࡩƄ ‫ࡩޢܻۑ‬Əܻ୐งযওߥࡩƄ‫ ݝ‬ხ჋ƎƄƍƀ߰ ‫ޢ‬༨༦ ࣇ ‫ࡦࣝ ࢲޟ‬ƁƂ ܸࢽ ߾  ࠣ‫ݽ‬ ࣝަ଍ଉƁ౭ౄ‫ݐ‬ ࡦƁƄ ܸ‫ࡦަޟܸހ‬Ɓቼቮܸၭ ። ጼକଐƢŽ ࠦ‫ࡷީ࣠ހ‬౰ే‫ݓ‬ ࡩƄƇ ܻ‫ࡩީޢܻރ‬Ƅቿቱܻၰ ፥ ጿ࣊ ‫ࡩ࣠ ࢵޢ‬Ƅƅ ܻࣀ ஊै ƣ ܻޯࡩܻၰ ߄ ၴ ࣠‫ކ‬ ،ࠦ‫ݹ‬ƌ࢖ߙ nบ คᇴᆝƥ xn‫ۑۑۑۑۑۑۑۑۑۑۑۑۑۑۑۑۑ‬، x3 ، x2 ܻ‫ݹ‬ƌ࢖ߙࠦ‫ީ࣠ހ‬ଐଌƄ౰ే‫ݓ‬ ࡩƄযওߥࡩƄƌ߄ ၮƣ ᇴᆝ Ƅƍƀ  ، x1 ،Ƈ Ƅ߄ ܻࠧࣃ ଐଌƄযওߘܻ୐෺ኆ ቱƣ ᇴᆝ Ƅƍƀ ‫ࠦۑ‬୆งଘଓƄƇ Ƅ࣐࣠ உै Ƅ݊ ީჴ჈࣊ ‫ࡩ࣠ ࢵޢ‬Ƅƅ ܻࣀ ࠒ ࡯ߙ ܻ‫ࡩޢ‬Əܻ୐งƥƀܻ‫ ߄ݹ‬ၮܻࢧ ‫ކ‬যওߥࡩƄว งमए ࠦ୆งଘଓƄࠦߘൎ൉‫ࡩݘ‬Ƅࡼ ࡷ࣊ ‫ࡩ࣠ ࢵޢ‬Ƅ‫ ݪ‬ყ჋मऒ߄ ၴ ࣠‫ރ‬ 1 Ƈ Ƅƌ࢖ߦଐଌƄƌ߄ၮ ࣊ ‫ࡩ࣠ ࢵޢ‬Ƅ‫ ݪ‬ყ჋मऒ 2 2 1 + n ࣊ ‫ࡩ࣠ ࢵޢ‬Ƅ‫ ݪ‬ყ჋मऒ ࣊ ‫ࡩ࣠ ࢵޢ‬Ƅ‫ ݪ‬ყ჋౰౔ࡩƆ࢖࣌ ܻެଐଌƄ፥ ፕ ƥ࣊ ‫ࡩ࣠ ࢵޢ‬Ƅࠦࡺ‫ ߄ߘޢ‬ၴ ࣠‫ۑރ‬III ܸ‫ ࠂޟ‬Ƣƌࠣ‫ݽ‬ ࣝަ଍ଉƁ౭ౄ‫ݐ‬ ࡦƁবঐߢࡦƁƉ߁ ၫƠ ᇱᆚ ƁƊſ 45 ܸୃกܸ଍ଉŽ–ໝ ແࡦफउ –ࠣ‫ݶ‬ Ɖܸࢤ ެߕକଐƁƢࠣ‫ࡦ ࢽޟܸ ࢲޟ‬ƁƢࠣ‫ޟ‬පඓ ൷Ɓ౭౑ࢲ କଐƁƄ ܸࢲ ƁƋኃ ᒔᐺ። ጏ࢓‫ݕ‬ ࡦƁ࣬ ࣊ Ɓ࢓ߢೕಹኃ ቮƁ೾೥࢓଍ଉƁƄ ƁƋƁ߁ ࢟ ſ .I .II ። ፎକଐ߁ ެࢲ କଐƁƢ। ्࢟ ࣝࡦƁŻܸࢤ ߾ ଔଐƁ ƛ ܸߢࢽ ࡦƁ߁ ዖዄ Ž૊ સၫƋࣝެ࠾ኃ ቮƁƊܸެࢲ ଘଐƁ ࠦࡺ‫ࡩߥޢ‬Ƅƥ n/2ܻᐱᐜყ჋मऒ཯དྷࡩƄࠦࡺ‫ࡩߥޢ‬Ƅ໠ ໄ‫ ࠅޢܻݹߥ ݊ހ‬ƥƏࠦ‫ީ࣠ހ‬ଐଌƄ౰ే‫ݓ‬ ࡩƄযওߥࡩƄƌ߄ ၮƣ ᇴᆝ Ƅƍƀ࣊ ‫ࡩ࣠ ࢵޢ‬Ƅ‫ ݪ‬ყ჋मऒ ‫ۑ‬໠ ໄޯࡺ‫ࡩߥޢ‬Ƅ໠ ໄ‫ࡷޯ࣠ ࢵ ܻ࣊ࠧކ‬፥ ፕ ࣊ ‫ࡩ࣠ ࢵޢ‬Ƅࠦࡺ‫ߘޢ‬ƥ‫ۑ‬1 + n /2 ܻᐱᐜყ჋मऒ཯དྷࡩƄ ௡ାଶ ଶ n/2 ơ ƥଘଓƄ࣊ ‫ࡩ࣠ ࢵޢ‬Ƅ‫ ݪ‬ყ჋मऒ = 1 + n /2 ፥ ጕܻ޲ࡩƄ࣊ ‫ࡩ࣠ ࢵޢ‬Ƅ‫ ݪ‬ყ჋मऒ ࠣୃคକଐƁবঐߢᑀᐺࣇ ‫ࡦࣝ ࢲޟ‬Ɓ߁ ၱ ƢƁ ‫ ێ‬ƞ ܸޯࡴ 22 ، 12 ، 8 ، 6 ، 10 ، 3 ، 15 ، 20 ࡠ இॅ Ɓ ܸ‫ݶ‬ ߁ ၫܸࢤ ‫ރ‬বঐߢࡦƁฤ คफऌ - ، 20 ، 15 ، 12 ، 10 ، 8 ، 6 ، 3 ࣇ ‫ࡦࣝ ࢲޟ‬Ɓ‫ ݧ‬ფ჈फए߁ ၱ ࣝ‫ ހ‬- ࣊ ‫ࡩ࣠ ࢵޢ‬Ƅ‫ ݪ‬ყ჋मऒƣ ܾߙࠦ‫ݹ‬ƌ࢖ߙযওߥࡩƄƣ ƀܻ೘ೃ 1 + n/2 ࣊ ‫ࡩ࣠ ࢵޢ‬Ƅ‫ ݪ‬ყ჋मऒ 4 = 2/ 8 = ଻ାଵ ଶ 10 ݇ ‫ݽ‬ Ɓ༜་Ɓ‫ ݧ‬ფ჈౭౑ࡦƁ࢓ࣉ ୍ܸคཬཏࡦƁࠣࡷ‫ࡦߢޟ‬Ɓ። ፒ Ƣࣇ ‫ࡦࣝ ࢲޟ‬Ɓࠣࡷ‫ ߁ߕޟ‬ၱ ࣝ‫ ހ‬ࠣୃคଘଐƁবঐߢᑀᐺࣇ ‫ࡦࣝ ࢲޟ‬Ɓ߁ ၱ ƢŽ ‫ ێ‬ƞ ܸޯࡴ 26 ، 50 ، 41 ، 16 ، 5 ، 13 ، 9 ، 17 ، 22 ، 32 ࡠ இॅ Ɓ ܸ‫ݶ‬ ߁ ၫܸࢤ ‫ރ‬বঐߢࡦƁฤ คफऌ - 50 ، 41 ، 32 ، 26 ، 22 ، 17 ، 16 ، 13 ، 9 ، 5 যওߥᑃᐽ໠ ໄ࣐ ‫ ࢵޢ‬ƥƠ ܻެࠧƣ ܾߙࠦ‫ ࠅޢ‬ƥƏযওߥࡩƄƣ ƀܻ೘ೃ ࣇ ‫ࡦࣝ ࢲޟ‬Ɓ‫ ݧ‬ფ჈फए߁ ၱ ࣝ‫ ހ‬- 5 = 2 / 10 = n /2 ơ ƥଘଓƄ࣊ ‫ࡩ࣠ ࢵޢ‬Ƅ‫ ݪ‬ყ჋मऒ 6 = 2 / (2+ 10 )= n+2/2 ፥ ጕܻ޲ࡩƄ࣊ ‫ࡩ࣠ ࢵޢ‬Ƅ‫ ݪ‬ყ჋मऒ 22 ‫ و‬17ܻዙ዗Ɛ ƌܻࣀ ࡩƄƥ ࢪ ࡷܻஉै Ƅ໠ ໄީყ჋౰౔ᑃᐽƆ࢖࣌ ܻެଐଌƄযওߥࡩƄ߄ ༬စ 46 ܸୃกܸ଍ଉŽ–ໝ ແࡦफउ –ࠣ‫ݶ‬ Ɖܸࢤ ެߕକଐƁƢࠣ‫ࡦ ࢽޟܸ ࢲޟ‬ƁƢࠣ‫ޟ‬පඓ ൷Ɓ౭౑ࢲ କଐƁƄ ܸࢲ ƁƋኃ ᒔᐺ። ጏ࢓‫ݕ‬ ࡦƁ࣬ ࣊ Ɓ࢓ߢೕಹኃ ቮƁ೾೥࢓଍ଉƁƄ ƁƋƁ߁ ࢟ ſ ƛ ܸߢࢽ ࡦƁ߁ ዖዄ Ž૊ સၫƋࣝެ࠾ኃ ቮƁƊܸެࢲ ଘଐƁ ። ፎକଐ߁ ެࢲ କଐƁƢ। ्࢟ ࣝࡦƁŻܸࢤ ߾ ଔଐƁ ‫ = ۑ‬2 /39 = 2 /( 22 + 17 ) ࣊ ‫ࡩ࣠ ࢵޢ‬Ƅ ࠣ‫ݽ‬ ࣝަ଍ଉƁƄ ܸ‫ަޟܸހ‬ᑀᐺࣇ ‫ࡦࣝ ࢲޟ‬Ɓ  ፥ ጘଘଓƄƣ ࣠‫ࡩߥܻރ‬ƄƢ߄ ႀ ޯఱ ఝࠦ‫ީ࣠ހ‬ଐଌƄƇ ܻ‫ީޢܻރ‬ᑃᐽ࣊ ‫ࡩ࣠ ࢵޢ‬Ƅƅ ܻࣀ ஊै ŗॻʠॻŪʦƃ¦ŗʭŽƃ¦¾ʦʟ x Ȗ ŕœ ʴ ƃ¦°¦ʙؒ ʯƃ¦ –ȊॻŪʦƃ¦ʔ ʻřʙř ŗॻʠॻŪʦƃ¦ŗʭŽƃ¦ŗǻ¦ʗŕ ȊॻŪʦƃ¦ Ȗ ŕœ ʴ ƃ¦°¦ʙؒ ʯƃ¦ –Ȗ šƚƃ¦°¦ʙؒ ʯƃ¦ -100 110 10 100-90 90-80 25 47 ࠣ‫ࡦެܸࡦޟ‬ƁƄ ܸ‫ަޟܸހ‬ᑀᐺࣇ ‫ࡦࣝ ࢲޟ‬Ɓ߁ ၱ ƢŽ ‫ ێ‬ƞ ܸޯࡴ 80-70 12 70-60 6 60-50 0 Ƈ ܻ‫ࡩߦޟ‬Ƅ ƎƄಒ౷ޯࡩƄ ࡠ இॅ Ɓ ፥ ጘଘଓƄơ ƥ߄ எै Ƅ፥ ጿ໠ ໄީࡷ࣠ࠧሯሡሁ ،߄ ၮܻࢧ ࡩƄ݊ ࡺࠖ ޯଐଌƄƧ ƎƄಒ౷ޯࡩƄơ ƥ߄ எै Ƅƣ ࣠ࡓ ‫ރ‬ ߄ၮܻࢢ ݊ ࡺࠖ ޯࡷƎƄಒ౷‫ކ‬ Ɔ߄ ၮܻࢢ ࠦ‫ࡷޯ ࠖࡺݘ‬Ƈ ܻ‫ߙޟ‬ ƎƄಒ౷ޯࡩƄ 6 70ࡼ ࡷࡣ ߘƀ 6 0 18 65 90 47 ࣇ ‫ࡦࣝ ࢲޟ‬ƁƂ ܸࢽ இॅ ƃ߁ ၫܸ࢟ ࠣ‫ݕ‬ ࡷࠓ ެࡴƄ ܸ‫ ێ ߖޜ‬ƞ Ƣ߁ ၱ 60ࡼ ࡷࡣ ߘƀ Ƈ ܻ‫ࡩߦޟ‬Ƅ 0 60-50 80ࡼ ࡷࡣ ߘƀ 12 80-70 90ࡼ ࡷࡣ ߘƀ 47 100ࡼ ࡷࡣ ߘƀ 25 70-60 90-80 100-90 ܸୃกܸ଍ଉŽ–ໝ ແࡦफउ –ࠣ‫ݶ‬ Ɖܸࢤ ެߕକଐƁƢࠣ‫ࡦ ࢽޟܸ ࢲޟ‬ƁƢࠣ‫ޟ‬පඓ ൷Ɓ౭౑ࢲ କଐƁƄ ܸࢲ ƁƋኃ ᒔᐺ። ጏ࢓‫ݕ‬ ࡦƁ࣬ ࣊ Ɓ࢓ߢೕಹኃ ቮƁ೾೥࢓଍ଉƁƄ ƁƋƁ߁ ࢟ ſ ƛ ܸߢࢽ ࡦƁ߁ ዖዄ Ž૊ સၫƋࣝެ࠾ኃ ቮƁƊܸެࢲ ଘଐƁ 100 ። ፎକଐ߁ ެࢲ କଐƁƢ। ्࢟ ࣝࡦƁŻܸࢤ ߾ ଔଐƁ 110ࡼ ࡷࡣ ߘƀ 10 110-100 50 = 2 /100 = 2 / Ƈ ƄƎƄಒ౷ޯࡩƄƖ ࣠ዙಣಝ ࣊ ‫ࡩ࣠ ࢵޢ‬Ƅ‫ ݪ‬ყ჋मऒ߄ ၴ ࣠‫ރ‬ ( 90- 80 ࠦ‫ࡩ࣠ ࢵޢ ࣐ޢ‬Ƅࠦ‫ࡩߦޟ‬Ƅ 80 ࠦ‫ࡩ࣠ ࢵޢ ࣐ޢ‬Ƅࠦ‫ࡩߦޟ‬Ƅࠦ‫ݹ‬Ƅਔ ৹ ( 65 ، 18 ໠ ໄ‫ܻކߥ ݊ހ‬ᐱ᏶ƀ߄ ༬စ ، 50݊ ߘ࣠ࡷࡼ ‫ݑ‬౰ేၯ ଘଓƄƌ࣠ࡺ‫ࡩݘ‬Ƅ፥ ጿ‫ ݟ‬ඳඝ ൷ 18 ࢇ ‫ࡩ ࣀܻހ‬ƄƎƄಒ౷ޯࡩƄ 65 ࢇ ࠁ ൎᑉᐽƄ ƎƄಒ౷ޯࡩƄ ࠦ‫ࡩߦޟ‬Ƅơ ࣠࣍ x ࢇ ‫ࡩ ࣀܻހ‬ƄƎƄಒ౷ޯࡩƄ –࣊ ‫ࡩ࣠ ࢵޢ‬Ƅ‫ ݪ‬ყ჋मऒ ࠦ‫ࡩ࣠ ࢵޢ ࣐ޢ‬Ƅࠦ‫ࡩߦޟ‬Ƅࠦ‫ݹ‬Ƅਔ ৹ ࣊ ‫ࡩ࣠ ࢵޢ‬ƀ ࢇ ‫ࡩ ࣀܻހ‬ƄƎƄಒ౷ޯࡩƄ –ࢇ ࠁ ൎᑉᐽƄƎƄಒ౷ޯࡩƄ 10 x 18- 50 + 80 ࣊ ‫ࡩ࣠ ࢵޢ‬Ƅ 18 -65 10 x 32 + 80 = 47 10 x ‫ ۑ‬+ 80 = ‫ ۑ‬+ 80 = ‫= ۑ‬ Mode] [ ƞ Ɓࣝީ଍ଉƁ ‫ێ‬ ‫ۑ‬Ƈ ܻ‫ࡩީޢܻރ‬Ƅࡼ ࡷࠦၮ࣠ዙಣಝ፥ ጿ (ܻၮ࣠‫ ࢸޢ‬౰౗ࡁ ଛଓƄ )ƎƄಒ౷‫ކ‬౰౗ࡁ ଛଓƄࠦࡺ‫ࡩߥޢ‬Ƅ࣠ࠧơ Ƅ࣠ެଐଌƄ ࠣ‫ݽ‬ ࣝަ଍ଉƁ౭ౄ‫ݐ‬ ࡦƁƄ ܸ‫ࡦަޟܸހ‬Ɓࡹ ࡴƞ Ɓࣝީ଍ଉƁƂ ܸࢽ ߾ କଐƢŽ ࠦ୆งଛଓƄযওߥᑃᐽơ Ƅ࣠ެଐଌƄ߄ ၴ ƥƀ : ( ‫ ێ‬ƞ ܸޯࡴ 6 ، 9، 11 ، 13 ، 15 ، ࡠ இॅ Ɓ 48 ܸୃกܸ଍ଉŽ–ໝ ແࡦफउ –ࠣ‫ݶ‬ Ɖܸࢤ ެߕକଐƁƢࠣ‫ࡦ ࢽޟܸ ࢲޟ‬ƁƢࠣ‫ޟ‬පඓ ൷Ɓ౭౑ࢲ କଐƁƄ ܸࢲ ƁƋኃ ᒔᐺ። ጏ࢓‫ݕ‬ ࡦƁ࣬ ࣊ Ɓ࢓ߢೕಹኃ ቮƁ೾೥࢓଍ଉƁƄ ƁƋƁ߁ ࢟ ſ ። ፎକଐ߁ ެࢲ କଐƁƢ। ्࢟ ࣝࡦƁŻܸࢤ ߾ ଔଐƁ ƛ ܸߢࢽ ࡦƁ߁ ዖዄ Ž૊ સၫƋࣝެ࠾ኃ ቮƁƊܸެࢲ ଘଐƁ Ꭸ፻যওߥࡩƄࡼ ࡷܻ‫ݹ‬ƀࠢ‫ރ‬ƀ߄ ༬စ ،Ƥൎ൉ၮƀƇ ܻ‫ࡩީޢܻރ‬Ƅࡼ ࡷ ،ƄƎƄಒ౷‫ކ‬౰౗ࡁ ଛଓƄࠦࡺ‫ࡩߥޢ‬Ƅࠢ‫ހܾރ‬ơ Ƅ࣠ެଐଌƄߎ ‫࣍ީߥܻࡩޯݘ࢖ݹ‬ ‫ۑ‬যওߥᑃᐽơ Ƅ࣠ެࡷ߄ ၴ ࣠‫ݹ‬ଘଓ፥ ፑܻޯࡩܻ‫ހ‬ƥƎಒ౷୓ข ơ Ƅ࣠ެଐଌƄ߄ ၴ ƥƀ : ( ‫ ێ‬ƞ ܸޯࡴ 8 ، 13 ، 7 ، 12 ، 8 ، 5 ،17 ، 8 ࡠ இॅ Ɓ 8፥ ፕ ܻၮ࣠‫ ࢸޢ‬ƄƎƄಒ౷‫ކ‬౰౗ࡁ ଘଓƄࠦࡺ‫ࡩߥޢ‬Ƅ ơ Ƅ࣠ެଐଌƄ߄ ၴ ƥƀ : ( ‫ ێ‬ƞ ܸޯࡴ 14 ، 8 ، 3 ، 10 ، 16 ، 14 ، 10 ، 7 :ࡠ இॅ Ɓ ౰౗ࡁ ƄƄƎಒ౷‫ࡺܻކ‬ᐱ᏶ଘଓ 14، 10ܻዙ዗ƣ ଘଓƄ࣠ެࡷ߄ ၴ ࣠‫ݹ‬ ࠣ‫ݽ‬ ࣝަ଍ଉƁবঐߢᑀᐺƞ Ɓࣝީ଍ଉƁƂ ܸࢽ ߾ ܸୃกܸ‫ކ‬ ࠦ‫ߙݘ‬Ƅ༟༎Ƅࠦߥ‫ࠧެܻ ࣍࢖ݹ‬Ƣ߄ ႀ ޯఱ ఝ‫ީ࣠ހࠦۑ‬ଐଌƄƇ ܻ‫ࡩީޢܻރ‬Ƅቿቱܻၰ ፥ ጿơ Ƅ࣠ެଐଌƄƌܻ༬࿻ ଗଓƆƌ߄ ‫ࡷޯݘ‬Ɵ ࢖࣍ ߄ ၴ ࣠‫ކ‬ ፥ ጘଘଓƄƣ ࣠‫ࡩߥܻރ‬ƄƢƄ߄ ႀ ޯࢵ ܻ‫ހ‬ơ Ƅ࣠ެଐଌƄƌܻ༬࿻ ƀࡼ ࡓ ೘಼ƥ ŗʭŽƃ¦¾ʦʟ x Ȗ šƚƃ¦°¦ʙؒ ʯƃ¦ ŗॻƃ¦ʦʹʸƃ¦ŗʭŽƃ¦ŗǻ¦ʗŕ ¾¦ʦʹʸƃ¦ (Ȗ šƚƃ¦°¦ʙؒ ʯƃ¦Ȗ ŕœ ʴ ƃ¦°¦ʙؒ ʯƃ¦ ƞ Ɓࣝީ଍ଉƁƉܸ༩࿸ ŽƄ Ɓࣝ࣍ ߽  ƄƎƄಒ౷‫ކ‬౰౗ࡁ ଘଓƄࠦ‫ࡩߦޟ‬Ƅࡣ ‫཯ކߥܻހ‬དྷࡩƄ፥ ፕ ƥࠦ‫ࡩޢ‬Ƅ࣠ެଐଌƄࠦ‫ࡩߦޟ‬Ƅ߄ ၴ ࣠‫ރ‬ ࠦ‫ࡩޢ‬Ƅ࣠ެଐଌƄࠦ‫ࡩߦޟ‬Ƅࠦ‫ݹ‬Ƅਔ ৹ƌ߄ ༨စ ࠦ‫ࡩޢ‬Ƅ࣠ެଐଌƄࠦ‫ࡩߦޟ‬ƄƎƄಒ౷ޯࡩࢇ ࠁ ൎᑉᐽƄƎƄಒ౷ޯࡩƄƥࢇ ‫ࡩ ࣀܻހ‬ƄƎƄಒ౷ޯࡩƄƌ߄ ༨စ ơ Ƅ࣠ެଐଌƄƅ ܻࣀ ஊै ƣ ࣠‫ࡩߥܻރ‬Ƅࢇ ީ࣐ ‫ރ‬ 49 ܸୃกܸ଍ଉŽ–ໝ ແࡦफउ –ࠣ‫ݶ‬ Ɖܸࢤ ެߕକଐƁƢࠣ‫ࡦ ࢽޟܸ ࢲޟ‬ƁƢࠣ‫ޟ‬පඓ ൷Ɓ౭౑ࢲ କଐƁƄ ܸࢲ ƁƋኃ ᒔᐺ። ጏ࢓‫ݕ‬ ࡦƁ࣬ ࣊ Ɓ࢓ߢೕಹኃ ቮƁ೾೥࢓଍ଉƁƄ ƁƋƁ߁ ࢟ ſ .I .II .III .IV ። ፎକଐ߁ ެࢲ କଐƁƢ। ्࢟ ࣝࡦƁŻܸࢤ ߾ ଔଐƁ ƛ ܸߢࢽ ࡦƁ߁ ዖዄ Ž૊ સၫƋࣝެ࠾ኃ ቮƁƊܸެࢲ ଘଐƁ Ɩ ࣠ࡺ഻വറƄ 50 45-40 4 ࠦ୆งଛଓƄƇ ܻ‫ࡩީޢܻރ‬Ƅࡼ ࡷơ Ƅ࣠ެଐଌƄ߄ ၴ ƥƀ : ( ‫ ێ‬ƞ ܸޯࡴ 40-35 35-30 10 21 Ƈ ƄƎƄಒ౷ޯࡩƄ 30-25 12 25-20 3 ƎƄಒ౷ޯࡩƄ ࠣ‫ݽ‬ ࣝަ଍ଉƁƄ ܸ‫ަޟܸހ‬ᑀᐺƞ Ɓࣝީ଍ଉƁƂ ܸࢽ ߾ ‫ ێ‬ƞ Ƣ߁ ၱ Ƈ ܻ‫ࡩߦޟ‬Ƅ 3 25-20 21 35-30 12 30-25 10 40-35 4 45-40 50 ࠦ‫ࡩߦޟ‬Ƅơ ࣠࣍ Ƈ ܻ‫ࡩߦޟ‬Ƅ Ɩ ࣠ࡺ഻വറƄ x ࢇ ࠁ ൎᑉᐽƄƎƄಒ౷ޯࡩƄ ࠦ‫ࡩޢ‬Ƅ࣠ެଐଌƄࠦ‫ࡩߦޟ‬Ƅࠦ‫ݹ‬Ƅਔ ৹ ơ Ƅ࣠ެଐଌƄ ࢇ ࠁ ൎᑉᐽƄƎƄಒ౷ޯࡩƄ ࢇ ‫ࡩ ࣀܻހ‬ƄƎƄಒ౷ޯࡩƄ 5 X 10 (10+12) + 30 ơ Ƅ࣠ެଐଌƄ ‫ ۑ = ۑ‬+ 30 = 22 / 50 + 30 ơ Ƅ࣠ެଐଌƄ ‫ࡦޯܸࡦݜ‬Ɓࡠ ࢤ ߣࡦƁ ௧ ோƋܸೕೆ Ƥܻ‫ރ‬ƌƀƧ ƎƄಒ౷ޯࡩƄơ ƥ߄ எै Ƅࡼ ࡷơ Ƅ࣠ެଐଌƄƥ࣊ ‫ࡩ࣠ ࢵޢ‬Ƅƥ፥ ጒܻࣀ ஊै Ƅ࣊ ࢵ ࣠ࡩƄ‫ ࠁ ࣀ ݪ‬Ƅ‫ێ‬1 Ƅ ܸ‫ࡦߣޜ‬Ɓ ƋƁಏ౴ެࡦƁ 13 – 16 15 19-22 51 10 – 13 16 -19 22-25 50 8 27 75 ܸୃกܸ଍ଉŽ–ໝ ແࡦफउ –ࠣ‫ݶ‬ Ɖܸࢤ ެߕକଐƁƢࠣ‫ࡦ ࢽޟܸ ࢲޟ‬ƁƢࠣ‫ޟ‬පඓ ൷Ɓ౭౑ࢲ କଐƁƄ ܸࢲ ƁƋኃ ᒔᐺ። ጏ࢓‫ݕ‬ ࡦƁ࣬ ࣊ Ɓ࢓ߢೕಹኃ ቮƁ೾೥࢓଍ଉƁƄ ƁƋƁ߁ ࢟ ſ ። ፎକଐ߁ ެࢲ କଐƁƢ। ्࢟ ࣝࡦƁŻܸࢤ ߾ ଔଐƁ ƛ ܸߢࢽ ࡦƁ߁ ዖዄ Ž૊ સၫƋࣝެ࠾ኃ ቮƁƊܸެࢲ ଘଐƁ 25 -28 54 31-34 18 28-31 36 34-37 9 37-40 7 ࠦ‫ݹ‬౮ే࡯༬စ ଘଓƄࠦ‫ݓ‬ ᑃᐽƄƆƌܻࡷ፥ ጿࡣ ࢧ ߦࡩƄơ ܻዙኸ ƀ፥ ጿܻީࡩܻ࣍ 52Ƈ ܻၴ Ǝƌࡣ ޲೘೉Ƥܻ‫ރ‬ƌƀƇ ܻ‫ࡩީޢܻރ‬Ƅ‫ۑ‬2 (5 ࠦ‫ࡩߦޟ‬Ƅơ ࣠࣍ Ƨ ƎƄಒ౷‫ކ‬ơ ƥ߄ ၴ ፥ ጿƇ ܻ‫ࡩީޢܻރ‬Ƅƅ ࣠‫ހ‬ ཯ཌࣀ ჱࡩƄƎƄಒ౷ޯࡩƄ߄ ၴ ƥƀ 19 18 15 35 32 30 35 37 30 39 25 24 18 17 17 8 26 8 17 24 18 19 12 29 19 21 21 27 18 24 29 25 5 30 36 10 16 30 28 26 31 22 ፥ ጒܻࣀ ஊै Ƅ࣊ ࢵ ࣠ࡩƄ߄ ၴ ƥƀ 12 21 28 28 42 15 19 21 35 22 : ࠦ‫ࡩޯܻࡩޢ‬ƄƇ ܻ‫ީޢܻރ‬ᑃᐽ࿃ ႓߄ ެಛಗƄ࣊ ࢵ ࣠ࡩƄ߄ ၴ ƥƀ ‫ێ‬3 ‫ ۑ‬، ‫ ۑ‬، ‫ ۑ‬، ‫ ۑ‬، ،7 ، 38 ، 1462 ، 125 :: ࠦ‫ࡩޯܻࡩޢ‬ƄƇ ܻ‫ީޢܻރ‬ᑃᐽ१ ॏߙƄ࣠ޯࡩƄ࣊ ࢵ ࣠ࡩƄ߄ ၴ ƥƀ‫ۑ‬4 ‫ ۑ‬، ‫ ۑ‬، ‫ ۑ‬، ‫ ۑ‬،5 ، 75 ، 475 ، 2574 51 ܸୃกܸ଍ଉŽ–ໝ ແࡦफउ –ࠣ‫ݶ‬ Ɖܸࢤ ެߕକଐƁƢࠣ‫ࡦ ࢽޟܸ ࢲޟ‬ƁƢࠣ‫ޟ‬පඓ ൷Ɓ౭౑ࢲ କଐƁƄ ܸࢲ ƁƋኃ ᒔᐺ። ጏ࢓‫ݕ‬ ࡦƁ࣬ ࣊ Ɓ࢓ߢೕಹኃ ቮƁ೾೥࢓଍ଉƁƄ ƁƋƁ߁ ࢟ ſ .I .II .III ƛ ܸߢࢽ ࡦƁ߁ ዖዄ Ž૊ સၫƋࣝެ࠾ኃ ቮƁƊܸެࢲ ଘଐƁ 52 ። ፎକଐ߁ ެࢲ କଐƁƢ। ्࢟ ࣝࡦƁŻܸࢤ ߾ ଔଐƁ ܸୃกܸ଍ଉŽˍ ໝ ແࡦफउˍ ࠣ‫ݶ‬ Ɖܸࢤ ެߕକଐƁƢࠣ‫ࡦ ࢽޟܸ ࢲޟ‬ƁƢࠣ‫ޟ‬පඓ ൷Ɓ౭౑ࢲ କଐƁƄ ܸࢲ ƁƋኃ ᒔᐺ። ጏ࢓‫ݕ‬ ࡦƁ࣬ ࣊ Ɓ࢓ߢೕಹኃ ቮƁ೾೥࢓଍ଉƁƄ ƁƋƁ߁ ࢟ ſ ። ፎକଐ߁ ެࢲ କଐƁƢ। ्࢟ ࣝࡦƁŻܸࢤ ߾ ଔଐƁ ƛ ܸߢࢽ ࡦƁ߁ ዖዄ Ž૊ સၫƋࣝެ࠾ኃ ቮƁƊܸެࢲ ଘଐƁ ࠦࡷܻၮƆಒ౷ߙܻެ‫཯ކݘ ࣐ޢ‬དྷࡩƄƥ ،‫ࡩ޲ܻࡩ ݟ‬Ƅࡣ ࢧ ߦࡩƄ፥ ጿܻᐳႩ Ƅ࢖‫ ࢵޯݘ‬ƄᎨ᎙ ،Ƈ ܻ࣐ ࢵ ࣠ޯଐଌƄࠦ‫ޢ‬ዙ዗ƀܻެ࣏ ࠁ ଘଓ ଘଓƥƇ ܻ‫ީޢܻރ‬ᑃᐽࠦ‫ݹ‬ഁ೨࢖ଐଌƄࠦၮ౮౑ࡩƄࢪ ყႾܻߥࡷ፥ ፕ Ƈ ܻ࣐ ࢵ ࣠ޯଐଌƄࡼ ࡓ ᚻࡩ‫ ࢵࠦۑ‬ƥƎ߄ ଐଌƄƇ ܻ‫ࡩީޢܻރ‬ƄƇ Ƅƌ࢖ߦࡷࡼ ‫ݑ‬ ‫ܻۑ‬ᐱᏹܻ‫ݹݘ‬Ə࣠‫ކ‬ᇯ ᇍ ࢸ ࡼ ‫ݑ‬Ɔಒ౷ߙƧ Ƅܻެ‫ކݘ ࣐ޢ‬ ƅ ƎܻߥޯࡩƄƥƀ߄ၮܻީޯࡩƄƦ ߄ ࡷࠦࢵ ƄƎƌ ،Ƈ ܻ‫ࡩީޢܻރ‬Ƅࡼ ࡷࠦၮ࣠ࡺ഻വറ [ Dispersion ]‫ ݝ‬ჴࣃ ჴࡩܻ‫ݹߥ ࢧ ߄ހ‬ Ƥ߃ ࠧบ คᇴᆝ ፥ ጒܻࣀ ஊै Ƅܻ࣐ࠩ ࢵ ƥࡼ ࡷ୍ࠦ෺࢖ߘƇ ܻ‫ࡩީޢܻރ‬Ƅบ คᇴᆝ ܻ൞ᇅᇍ ߙ ،፥ ጒܻࣀ ஊै Ƅܻ࣐ࠩ ࢵ ƥࡼ ‫ݑ‬Ƈ ܻ‫ࡩީޢܻރ‬ƄƤ߃ ಛಗ ‫ޯࠦۑ‬ჴࣃ ࡷƇ ܻ‫ࡩީޢܻރ‬Ƅ Ƥ߃ ࠧบ คᇴᆝ ፥ ጒܻࣀ ஊै Ƅܻ࣐ࠩ ࢵ ƥࡼ ‫ݑ‬Ɔ߄ ‫ހݘޢ‬Ƈ ܻ‫ࡩީޢܻރ‬Ƅบ คᇴᆝ ܻ൞ᇅᆝƥࠦޯჴࣃ ࡷ౰ేၩƇ ܻ‫ࡩީޢܻރ‬Ƅ Ƈ ܻ‫ݹܻࢠޢ‬༟༎ƄƟ ܻࣀ ࡷ፥ ጿƅ ൎ൉࣐ ࡩƄ ࡼ ࡷ໠ ໄޯ‫࣠ݑ‬ዙಣಝƇ ܻၴ Ǝƌ໠ ໄ୍෺Ƨ ኅ ቱƄ ،፥ ጘଘଓƄơ ܻ޲ଐଌƄܻ୐෺ኆ ቱࡼ ࡓ ‫ࡩޢ‬ 69 100 68 100 66 85 66 75 63 75 63 70 63 65 62 60 60 50 10 10 Aࠦၮ࣠ࡺ഻വറƄ B ࠦၮ࣠ࡺ഻വറƄ ܻࡺᐱᐟࡷᇯ ᆝ ࣊ ࢵ ࣠ޯࡷƥ‫ۑ‬640፥ ፑƄ࣠ޯࡩƄે ઻ၮ B ، A ࠦၮ࣠ࡺ഻വറƄࡼ ࡷᇯ ᇍ ࡩƇ ܻၴ Ǝኆ ቱƄƖ ࣠ዙಣಝƣ ƀࣉ ࠁ ൎ൉‫ރ‬ ࠦၮ࣠ࡺ഻വറƄƇ ܻ‫ܻރ‬୆ก፥ ጿܻࡺެყჅ‫ۑ‬69 ‫ و‬60໠ ໄ‫ހ‬ƆƎ࣠ࢧ ༨ீ A ࠦၮ࣠ࡺࠖ ൞ᒑᐽƇ ܻ‫ࡩީޢܻރ‬Ƅƣ Ƅ‫ۑ‬64Ƨ ƥܻఱ ఓ ‫ و‬10໠ ໄ‫ކߥ ݊ހ‬Ƈ ܻ‫ࡩީޢܻރ‬Ƅƣ ƀƧ ƀ ،Əܻޯያዪࠦၴ Ǝƌે ઻ၮൎ൉ࢧ ࠁ ໠ ໄީࡩܻ࣍ ƥƅ ࣠ࢵ ƎܻࡺᐱᏬኆ ቱ໠ ໄީࡩܻ࣍ ، B Ƅ߃ ࠧࠦࢵ ƄƎኆ ቱƥ‫ۑ‬B ࠦၮ࣠ࡺ഻വറܻ‫ރࠦހ‬Ǝܻߥࡷܻޯჴఴ ఠࡣ ߘƄ፟ ፑƥଘଓƄࠦၮ࣠ࡺ഻വറƄƇ ܻ‫ܻރ‬୆กƣ ƀࣉ ࠁ ൎ൉‫ރ‬ƥ‫ ۑ‬100 ‫ و‬،ࠦ୆෷ܻࢧ ࠁ ଘଓƄࢪ ყႾܻߥଐଌƄ࢛ ‫ހݘ‬Ƣ߄ ႀ ޯఱ ఝ‫ۑ‬ƞ ൎ൉ޯࠀ ଘଓƄƥƀ‫ ݝ‬ჴࣃ ჴࡩƄƥƀᄅ ྈܻႅ ޯࡩƄƢ߄ ၮࡼ ࡷƖ ࣠ެࡩƄ ፥ ጘଘଓᇴᆝ ܻᐳႩ Ƅ࢖‫ ࢵޯݘ‬Ƅࡼ ࡓ ೘಼ [ Range ]ƣ ߁ ଍ଉƁ ࠦࡺ‫ݓ࢖ߘޢ‬ ࢢ ƀƥࠦࡺ‫ߘޢ‬౰౎ࡁ ƀ໠ ໄ‫ހ‬Ɵ ࢖ߦࡩƄ࣠ࠧ ࠣࡷ‫ݐ࢓ߕޟ‬ ࢟ Ž–ࠣࡷ‫ߕޟ‬౭ో࠾ Ž [ R ]ƣ ߁ ଍ଉƁ ࠦߙܻࣀ ൞ᒑᐽ ƤƎܻࠩ࣌ ƀơ ൎ൉ၯ ࡼ ࡷࠦࡷܻၮƆಒ౷ߙ࣯ ࣐ ‫ݹݘ‬ƥ‫࡯ܻࠩۑ‬ᐳ႓Ƅƥ‫ ݝ‬ჴࣃ ჴࡩƄࢪ ყႾܻߥࡷ࣊ ఱ చƀࡼ ࡷƦ ߄ ଐଌƄ߄ ‫ݹݘ‬ƥ ‫ࠦۑ‬୆෷ܻࢧ ࠁ ଘଓƄቿቷࣀ ࡯ࣀ ࡩƄ፥ ጿ࢖࣍ ໠ ໄ‫ހ‬ቿቷࢢ ܻߦࡩƄ Ƈ ܻ‫ݹܻࢠޢ‬༟༎ƄƆƌܻࡷ፥ ጿƅ ൎ൉࣐ ࡩƄƇ ܻၴ Ǝƌࡣ ޲೘೉཯དྷࡩƄযওߥࡩƄ཯དྷ‫ࡺ࣠ݑ‬഻വറƦ ߄ ଐଌƄ߄ ၴ ƥƀ ( ‫ ێ‬ƞ ܸޯࡴ 66 ، 62 ، 63 ، 63 ، 63 ، 66 ، 68 ، 60 ، 60 ، 69 60 ، 10 ، 85 ، 75 ، 75 ، 70 ، 65 ، 0 ، 100 ، 100 53 ܸୃกܸ଍ଉŽ–ໝ ແࡦफउ –ࠣ‫ݶ‬ Ɖܸࢤ ެߕକଐƁƢࠣ‫ࡦ ࢽޟܸ ࢲޟ‬ƁƢࠣ‫ޟ‬පඓ ൷Ɓ౭౑ࢲ କଐƁƄ ܸࢲ ƁƋኃ ᒔᐺ። ጏ࢓‫ݕ‬ ࡦƁ࣬ ࣊ Ɓ࢓ߢೕಹኃ ቮƁ೾೥࢓଍ଉƁƄ ƁƋƁ߁ ࢟ ſ 4.1 ƛ ܸߢࢽ ࡦƁ߁ ዖዄ Ž૊ સၫƋࣝެ࠾ኃ ቮƁƊܸެࢲ ଘଐƁ 69 68 66 100 100 85 66 75 ። ፎକଐ߁ ެࢲ କଐƁƢ। ्࢟ ࣝࡦƁŻܸࢤ ߾ ଔଐƁ ܻ‫ ߄ݹ‬ၮܻࢧ ‫ކ‬Ƈ ܻ‫ࡩީޢܻރ‬Ƅว งमए 63 63 63 62 60 60 ࠦၮ࣠ࡺ഻വറƄ ፟ ፑƥଘଓƄ 75 70 65 60 10 10 ࠦ୆คܻ޲ࡩƄࠦၮ࣠ࡺ഻വറƄ ࠦࡺ‫ݓ࢖ߘޢ‬ ࢢ ƀ –ࠦࡺ‫ߘޢ‬౰౎ࡁ ƀ Ʀ ߄ ଐଌƄ 9 = 60 -69 Ʀ ߄ ଐଌƄ ፟ ፑƥଘଓƄࠦၮ࣠ࡺ഻വറƄ 90 = 10 -100 Ʀ ߄ ଐଌƄ ࠦ୆คܻ޲ࡩƄࠦၮ࣠ࡺ഻വറƄ ࠦ୆คܻ޲ࡩƄࠦၮ࣠ࡺ഻വറƄযওߘ፥ ጿ౰౎ࡁ ƀƦ ߄ ଐଌƄƥ‫ۑ‬໠ ໄޯߙ࢖࣐ ޯଐଌƄ໠ ໄޯࡺ‫୍ ࣊ހܻࡩߥޢ‬ง୹ ୪Ʀ ߄ ଐଌƄƣ ƀ໠ ໄࡩܻ޲ଐଌƄࡼ ࡷࣉ ࠁ ൎ൉‫ރ‬ ‫ࠦۑ‬୆คܻ޲ࡩƄࠦၮ࣠ࡺࠖ ൞ᒑᐽࠦީࣀ ჱࡩܻ‫ޯܻހ‬ჴఴ ఠ౰౗ࡁ ƀযওߥࡩƄƣ ƀ཯ཏ‫ݹݘ‬Ƅ߃ ࠧƥ‫ۑ‬፟ ፑƥଘଓƄࠦၮ࣠ࡺ഻വറƄযওߥ‫ރࠦހ‬Ǝܻߥࡷ ( Semi-Inter Quartile Range) പ ୥ୃ෾༜་Ɓƣ ߁ ଍ଉƁߋ ࢤ ‫ێ ێހ‬ Ɔƌ࣠ࠅ ࣠ଐଌƄƅ ࣠‫઻ ે اࡩݘޢ‬ၮ‫ݓ࡯ ݪ‬ ޯᑃᐽƢ߄ ႀ ޯఱ ఓƥள ஭࣐ ଐଌƄ‫ ݝ‬ჴࣃ ჴࡩƄࢪ ყႾܻߥࡷ߄ ၰ ƀ࣠ࠧഭ ୨୆ก༟༎ƄƦ ߄ ଐଌƄߎ ࢧ ‫ރ‬ ‫ۑ‬໠ ໄߙ࢖࣐ ࡩƄࡼ ࡷࠦߙ࢖࣐ ޯଐଌƄযওߥࡩƄ߄ ‫ީݘ‬ჴఱ ఓࠢ‫ރ‬ଛଓᑑ ᐽƍƥƦ ߄ ଐଌƄ፥ ጿ ቻቱ୷ ୪मअƥഭ ୨୆ก༟༎Ƅƞ Ƅ࢖༨စ ଘଓܻ‫ݹ ࢥܻހ‬ƀƞ ࢖‫ݹݘ‬ƥơ ƥଛଓƄƥ‫ࡩ޲ܻࡩ ݟ‬Ƅ໠ ໄ‫ݘޢ‬୆ก༟༎Ƅ໠ ໄ‫ހ‬Ɵ ࢖ߦࡩƄ࣊ ࢵ ࣠ޯࡷ࣠ࠧƥ ‫ۑ‬Q୷ ୪༟༎ܻ‫ހ‬ ،Ƣܻࣀ ߘƀ ࠦ‫ހݘ‬ƎƄ፟ ፑƄᄎ ႞ߥ‫ކ‬ƥƨܻ‫ࡩޢ‬Əܻ୐งƥƀƨܻ‫ ߄ݹ‬ၮܻࢧ ‫ކ‬Ƈ ܻ‫ࡩީޢܻރ‬Ƅว งमऒ ഭ୨୆ก༟༎ƄƦ ߄ ଐଌƄߎ ࢧ ‫ރ‬ƅ ܻࣀ ஊै ƥ ߄‫ީݘ‬ჴఱ ఓƧ ƀ Q1 ୷୪༟༎ܻ‫ހ‬ቻቱ୷ ୪मअƥ Lower Quartile ፟ ጕƌଛଓƄƥƀơ ƥଛଓƄ݊ ‫ހ‬༟༎Ƅࠦ‫ܻݹ‬ᐱᐟ᠊ܻᐱᐟࡷযঠᐱ᏶ ߄‫ީݘ‬ჴఱ ఓƧ ƀ Q3 ୷୪༟༎ܻ‫ހ‬ቻቱ୷ ୪मअƥ Upper Quartile ે ઻ၮଛଓƄƥƀ݊ ‫ހ‬Ƅ༟༎Ƅ݊ ‫ހ‬༟༎Ƅࠦ‫ݹ‬Ƅਔ ৹ƥơ ƥଛଓƄ݊ ‫ހ‬༟༎Ƅ ᇯ ᇍ ࣃ ࡩƄࠢެ‫ޢ‬ხႾሯሡሁ যওߥࡩƄƖ ܻ‫ހ‬Ǝƀࠦ‫މ‬ൎ൉‫މ‬ƥযওߥࡩƄ݊ ‫ހ‬Ǝ߄ ެ‫ݑ‬Q1 ‫ و‬Q3 ࠢ‫ ࣀܻހ‬ஊै Ɖ ܻޯ༨စ ܻࡷ࣠ࠧƥ݊ ‫ހ‬Ƅ༟༎Ƅ݊ ‫ހ‬༟༎Ƅ ፥ ጘଜଓƄ 54 ܸୃกܸ଍ଉŽ–ໝ ແࡦफउ –ࠣ‫ݶ‬ Ɖܸࢤ ެߕକଐƁƢࠣ‫ࡦ ࢽޟܸ ࢲޟ‬ƁƢࠣ‫ޟ‬පඓ ൷Ɓ౭౑ࢲ କଐƁƄ ܸࢲ ƁƋኃ ᒔᐺ። ጏ࢓‫ݕ‬ ࡦƁ࣬ ࣊ Ɓ࢓ߢೕಹኃ ቮƁ೾೥࢓଍ଉƁƄ ƁƋƁ߁ ࢟ ſ ። ፎକଐ߁ ެࢲ କଐƁƢ। ्࢟ ࣝࡦƁŻܸࢤ ߾ ଔଐƁ ƛ ܸߢࢽ ࡦƁ߁ ዖዄ Ž૊ સၫƋࣝެ࠾ኃ ቮƁƊܸެࢲ ଘଐƁ പ ୥ୃ෾༜་Ɓƣ ߁ ଍ଉƁߋ ࢤ ‫ހ‬Ƃ ܸࢽ ߾ Ƅ Ɓࣝ࣍ ߽  ƨܻ‫ࡩޢ‬Əܻ୐งƥƀ ƨܻ‫ ߄ݹ‬ၮܻࢧ ‫ކ‬যওߥࡩƄว งमऒ .i ે ઻ၮଛଓƄ ‫ࡩ޲ܻࡩ ݟ‬Ƅ݊ ୆ก༟༎Ƅ୍ࠦงƎƥ ፟ ጕƌଛଓƄ ơ ƥଛଓƄ݊ ୆ก༟༎Ƅ୍ࠦงƎƌܻ༬࿻ Ƃ .ii ࠦ୆งଘଓƄࠦߘൎ൉‫ࡩݘ‬ƄƢƄ߄ ႀ ޯࢵ ܻ‫ހ‬ n+1 /4 ፟ ጕƌଛଓƄ݊ ୆ก༟༎Ƅ୍ࠦงƎ 3( n+1 ) /4 ે ઻ၮଘଓƄ݊ ୆ก༟༎Ƅ୍ࠦงƎ Q3 ، Q1 যওߘƌܻ༬࿻ Ƃ .iii ፥ ጘଘଓƄƣ ࣠‫ࡩߥܻރ‬ƄƢƄ߄ ႀ ޯࢵ ܻ‫ ހ‬ഭ ୨୆ก༟༎ƄƦ ߄ ଐଌƄߎ ࢧ ‫ ރ‬ഭ ୨୆ก༟༎Ƅƞ Ƅ࢖༨စ ଘଓƄ߄ ၴ ࣠‫ ރ‬.iv Q = (Q3 – Q1)/2 Ƈ ܻၴ Ǝƌࡣ ޲೘೉཯དྷࡩƄ 1 ơ ܻ޲ଐଌƄ፥ ጿƆƌƎƄ࣠ࡩƄƇ ܻ‫ީޢܻރ‬ᑃᐽഭ ୨୆ก༟༎ƄƦ ߄ ଐଌƄߎ ࢧ ‫ ߄ރ‬ၴ ƥƀ ‫ ۑ‬ơ ܻ޲ࡷ ‫ۑ‬Ƈ ܻ‫ݹܻࢠޢ‬༟༎ƄƟ ܻࣀ ࡷ፥ ጿƅ ൎ൉࣐ ࡩƄࡼ ࡷ໠ ໄޯ‫࣠ݑ‬ዙಣಝ 66 ، 62 ، 63 ، 63 ، 63 ، 66 ، 68 ، 60 ، 60 ، 69 ፟ ፑƥଘଓƄࠦၮ࣠ࡺ഻വറƄƇ ܻ‫ܻރ‬୆ก 60، 10 ، 85 ، 75 ، 75 ، 70 ، 65 ، 0 ، 100 ، 100ࠦ୆คܻ޲ࡩƄࠦၮ࣠ࡺ഻വറƄƇ ܻ‫ܻރ‬୆ก 69 100 68 100 66 85 66 75 63 75 = (n+1)/4 = (10+1)/4 = 11/4 = 2.75 63 70 ܻ‫ ߄ݹ‬ၮܻࢧ ‫ކ‬Ƈ ܻ‫ࡩީޢܻރ‬Ƅว งमएࡣ ஊै Ƅ 63 65 62 60 60 60 10 10 : ፜ ፎƢକଐƁࠣၫࣝࡷസലമƁƄ ܸ‫ୃܸހ‬෾ : ፟ ፑƥଘଓƄࠦၮ࣠ࡺ഻വറƄ ፟ ጕƌଛଓƄ݊ ୆ก༟༎Ƅ୍ࠦงƎ = 3 (n+1)/4 ે ઻ၮଘଓƄ݊ ୆ก༟༎Ƅ୍ࠦงƎ = 3(10+1)/4 = 33/4= 8.25 (‫ࡩ޲ܻࡩ ݟ‬Ƅƥ፥ ጕܻ޲ࡩƄࡴ ߘ༟༎Ƅ໠ ໄ‫ހ‬Ɵ ࢖ߦࡩƄ ࠦࡺ‫ ߘޢ‬0.75 + 2ࠢީყ჋मऒƧ ኅ ቱƄࡴ ߘ༟༎Ƅࠦࡺ‫ ߘޢ‬፟ ጕƌଘଓƄ݊ ୆ก༟༎Ƅࠦࡺ‫ߘޢ‬ Q1 = 60 +0.75 ( 62-60 ) = 60 + 1.5 = 61.5 ࡼ ࡷܻ޲ࡩƄ໠ ໄޯࡺ‫ࡩߥޢ‬Ƅ໠ ໄ‫ހ‬Ɵ ࢖ߦࡩƄ ‫ࡩ޲ܻࡷ ࡼ ۑ‬Ƅࠢީყ჋मऒƧ ኅ ቱƄࡴ ߘ༟༎Ƅࠦࡺ‫઻ ે ߘޢ‬ၮଘଓƄ݊ ୆ก༟༎Ƅࠦࡺ‫ߘޢ‬ Q3 = 66 + 0.25 ( 86-66) = 66 + 0.5 = 66.5 55 ݊ ࢵ ܻޯࡩƄƥ ܸୃกܸ଍ଉŽ–ໝ ແࡦफउ –ࠣ‫ݶ‬ Ɖܸࢤ ެߕକଐƁƢࠣ‫ࡦ ࢽޟܸ ࢲޟ‬ƁƢࠣ‫ޟ‬පඓ ൷Ɓ౭౑ࢲ କଐƁƄ ܸࢲ ƁƋኃ ᒔᐺ። ጏ࢓‫ݕ‬ ࡦƁ࣬ ࣊ Ɓ࢓ߢೕಹኃ ቮƁ೾೥࢓଍ଉƁƄ ƁƋƁ߁ ࢟ ſ ። ፎକଐ߁ ެࢲ କଐƁƢ। ्࢟ ࣝࡦƁŻܸࢤ ߾ ଔଐƁ ƛ ܸߢࢽ ࡦƁ߁ ዖዄ Ž૊ સၫƋࣝެ࠾ኃ ቮƁƊܸެࢲ ଘଐƁ : Q പ ୥ୃ෾༜་Ɓƛ Ɓ࢓༥ဂ କଐƁ പ ୥ୃ෾༜་Ɓƣ ߁ ଍ଉƁߋ ࢤ ‫ހ‬ Q =( Q3 – Q1 )/2 = (66.5 – 61.5) /2 = 5/2 = 2.5 = (n+1)/4 = (10+1)/4 = 11/4 = 2.75 ࠦ୆คܻ޲ࡩƄࠦၮ࣠ࡺ഻വറƄƇ ܻ‫ ࣀީࠦࡩީޢܻރ‬ჱࡩܻ‫ހ‬ : ࠦ୆คܻ޲ࡩƄࠦၮ࣠ࡺ഻വറƄ ፟ ጕƌଛଓƄ݊ ୆ก༟༎Ƅ୍ࠦงƎ ે ઻ၮଘଓƄ݊ ୆ก༟༎Ƅ୍ࠦงƎ = 3 (n+1)/4 = 3(10+1)/4 = 33/4= 8.25 (‫ࡩ޲ܻࡩ ݟ‬Ƅƥ፥ ጕܻ޲ࡩƄࡴ ߘ༟༎Ƅ໠ ໄ‫ހ‬Ɵ ࢖ߦࡩƄ ࠦࡺ‫ ߘޢ‬0‫ۑ‬75 + 2ࠢީყ჋मऒƧ ኅ ቱƄࡴ ߘ༟༎Ƅࠦࡺ‫ ߘޢ‬፟ ጕƌଘଓƄ݊ ୆ก༟༎Ƅࠦࡺ‫ߘޢ‬ (݊ ࢵ ܻޯࡩƄƥࡼ ࡷܻ޲ࡩƄࡴ ߥࡩƄ໠ ໄ‫ހ‬Ɵ ࢖ߦࡩƄ ‫ࡩ޲ܻࡷ ࡼ ۑ‬Ƅࠢީყ჋मऒƧ ኅ ቱƄࡴ ߘ༟༎Ƅࠦࡺ‫઻ ે ߘޢ‬ၮଘଓƄ݊ ୆ก༟༎Ƅࠦࡺ‫ߘޢ‬ Q1 = 10 + 0.75(60-10 ) = 10 + 0.75 (50) = 10 + 37.5 =47.5 Q3 = 85 + 0.25( 100-85 ) = 85 + 0.25(15) = 85 + 3.75 = 88.75 (Q) ഭ ୨୆ก༟༎Ƅƞ Ƅ࢖༨စ ଘଓƄ ഭ ୨୆ก༟༎ƄƦ ߄ ଐଌƄߎ ࢧ ‫ރ‬ƣ ƍƀ Q = (Q3 – Q1) /2 =( ‫ ۑ‬-47.5)/2 = 20.6 ࠣ‫ݽ‬ ࣝަ଍ଉƁƄ ܸ‫ަޟܸހ‬ᑀᐺപ ୥ୃ෾༜་Ɓƣ ߁ ଍ଉƁߋ ࢤ ‫ ހ‬ ࠦ୆งଘଓƄࠦ‫ީ࣠ހ‬ଐଌƄƇ ܻ‫ީޢܻރ‬ᑃᐽ ഭ ୨୆ก༟༎Ƅƞ Ƅ࢖༨စ ଘଓƄ ഭ ୨୆ก༟༎ƄƦ ߄ ଐଌƄߎ ࢧ ‫ ߄ރ‬ၴ ƥƀ ( ‫ ێ‬ƞ ܸޯࡴ 100-80 11 80-60 20 15 40-20 20 – 0 10 4 Ǝܻዙኸ ଘଓƄƇ ܻ‫ߙޟ‬ ƒ ܻ༧ೞ ଘଓƄƌ߄ ၮ ࡠ இॅ Ɓ ߁ ၫܸࢤ ࡦƁ݇ ࡷࠓ ެ଍ଉƁƋƁಏ౴ެࡦƁ ߄ၮܻࢢ ݊ ࡺࠖ ޯࡷƎƄಒ౷‫ކ‬ ƒ ܻ༧ೞ ଘଓƄƌ߄ ၮ 14 10 40-20 49 20 80-60 4 29 60 56 60-40 4 15 11 Ǝܻዙኸ ଘଓƄƇ ܻ‫ߙޟ‬ 20 -0 60-40 100-80 ܸୃกܸ଍ଉŽ–ໝ ແࡦफउ –ࠣ‫ݶ‬ Ɖܸࢤ ެߕକଐƁƢࠣ‫ࡦ ࢽޟܸ ࢲޟ‬ƁƢࠣ‫ޟ‬පඓ ൷Ɓ౭౑ࢲ କଐƁƄ ܸࢲ ƁƋኃ ᒔᐺ። ጏ࢓‫ݕ‬ ࡦƁ࣬ ࣊ Ɓ࢓ߢೕಹኃ ቮƁ೾೥࢓଍ଉƁƄ ƁƋƁ߁ ࢟ ſ ƛ ܸߢࢽ ࡦƁ߁ ዖዄ Ž૊ સၫƋࣝެ࠾ኃ ቮƁƊܸެࢲ ଘଐƁ ። ፎକଐ߁ ެࢲ କଐƁƢ। ्࢟ ࣝࡦƁŻܸࢤ ߾ ଔଐƁ 15 = 4 / 60 = 4 / n ፟ ጕƌଘଓƄ݊ ୆ก༟༎Ƅ‫ ݪ‬ყ჋मऒ 45 = ( 4 / 60 ) 3 = (4 / n ) 3 ે ઻ၮଘଓƄ݊ ୆ก༟༎Ƅ‫ ݪ‬ყ჋मऒ (60 – 40) = 15ƎƄಒ౷ޯࡩƄࡣ ‫཯ކߥܻހ‬དྷࡩƄࠦ‫ࡩߦޟ‬Ƅ ( Q1 )፟ ጕƌଘଓƄ݊ ୆ก༟༎Ƅࠦࡺ‫ߘޢ‬ Q1 = L + N/4 – c.f x i f Q3 = L + 3N/4 – c.f x i f Q1 = 40 + ( 15 – 14 ) X 20 15 Q1 = 40 + 1/15 (20) Q1 = 41.33 Q3 = 60 + ( 45-29)/20 = 60 + 16 = 76 Q = Q3 – Q1 / 2 Q = (76 – 41.33 )/2 = 17.33 Coefficient of Q =( Q3 – Q1 )/ (Q3 + Q1) =( 76- 41.33)/ 76+ 41.33 = 34.67/117.33 = 0.29 ‫ ۑ‬ഭ ୨୆ก༟༎Ƅƞ Ƅ࢖༨စ ଘଓƄࡣ ࡷܻ‫ࡷݘ‬ ࣇ ࢲ ࣝެ଍ଉƁƛ Ɓ࢓༥ဂ କଐƁ‫ێ ێ‬ (‫࣠ ࠅ ݪ‬ଐଌƄ ၖ࿟ ܻެࡩƄ ࠦߥ࡯࣐ ଐଌƄ ࠦࡺ‫ࡩߥޢ‬Ƅ Ɩ ࣠ዙಣಝ ၖ࿟ ܻ‫ ࠧ࣠ ރ‬Ƈ ܻߙƄ࢖༨စ ଘଓƄ ࣊ ࢵ ࣠ޯࡷ ƥƀ ࣊ ࢵ ࣠ޯଐଌƄ ƞ Ƅ࢖༨စ ଘଓ‫أ‬ Dm ୷ ୪༟༎ܻ‫ހ‬ቻቱ୷ ୪मअƥūܻࠧƌ߄ၮે ઻ၮܻࡷ࣠ࣀ ߥࡷ፥ ጒܻࣀ ஊै Ƅܻ࣐ࠩ ࢵ ƥࡼ ‫ݑ‬যওߥࡩƄƇ ܻߙƄ࢖༨စ ଘଓ ፥ ጘଘଓƄƣ ࣠‫ࡩߥܻރ‬ƄƢƄ߄ ႀ ޯࢵ ܻ‫ީ࣠ހࠦހ‬ଐଌƄ౰ే‫ݓ‬ ࡩƄƇ ܻ‫ީޢܻރ‬ᑃᐽ࣊ ࢵ ࣠ޯଐଌƄƞ Ƅ࢖༨စ ଘଓƄ‫ ࣀ ݪ‬༨࿻ ƥ Dm = ∑ │ Xi - X̅│/ n ، i = 1 ،2 ،3 ،4 ، …..، n ƣ ƀ‫ࠁޢ ݟ‬ ቿቷࣀ ࡯ࣀ ࡩƄযওߘ X 57 ܸୃกܸ଍ଉŽ–ໝ ແࡦफउ –ࠣ‫ݶ‬ Ɖܸࢤ ެߕକଐƁƢࠣ‫ࡦ ࢽޟܸ ࢲޟ‬ƁƢࠣ‫ޟ‬පඓ ൷Ɓ౭౑ࢲ କଐƁƄ ܸࢲ ƁƋኃ ᒔᐺ። ጏ࢓‫ݕ‬ ࡦƁ࣬ ࣊ Ɓ࢓ߢೕಹኃ ቮƁ೾೥࢓଍ଉƁƄ ƁƋƁ߁ ࢟ ſ ። ፎକଐ߁ ެࢲ କଐƁƢ। ्࢟ ࣝࡦƁŻܸࢤ ߾ ଔଐƁ ƛ ܸߢࢽ ࡦƁ߁ ዖዄ Ž૊ સၫƋࣝެ࠾ኃ ቮƁƊܸެࢲ ଘଐƁ যওߥࡩƄƤ߃ ಛಗ፥ ጒܻࣀ ஊै Ƅ࣊ ࢵ ࣠ࡩƄ X̅ ࠦ୆งଘଓƄযওߥᑃᐽ࣊ ࢵ ࣠ޯଐଌƄƞ Ƅ࢖༨စ ଘଓƄ‫ ࠁ ࣀ ݪ‬ƀ ( ‫ ) ۑ‬ơ ܻ޲ࡷ 34 ، 25 ، 23 ، 15 ، 13 ࡣ ஊै Ƅ X̅ = ∑ X / n = 110 /5 = 22 X X- X̅ 15 -7 13 যওߥᑃᐽ፥ ጒܻࣀ ஊै Ƅ࣊ ࢵ ࣠ࡩƄଘଓƥƀ߄ ၴ ࣠‫ރ‬ .i ፥ ጒܻࣀ ஊै Ƅ࣊ ࢵ ࣠ࡩƄࡼ ‫ݑ‬যওߥࡩƄƇ ܻߙƄ࢖༨စ Ƅ ߄ ၴ ࣠‫ރ‬ .ii │X - X̅ │ -9 23 9 7 1 25 1 3 34 3 12 12 0 ∑ 32 ࣊ ࢵ ࣠ޯଐଌƄƞ Ƅ࢖༨စ ଘଓƄ߄ ၴ ࣠‫ۑރ‬iii Dm = ∑ │ Xi - X̅ │/ n Dm = 32/5 = 6.4 Ƈ ܻၴ Ǝኆ ቱƧ ƎƄಒ౷ޯࡩƄ݊ ‫ݹ‬Ə࣠ޯࡩƄࡣ ޲೘೉཯དྷࡩƄࠦ‫ࡩޯܻࡩޢ‬ƄƇ ܻ‫ࡩީޢܻރ‬Ƅࡼ ࡷ࣊ ࢵ ࣠ޯଐଌƄƞ Ƅ࢖༨စ ଘଓƄ߄ ၴ ƥƀ ( ‫ ێ‬ƞ ܸޯࡴ 60 – 50 50 – 40 40 – 30 30 – 20 5 58 11 19 10 ‫ۑ‬ƆƎƄƌଗଓƄƃ ƌܻީࡷƟ ܻࣀ ࡷ፥ ጿܻީࡩܻ࣍ 50 20- 10 4 10 – 0 1 Ƈ ܻၴ Ǝኆ ቱƄ Ƈ ƄƎƄಒ౷ޯࡩƄ ܸୃกܸ଍ଉŽ–ໝ ແࡦफउ –ࠣ‫ݶ‬ Ɖܸࢤ ެߕକଐƁƢࠣ‫ࡦ ࢽޟܸ ࢲޟ‬ƁƢࠣ‫ޟ‬පඓ ൷Ɓ౭౑ࢲ କଐƁƄ ܸࢲ ƁƋኃ ᒔᐺ። ጏ࢓‫ݕ‬ ࡦƁ࣬ ࣊ Ɓ࢓ߢೕಹኃ ቮƁ೾೥࢓଍ଉƁƄ ƁƋƁ߁ ࢟ ſ ። ፎକଐ߁ ެࢲ କଐƁƢ। ्࢟ ࣝࡦƁŻܸࢤ ߾ ଔଐƁ ƛ ܸߢࢽ ࡦƁ߁ ዖዄ Ž૊ સၫƋࣝެ࠾ኃ ቮƁƊܸެࢲ ଘଐƁ ࡠ இॅ Ɓ ፥ ጘଘଓƄƣ ࣠‫ࡩߥܻރ‬ƄƢ߄ ႀ ޯఱ ఝ࣊ ࢵ ࣠ޯଐଌƄƞ Ƅ࢖༨စ ଘଓƄƅ ܻࣀ ஊै Dm = ∑ f │Xi- X̅ │/ ∑ f Ơ Ž‫߾ޟݜ‬ Ƈ ƄƎƄಒ౷ޯࡩƄƖ ࣠ዙಣಝ Ƈ ܻ‫ࡩߦޟ‬Ƅ 0 – 10 1 f 10 – 20 4 20 – 30 10 30 – 40 19 40 – 50 11 50 – 60 5 ∑ 50 X̅ = ∑ f X / ∑ f ∑f ࠣ‫ݽ‬ ࣝަ଍ଉƁƄ ܸ‫ަޟܸހ‬ᑀᐺࣇ ࢲ ࣝެ଍ଉƁƛ Ɓ࢓༥ဂ କଐƁƂ ܸࢽ ߾ 4.1 ƞ Ƣ߁ ၱ ( X) ࠦ‫ࡩߦޟ‬Ƅഁ೨୹ ୪ 5 15 25 35 45 55 - f*X X - X̅ │X - X̅ │ f│X- X̅ │ 60 -20 20 80 5 250 665 495 275 -30 -10 0 10 20 1750 - 30 10 0 10 20 - 30 100 0 110 100 420 = 1750 / 50 = 35 Dm = ∑ f │Xi - X̅ │/ ∑ f = 420 / 50 =8.4 [ Standard Deviation ]Ƥ Ƌܸ‫ݕޟ‬ ଍ଉƁƛ Ɓ࢓༥ဂ କଐƁ‫ ێ‬1.4 ࠢߦࢢ ƥƥࠢޯߘኆ ቱƄ࢖࣏ ‫ ރ‬،ܻࡷƄ߄ ႀ ޯࢵ Ƅܻࠧ౰౗ࡁ ƀƥ‫ ݝ‬ჴࣃ ჴࡩƄࢪ ყႾܻߥࡷࡴ ࠧƀࡼ ࡷƧ Ǝܻ‫ݘޢ‬ଐଌƄƞ Ƅ࢖༨စ ଘଓƄ౰౎ޯ‫ݹݘ‬ ‫ ࢵࠦۑ‬ƥƎ߄ ଐଌƄƆ࢖ܻ࣏ࠧ ࡩƄযওߘ‫ ݝ‬ჴࣃ ჴࡩ߰ ‫ࡩ ࢧ ࠒޢ‬Ƅ [ s]୷ ୪༟༎ܻ‫ހ‬ቻቱ୷ ୪मअƥ௪ ௎ܻީޯᑃᐽഭ ୨୆ก౰౔ࡩƄƎ߃ எै Ƅ࣠ࠧ Ƥ Ƌܸ‫ݕޟ‬ ଍ଉƁƛ Ɓ࢓༥ဂ କଐƁߋ ‫ݕ࢓ݶ‬ ‫ރ‬ 59 ܸୃกܸ଍ଉŽ–ໝ ແࡦफउ –ࠣ‫ݶ‬ Ɖܸࢤ ެߕକଐƁƢࠣ‫ࡦ ࢽޟܸ ࢲޟ‬ƁƢࠣ‫ޟ‬පඓ ൷Ɓ౭౑ࢲ କଐƁƄ ܸࢲ ƁƋኃ ᒔᐺ። ጏ࢓‫ݕ‬ ࡦƁ࣬ ࣊ Ɓ࢓ߢೕಹኃ ቮƁ೾೥࢓଍ଉƁƄ ƁƋƁ߁ ࢟ ſ ። ፎକଐ߁ ެࢲ କଐƁƢ। ्࢟ ࣝࡦƁŻܸࢤ ߾ ଔଐƁ ƛ ܸߢࢽ ࡦƁ߁ ዖዄ Ž૊ સၫƋࣝެ࠾ኃ ቮƁƊܸެࢲ ଘଐƁ ୷୪मअƥ፥ ጒܻࣀ ஊै Ƅܻ࣐ࠩ ࢵ ƥࡼ ‫ݑ‬যওߥࡩƄƇ ܻߙƄ࢖༨စ ƄƇ ܻ‫ ୹ހݘ‬୪࣊ ࢵ ࣠ޯࡷ࣠ࠧ [ Variance]௧ ோܸަެࡦƁߋ ‫ݕ࢓ݶ‬ ‫ރ‬ [ s2 ]୷ ୪༟༎ܻ‫ހ‬ቻቱ Ƥ Ƌܸ‫ݕޟ‬ ଍ଉƁƛ Ɓ࢓༥ဂ କଐƁƂ ܸࢽ ߾  ፥ ጘଘଓƄƣ ࣠‫ࡩߥܻރ‬ƄƢ߄ႀ ޯఱ ఝࠦ‫ީ࣠ހ‬ଐଌƄ౰ే‫ݓ‬ ࡩƄƇ ܻ‫ީޢܻރ‬ᑃᐽƧ Ǝܻ‫ݘޢ‬ଐଌƄƞ Ƅ࢖༨စ ଘଓƄƅ ܻࣀ ஊै S=ට S=ට ∑(ࢄିࢄҧ) ࢔ ፥ ጘଘଓƄƣ ࣠‫ࡩߥܻރ‬ƄƢ߄ႀ ޯఱ ఝࠦ‫ީ࣠ހ‬ଐଌƄƇ ܻ‫ީޢܻރ‬ᑃᐽƧ Ǝܻ‫ݘޢ‬ଐଌƄƞ Ƅ࢖༨စ ଘଓƄƅ ܻࣀ ஊै ∑௙(௑ି௑ҧ) ∑௙ ࠦ୆งଘଓƄযওߥᑃᐽƧ Ǝܻ‫ݘޢ‬ଐଌƄƞ Ƅ࢖༨စ ଘଓƄ߄ ၴ ƥƀ ‫) ێ‬ƞ ܸޯࡴ 22 ، 15 ، 13 ، 12 ، 8 ࡠ இॅ Ɓ ࠣ‫ݽ‬ ࣝަ଍ଉƁ౭ౄ‫ݐ‬ ࡦƁ Ƅ ܸ‫ަޟܸހ‬ᑀᐺƤ Ƌܸ‫ݕޟ‬ ଍ଉƁƛ Ɓ࢓༥ဂ କଐƁƂ ܸࢽ ߾ 4.2 ƞ Ƣ߁ ၱ X 8 13 -2 4 -1 15 X̅ = (X -X̅)2 -6 12 22 X - X̅ 1 ∑X = 70 8 ∑ ( X - X̅ ) = 0 36 1 1 64 ∑(X - X̅)2 = 106 ∑X/n = 70/5 = 14 S2 = ∑ ( X - X̅)2 / n 60 ܸୃกܸ଍ଉŽ–ໝ ແࡦफउ –ࠣ‫ݶ‬ Ɖܸࢤ ެߕକଐƁƢࠣ‫ࡦ ࢽޟܸ ࢲޟ‬ƁƢࠣ‫ޟ‬පඓ ൷Ɓ౭౑ࢲ କଐƁƄ ܸࢲ ƁƋኃ ᒔᐺ። ጏ࢓‫ݕ‬ ࡦƁ࣬ ࣊ Ɓ࢓ߢೕಹኃ ቮƁ೾೥࢓଍ଉƁƄ ƁƋƁ߁ ࢟ ſ ። ፎକଐ߁ ެࢲ କଐƁƢ। ्࢟ ࣝࡦƁŻܸࢤ ߾ ଔଐƁ ƛ ܸߢࢽ ࡦƁ߁ ዖዄ Ž૊ સၫƋࣝެ࠾ኃ ቮƁƊܸެࢲ ଘଐƁ = 106/5 = 21.2 -------------------------˃ [Variance] S = √ ∑ ( X - X̅)2 / n = √106 /5 = √21.2 = 4.6 ------˃ Standard deviation (S.D ) ƞ ൎ൉ޯࠀ ଘଓƄࡣ ࡷܻ‫ࡷݘ‬ƥƧ Ǝܻ‫ݘޢ‬ଐଌƄƞ Ƅ࢖༨စ ଘଓƄƥ௪ ௎ܻީޯࡩƄ߄ ၴ ƥƀ ‫ߘ ࡴ ۑ‬Ǝơ ܻ޲ଐଌƄࡼ ࡷ ( ‫ ێ‬ƞ ܸޯࡴ ࡠ இॅ Ɓ Ƈ ܻ‫ࡩߦޟ‬Ƅ f 0 – 10 1 10 – 20 4 20 – 30 10 30 – 40 19 40 – 50 11 50 – 60 5 50 ∑ X̅ = ∑ࢌࢄ ࠣ‫ݽ‬ ࣝަ଍ଉƁƄ ܸ‫ަޟܸހ‬ᑀᐺƤ Ƌܸ‫ݕޟ‬ ଍ଉƁƛ Ɓ࢓༥ဂ କଐƁƢ௧ ோܸަެࡦƁƂ ܸࢽ ߾ ‫ ێ‬ƞ Ƣ߁ ၱ ( X ࠦ‫ࡩߦޟ‬Ƅഁ೨୹ ୪ f * X X - X̅ (X- X̅ )2 f (X- X̅ )2 15 -20 400 5 5 25 250 35 45 55 - 60 665 495 275 -30 -10 0 10 20 1750 - 900 900 100 1000 0 100 400 - 1600 0 1100 2000 6600 ∑ࢌ X̅ = 1750 / 50 = 35 S2 = ට ∑ࢌ(ࢄିࢄҧ) ∑ࢌ S2 = 6600/50 = 132 -------------------˃ [ variance ] S = √ 6600/50 =√ 132 = 11.48 (S.D) C.V = 61 ఙ x 100 ଡ଼ጟ ௪ ௎ܻީޯࡩƄ Ƨ Ǝܻ‫ݘޢ‬ଐଌƄƞ Ƅ࢖༨စ ଘଓƄ : ፥ ጘଘଓƄƣ ࣠‫ݹݘ ࣐ ࣩހܻࡩߥܻރ‬ƞ ൎ൉ޯࠀ ଘଓƄࡣ ࡷܻ‫ࡷݘ‬ : ƣ ƀ‫ࠁޢ ݟ‬ ƞ ൎ൉ޯࠀ ଘଓƄࡣ ࡷܻ‫ ࡷݘ‬coefficient of variation ) C.V ܸୃกܸ଍ଉŽ–ໝ ແࡦफउ –ࠣ‫ݶ‬ Ɖܸࢤ ެߕକଐƁƢࠣ‫ࡦ ࢽޟܸ ࢲޟ‬ƁƢࠣ‫ޟ‬පඓ ൷Ɓ౭౑ࢲ କଐƁƄ ܸࢲ ƁƋኃ ᒔᐺ። ጏ࢓‫ݕ‬ ࡦƁ࣬ ࣊ Ɓ࢓ߢೕಹኃ ቮƁ೾೥࢓଍ଉƁƄ ƁƋƁ߁ ࢟ ſ ። ፎକଐ߁ ެࢲ କଐƁƢ। ्࢟ ࣝࡦƁŻܸࢤ ߾ ଔଐƁ ƛ ܸߢࢽ ࡦƁ߁ ዖዄ Ž૊ સၫƋࣝެ࠾ኃ ቮƁƊܸެࢲ ଘଐƁ C.V = C.V = ఙ x 100 ଡ଼ጟ ଵଵ.ସ଼ ଷହ . 100 = 32.8 ݇‫ݽ‬ Ɓ༜་Ɓࡠ ࢤ ߣࡦƁ௧ ோƋܸೕೆ Ƈ ܻ‫ީޢܻރ‬ᑃᐽഭ ୨୆ก༟༎Ƅƞ Ƅ࢖༨စ ଘଓƄࡣ ࡷܻ‫ࡷݘ‬ƥ ഭ ୨୆ก༟༎Ƅƞ Ƅ࢖༨စ ଘଓƄ ഭ ୨୆ก༟༎ƄƦ ߄ ଐଌƄߎ ࢧ ‫ ߄ރ‬ၴ ƥƀ (1) : ࠦ୆งଘଓƄࠦ‫ީ࣠ހ‬ଐଌƄ -70 -60 -50 -40 -30 -20 -10 -0 Ƈ ܻ‫ߙޟ‬ 14 16 40 10 0 20 40 10 ƎƄಒ౷ޯࡩƄ 80 70 60 50 40 30 20 10 Ǝܻዙኸ ଘଓƄ Ƥܻ‫ރ‬ƌƄƇ ܻ‫ࡩީޢܻރ‬Ƅࡼ ࡷ (2 ( 10 ࠦ‫ࡩߦޟ‬Ƅơ ࣠࣍ Ƨ ƎƄಒ౷‫ކ‬ơ ƥ߄ ၴ ፥ ጿƇ ܻ‫ࡩީޢܻރ‬Ƅƅ ࣠‫ހ‬ ࣊ ࢵ ࣠ޯଐଌƄƞ Ƅ࢖༨စ ଘଓƄ߄ ၴ ƥƀ ௪ ௎ܻީޯࡩƄ‫ ࠁ ࣀ ݪ‬ƀ Ƨ Ǝܻ‫ݘޢ‬ଐଌƄƞ Ƅ࢖༨စ ଘଓƄ‫ ࠁ ࣀ ݪ‬ƀ 72 74 40 60 82 115 41 61 65 83 68 69 104 80 79 52 73 59 81 72 50 53 110 66 62 ƞ ൎ൉ޯࠀ ଘଓƄࡣ ࡷܻ‫ ߄ࡷݘ‬ၴ ƥƀ 49 76 77 79 84 90 50 67 84 76 52 103 79 96 78 42 51 79 64 86 56 69 78 65 70 94 ܸୃกܸ଍ଉŽ–ໝ ແࡦफउ –ࠣ‫ݶ‬ Ɖܸࢤ ެߕକଐƁƢࠣ‫ࡦ ࢽޟܸ ࢲޟ‬ƁƢࠣ‫ޟ‬පඓ ൷Ɓ౭౑ࢲ କଐƁƄ ܸࢲ ƁƋኃ ᒔᐺ። ጏ࢓‫ݕ‬ ࡦƁ࣬ ࣊ Ɓ࢓ߢೕಹኃ ቮƁ೾೥࢓଍ଉƁƄ ƁƋƁ߁ ࢟ ſ .I .II .III .IV .V ። ፎକଐ߁ ެࢲ କଐƁƢ। ्࢟ ࣝࡦƁŻܸࢤ ߾ ଔଐƁ ƛ ܸߢࢽ ࡦƁ߁ ዖዄ Ž૊ સၫƋࣝެ࠾ኃ ቮƁƊܸެࢲ ଘଐƁ 63 ܻߥࢵ ܻ୐ง౰౗ࡁ Ƅࠦ‫ࡩޯܻࡩޢ‬ƄƇ ܻ‫ࡩީޢܻރ‬ƄƧ Ƅƞ ൎ൉ޯࠀ ଘଓƄࡣ ࡷܻ‫ࡷݘ‬ƢƄ߄ ႀ ޯࢵ ܻ‫ ހ‬.3 A 32 28 47 63 71 39 10 60 96 14 B 19 31 48 53 67 90 10 62 40 80 ܸୃกܸ଍ଉŽ–ໝ ແࡦफउ –ࠣ‫ݶ‬ Ɖܸࢤ ެߕକଐƁƢࠣ‫ࡦ ࢽޟܸ ࢲޟ‬ƁƢࠣ‫ޟ‬පඓ ൷Ɓ౭౑ࢲ କଐƁƄ ܸࢲ ƁƋኃ ᒔᐺ። ጏ࢓‫ݕ‬ ࡦƁ࣬ ࣊ Ɓ࢓ߢೕಹኃ ቮƁ೾೥࢓଍ଉƁƄ ƁƋƁ߁ ࢟ ſ ƛ ܸߢࢽ ࡦƁ߁ ዖዄ Ž૊ સၫƋࣝެ࠾ኃ ቮƁƊܸެࢲ ଘଐƁ ļ [ Probabilities] ņ ʆſ 64 ። ፎକଐ߁ ެࢲ କଐƁƢ। ्࢟ ࣝࡦƁŻܸࢤ ߾ ଔଐƁ ļ ŅųȧŌ ſʆ ń ܸୃกܸ଍ଉŽˍ ໝ ແࡦफउˍ ࠣ‫ݶ‬ Ɖܸࢤ ެߕକଐƁƢࠣ‫ࡦ ࢽޟܸ ࢲޟ‬ƁƢࠣ‫ޟ‬පඓ ൷Ɓ౭౑ࢲ କଐƁƄ ܸࢲ ƁƋኃ ᒔᐺ። ጏ࢓‫ݕ‬ ࡦƁ࣬ ࣊ Ɓ࢓ߢೕಹኃ ቮƁ೾೥࢓଍ଉƁƄ ƁƋƁ߁ ࢟ ſ ƛ ܸߢࢽ ࡦƁ߁ ዖዄ Ž૊ સၫƋࣝެ࠾ኃ ቮƁƊܸެࢲ ଘଐƁ ። ፎକଐ߁ ެࢲ କଐƁƢ। ्࢟ ࣝࡦƁŻܸࢤ ߾ ଔଐƁ Ƅ କଐܸࡷެ߾ କଐƁ። ጼࠣ‫ ࢲܸ ࢲޟ‬Žবঐࠤܸߣࡴ 1 ࡼ ࡷƇ ߃ၯ ƀၖ࿏ ܻ୓คƥƀࠦߙ߄ ࢧ ᑃᐽࠦႅ ‫ޢ‬ჴެࡁ ƈ ߄ ༨ဈ ཯དྷࡩƄƈ ƌƄ࣠ஊै ƄƇ ଘଓܻࡺޯࠁ ଘଓƄࠦ‫ރ ࣏࢖ݹ‬ơ ƥܻެჴ჆ ‫ۑ‬ƅ Ǝܻ༬ဈ žƄ༇໮ ƀࠦႅ ‫ޢ‬ჴ჈ƥƀƇ ܻެ‫ݑޢ‬ ࠦႅ ‫ޢ‬ჴ჈፟ ፑƄƧ ƌ࣓‫཯ކ‬དྷࡩƄƇ ƄžƄ༇໮ ଘଓƄࠦၮ࣠ዙಣಝƥƀࠦ‫ࡩݘࡺ࡯ޢ‬Ƅ፥ ፕ : [ Experiment ] ࠣ‫ݽ‬ ࢓ࠓ ެࡦƁ .I ࠦ‫࢖ހ‬༬ဈ žƄ༇໮ ܾ‫ެܻހ‬ዙኼƄƍƄ،‫ۑ‬ƈ ƌƄ࣠ࠁ ܻᐱᐕࡺఱ ఝƣ ƀࡼ ࡓ ೘಼౰౗ࡁ ƀƥƀࠦႅ ‫ޢ‬ჴ჈፟ ፑƄƧ ƌ࣓‫࢖ހࠦކ‬༬ဈ ᇯ ᆝƥ ‫ࡷܻۑ‬ ࣓ީჱޯࡩƄࡼ ࡓ ೘಼Ƨ ƀ ،ࠦߙƥ࢖‫ࡷݘ‬ƥƆቧሼ࣓ࡷƣ ࣠ࡓ ‫ࠦކ‬ႅ ‫ޢ‬ჴެࡩƄƣ ƀ፥ ፕ ፟ ፑƥଘଓƄ ،໠ ໄޯެࡓ ያዪ໠ ໄޯࠖ ‫ޢ‬ჴ჈Ơ ܻެࠧƣ ܾߙ ‫ࡩ ࣏ܻࠧ࢖ة‬Ƅࣩ ࡺఱ ఠƥ‫ۑ‬Ɔቧሼ࣓ࡷƆƎ࣠ࢧ ‫ࠦހ‬ႅ ‫ޢ‬ჴެࡩܻ‫ީ࣓ހ‬ჱޯࡩƄࡼ ࡓ ೘಼ଘଓࠢ‫ރ‬ƀ፥ ፕ ࠦ୆คܻ޲ࡩƄƥ‫ۑ‬Ɔቧሼ࣓ࡷƆƎ࣠ࢧ ‫ࠦހ‬ႅ ‫ޢ‬ჴެࡩܻ‫ހ‬ Ɔ୹ ୪ƌ౰౑ࡩƄ࢖ᄽ஛ ፥ ጹमऌܻެዙኼƄƍƀ‫ۑ‬ƌ౰౑ࡩƄ࢖ᄽ஛ ፥ ጹƎࠦ‫࢖ހ‬༬ဈ ፥ ጿᑑ ᐽƍે ઻ၮơ ܻ޲ࡷƥ‫ ࠁޯࡺܻࡩޢࠦۑ‬Ƅቿቱܻஊै ƄƤ߃ ࠧ፥ ጿ 4ࡴ ߘ༟༎ƄƥƄൎ൉޲ࡷ 6ࡴ ߘ༟༎ƄƎ࣠ࠩ࣌ ƣ ࣠ࡓ ޯࢵ ࠦႅ ‫ޢ‬ჴެࡩƄƣ ܾ‫ހ‬Ɔቧሼ࣓ࡷƆƎ࣠ࢧ ‫ެީܾހ‬ჴ჈ƣ ƀ݊ ‫ޯ ࣐ޢ‬ఱ ఝଘଓƆ߄ ၰ Ƅƥ ଘଓ‫ ࠁޯࡺܻࡩޢࠦ ࠁޢ ݟ‬ଘଓƄƅ Ǝܻႅ ޯࡩƄે ઻ၮഁ೨౰౔‫ࠧޯࡺܻࡷެܻݹ‬Ƅƣ ܾߙƇ ଘଓܻࡺޯࠁ ଘଓƄࠦࢵ ƄƎƌ፥ ጿ‫ܻࠧۑ‬౰ేၩƥƀ 3 ‫أو‬ ‫ߙࠦ ࢵ࡯ߦܻۑ‬ƥ࢖‫ࡩޯ ࠖ࢖ހࠦࡷݘ‬Ƅၖ࿏ ܻ୓คƣ ࣠ࡓ ‫ކ‬ Ɔ߄ ၰ Ƅƥࠦႅ ‫ޢ‬ჴ჈࢖࣏ࠩ ‫ކ‬Ɔ߄ ၰ ƄƥƆ୹ ୪ƌ౰౑ࡩƄ࢖ᄽ஛ ፥ ጹ༟༎ࠦ‫࢖ހ‬༬ဈ žƄ༇໮ ƀ߄ ެ‫ [ ݑ‬Event ]ƅ ߁ இॅ Ɓ .II ‫ ߄މܻۑ‬ၰ ࣩ ࡺఱ ఠࠦެࡓ ያዪၖ࿏ ܻ୓คƆ߄ ၮ໠ ໄ‫ࡷ ࡼހ‬ ࠦ‫ࡩޯܻࡩޢ‬Ƅࠦެࡓ ࡺଐଌƄၖ࿏ ܻޯެࡩƄ࢖࣏ࠩ ‫ކ‬Ɔ߄ ၰ ƄƥƆ୹ ୪ƌ౰౑ࡩƄ࢖ᄽ஛ ፥ ጹƎࠦ‫࢖ހ‬༬ဈ ( ‫ ێ‬ƞ ܸޯࡴ { 6 ، 5 ، 4 ، 3 ، 2 ،1 } {ࠦ‫ૻܻހ‬૓ƥƀƆƎ࣠ࢢ } ࠦ‫ࡩޯܻࡩޢ‬Ƅࠦެࡓ ࡺଐଌƄၖ࿏ ܻޯެࡩƄ࢖࣏ࠩ ‫ކ‬ƌ࣠ߥެࡩƄࠦ‫ߘ ࣐ݘ‬፥ ጹƎࠦ‫࢖ހ‬༬ဈ ( ‫ ێ‬ƞ ܸޯࡴ ‫ ߄މܻۑ‬ၰ Ƥൎ൉ၮƀ໠ ໄ୓ก࢖ࠖ ޯࡩƄၖ࿏ ܻ୓คࡼ ࡷࠦႅ ‫ޢ‬ჴ჈Ƨ Ƅࣩ ࡺఱ ఠƥ ࠦ‫ࡩޢ‬ƥଘଓƄƈ Ƅ߄ ၰ ଘଓƄࠦၮ࣠ዙಣಝࠦެ‫ࡩݘޢ‬Ƅ Ɨ Ƅ࢖ߙ žܻࢥ ߙࡣ ޲೘಼ [ Sample space]ࠣީ‫ݕޟ‬ ࡦƁŻܸࢢ ߖ .III ‫ޯ ࠖ࢖ހࠦۑ‬ᑃᐽ ፥ ፕ Ɔ߄ ၰ ƄƥƆ୹ ୪ƌ౰౑ࡩƄ࢖ᄽ஛ ፥ ጹƎࠦ‫཯ࡩޯ ࠖ࢖ހ‬ཏ‫ࡩݘޢ‬Ƅžܻࢥ ߦࡩƄ ( ‫ ێ‬ƞ ܸޯࡴ { E6 ، E5 ، E4 ، E3 ، E2 ، E1 } E1 E4 65 E2 E5 E3 E6 ܸୃกܸ଍ଉŽ–ໝ ແࡦफउ –ࠣ‫ݶ‬ Ɖܸࢤ ެߕକଐƁƢࠣ‫ࡦ ࢽޟܸ ࢲޟ‬ƁƢࠣ‫ޟ‬පඓ ൷Ɓ౭౑ࢲ କଐƁƄ ܸࢲ ƁƋኃ ᒔᐺ። ጏ࢓‫ݕ‬ ࡦƁ࣬ ࣊ Ɓ࢓ߢೕಹኃ ቮƁ೾೥࢓଍ଉƁƄ ƁƋƁ߁ ࢟ ſ ። ፎକଐ߁ ެࢲ କଐƁƢ। ्࢟ ࣝࡦƁŻܸࢤ ߾ ଔଐƁ ƛ ܸߢࢽ ࡦƁ߁ ዖዄ Ž૊ સၫƋࣝެ࠾ኃ ቮƁƊܸެࢲ ଘଐƁ ၖᆆ ƌ፥ ጿƢ߄ ႀ ޯఱ ఠ཯དྷࡩƄƇ ܻၮ࣠ࡺ഻വറƄે ઻ၮࠦ‫ࡩݘࡺ࡯ޢ‬Ƅ፟ ፑƂ∪ƌܻ༨ဈ ଘଓƄ߰ ࡯࣐ ࢧ ࡷ౰ేఴ ఓ Ɖܸ༥စ କଐƁ .IV ྲ Ⴋ ܻެ‫ݑ‬Ƨ ࣠༨ဈ Ɔਔ ৲߄ ၴ ࠦၮ࣠ዙಣಝે ઻ၮơ ࣠ࢧ ॶॱ५໠ ໄޯ‫࣠ݑ‬ዙಣಝ ્ ઻‫ܻݹ‬೘ೃࡣ ޲๻๬ใ B ، Aƈ Ƅ߄ ၰ ଘଓƄƣ ܾߙƌ౰౑ࡩƄ࢖ᄽ஛ ܻެ‫ࡷޢ‬ƎƄƍƄ ( ‫ ێ‬ƞ ܸޯࡴ A={4،6،2} B={1،2،3} A U B = { 1، 2 ، 3 ، 4 ، 5 ، 6 } ݊ ࣍ ܻߥ‫ކ‬፟ ፑƂƎܻఴƬఓ‫ۑ‬໠ ໄޯ‫࣠ݑ‬ዙಣಝ໠ ໄ‫ހ‬በሼ౰౔ࣃ ଐଌƄྲ Ⴋ ܻެ‫ࡩݘ‬Ƅࠦၮ࣠ዙಣಝ࣠ࠧ : ( Intersection )݇ ࣊ ܸߢެࡦƁ .V ∩୷ ୪༟༎ܻ‫ ހ‬໠ ໄޯ‫ࡺ࣠ݑ‬഻വറƄ B ، A ݊ ࣍ ܻߥ‫ކ‬ƣ ܻߙ ‫ ۑ‬ơ ܻ޲ଐଌƄ፥ ጿ 2 }، 6 ،A = { 4 3 }، 2 ، B = { 1 A ∩B={2} [ Mutually Exclusive Evens ]ࠣ‫ެީܸߖޟ‬଍ଉƁƅ Ɓ߁ ၭ କଐƁ .VI ે ઻ၮƧ ࣠ޯ༨࿻ ଘଓ B ∩ A ݊ ࣍ ܻߥޯࡩƄƣ ᇴᆝ ƄƍƄࠦ‫މܻࡷޯެܻߙޢ‬Ƅ߄ ၰ Ƅ B ‫ و‬A ƈ Ƅ߄ ၰ ଘଓƄƣ ࣠ࡓ ‫ކ‬ ‫ۑ‬ƈ Ƅ߄ ၰ Ƅࠦ‫ݹ‬Ƅ 4 } ،A = { 3 : บ คᇴᆝ Ƅƍ ‫ ێ ( ا‬ƞ ܸޯࡴ 9 } ، 8 ،B = { 7 A ∩ B ={∅} A ، B ࠦ‫ࡷޯެܻߙޢ‬ƈ Ƅ߄ ၰ ƀ ơ ܻࡺޯࠁ ଘଓƄƥƀࠦࢢ ࢖ߦࡩƄࢪ ߦ‫ܻރ‬ಛಗ཯དྷࡩƄƈ Ƅ߄ ၰ ଘଓƄ፥ ፕ ቿቷ‫ޯࡺܻމ‬ଐଌƄƈ Ƅ߄ ၰ ଘଓƄ ቼቴ‫ެࡷܸކ‬଍ଉƁƅ Ɓ߁ ၭ କଐƁ .VII ‫࢖ހࠦࡷܻۑ‬༬ဈ žƄ༇໮ ƀ߄ ެ‫ݑ‬Ǝ࣏࣠ࠩ ࡩƄ፥ ጿ ࣩ ࡺఱ ఠ ،ࠦ‫ૻܻހ‬૓ƥƀƆƎ࣠ࢢ ໠ ໄ‫ ߄މ‬ஊै Ƅ‫ ࠁޢ ݟ‬،Ɔ߄ ၰ ƄƥƆ୹ ୪ƌ࣠ߥެࡩƄࠦ‫ߘ ࣐ݘ‬፥ ጹƎ߄ ެ‫ ێ (ݑ‬ƞ ܸޯࡴ 1/2Ƨ ƥܻఱ ఓܻᐱᐟࡷᇯ ᆝƥࠦ‫ܻހ‬ຨ຀ࡩƄƎ࣠ࠩ࣌ ơ ܻࡺޯࠁ ƄƧ ƥܻఱ ఓƆƎ࣠ࢧ ࡩƄƎ࣠ࠩ࣌ ơ ܻࡺޯࠁ Ƅƣ ଘଓቿቷ‫މܻࡷޯࡺܻމ‬Ƅ߄ ၰ ƀ ƢܻߘƎଘଓƄࡣ ޲೘೉Ƈ ଘଓܻၰ ࠦޯࢵ Ơ ܻެࠧƣ ܻߙƆ߄ ၰ ƄƥƆ୹ ୪ƌ౰౑ࡩƄࠦ‫ߘ ࣐ݘ‬፥ ጹƎ߄ ެ‫ ێ (ݑ‬ƞ ܸޯࡴ ቿቷ‫މܻࡷޯࡺܻމ‬Ƅ߄ ၰ ƀࣩ ࡺఱ ఠƥ ، 1/6Ƨ ƥܻఱ ఠܻࡺᐱᐟࡷᇯ ᆝ Ǝ࣠ࠩ࣌ ơ ܻࡺޯࠁ Ƅƣ ܻߙ ،ƌ౰౑ࡩƄ࢖ᄽ஛ ે ઻ၮƆ࢖ܻ࣏ࠧ ࡩƄ 66 ܸୃกܸ଍ଉŽ–ໝ ແࡦफउ –ࠣ‫ݶ‬ Ɖܸࢤ ެߕକଐƁƢࠣ‫ࡦ ࢽޟܸ ࢲޟ‬ƁƢࠣ‫ޟ‬පඓ ൷Ɓ౭౑ࢲ କଐƁƄ ܸࢲ ƁƋኃ ᒔᐺ። ጏ࢓‫ݕ‬ ࡦƁ࣬ ࣊ Ɓ࢓ߢೕಹኃ ቮƁ೾೥࢓଍ଉƁƄ ƁƋƁ߁ ࢟ ſ ። ፎକଐ߁ ެࢲ କଐƁƢ। ्࢟ ࣝࡦƁŻܸࢤ ߾ ଔଐƁ ƛ ܸߢࢽ ࡦƁ߁ ዖዄ Ž૊ સၫƋࣝެ࠾ኃ ቮƁƊܸެࢲ ଘଐƁ Ƅ କଐܸࡷެ߾ କଐƁ߁ ၫƁࣝߕ Ɖ࢓ߣީࡴƅ ߁ ၭ ƞ ܸࡷެ߾ Ɓ 1.2 .1 ࠦࢢ ࢖ߦࡩƄࠦ‫ݹ‬ƥܻࣀ ჴࡷၖ࿏ ܻ୓ค໠ ໄ‫ ࡷ ࡼހ‬nA ܻࠧƌ߄ ၮƟ ࢖࣐ ‫ހ‬ƈ ߄ ༨࿻ ƣ ƀࡼ ࡓ ೘಼ Aƈ ߄ ஊै Ƅƣ ᇴᆝ ƄƍƄ P(A) = nA / N : A ơ ܻࡺޯࠁ Ƅƣ ܾߙ‫ۑ‬Nܻࠧƌ߄ ၮƖ ࣠ߘ࣠ࡩƄ፥ ጿ Ơ Ž‫߾ޟݜ‬ Aƈ ߄ ஊै ƄƖ ࣠ߘƥơ ܻࡺޯࠁ Ƅ P(A) AܻᐱᏳ݊ ߥ‫ݹ‬ƣ Ƅࡼ ࡓ ೘಼཯དྷࡩƄƟ ࢖࣐ ࡩƄƌ߄ ၮ n Ǝ࣏࣠ࠩ ࡩƄ፥ ጿࠦࢢ ࢖ߦࡩƄࠦ‫ݹ‬ƥܻࣀ ჴଐଌƄၖ࿏ ܻޯެᑃᐽ્ ᇣᇍ ࡩƄƌ߄ ‫ࡩݘ‬Ƅ N ƣ ܻ‫ ߄މ‬ၰ ƣ ൎ൉޲೘಼ࠦ‫ܻހ‬ຨ຀ࡩƄƥƆƎ࣠ࢧ ࡩƄƣ ܾߙƆ߄ ၰ ƄƥƆ୹ ୪ƌ࣠ߥެࡩƄࠦ‫ߘ ࣐ݘ‬žܻߥࡩƄ߄ ެ‫ ێ (ݑ‬ƞ ܸޯࡴ P ( H ) = nH / N = 1/2 ƣ ƀƧ ƀ‫ۑ‬Ɩ ࣠ߘ࣠ࡩƄࠦࢢ ࢖ߘࢪ ߦ‫ࡺܻރ‬ಛಗƣ ܻ༬ဈ ܻ‫ރ‬ P ( T ) = nT /N = 1/2 P (H) + P(T) = 1 ࠣႂ ‫ޟ‬ჱჅ  P(A) = 1 บ คᇴᆝ ƄƍƄ‫ݹߥ ݊ۑ‬ƣ ƀࡼ ࡓ ೘಼ଘଓ A ƈ ߄ ஊै Ƅƣ ܾߙ ، P(A) = 0 บ คᇴᆝ Ƅƍƀ P(A′ ୷ ୪༟༎ܻ‫ ހ‬A ƈ ߄ ஊै ƄƖ ࣠ߘƥƢ߄ ၮơ ܻࡺޯࠁ ଘଓܻ‫ ୷ރ‬୪ƎƄƍƀƥ‫ۑ‬Ɩ ࣠ߘ࣠ࡩƄቧሼ࣓ࡷ A ƈ ߄ ஊै Ƅƣ ܾߙ P ( A ) + P(A′ ) = 1 1 ≤ P(A) ƣ ܾߙ) 1 ‫ و‬0໠ ໄ‫ ހ‬P(A) ࠦࡺ‫ߘޢ‬Ɗ ƥƄ౰౔‫ކ‬ƥ ≤1 : ‫أي أن‬ ࠦ‫ݹ‬ƥܻࣀ ჴࡷၖ࿏ ܻ୓คࠦޯࢵ Ơ ܻެࠧƣ ܾߙƆ߄ ၰ ƄƥƆ୹ ୪౮ేႁ ޯࡷ౰ేၩƌमए࢖ᄽ஛ žܻߥࡩƄ߄ ެ‫ ێ (ݑ‬ƞ ܸޯࡴ ƣ ܾߙᎨ᎜ࡼ ࡷƥ 6 ، 5 ، 4 ، 3 ، 2 ، 1 ،፥ ፕ ƥƖ ࣠ߘ࣠ࡩƄ፥ ጿࠦࢢ ࢖ߦࡩƄ P(1) + P(2) + P(3) + P(4) + P(5) + P(6) =1 P(1) = P(2) = P(3) = P(4) = P(5) = P(6) = 1/6 67 ܸୃกܸ଍ଉŽ–ໝ ແࡦफउ –ࠣ‫ݶ‬ Ɖܸࢤ ެߕକଐƁƢࠣ‫ࡦ ࢽޟܸ ࢲޟ‬ƁƢࠣ‫ޟ‬පඓ ൷Ɓ౭౑ࢲ କଐƁƄ ܸࢲ ƁƋኃ ᒔᐺ። ጏ࢓‫ݕ‬ ࡦƁ࣬ ࣊ Ɓ࢓ߢೕಹኃ ቮƁ೾೥࢓଍ଉƁƄ ƁƋƁ߁ ࢟ ſ ። ፎକଐ߁ ެࢲ କଐƁƢ। ्࢟ ࣝࡦƁŻܸࢤ ߾ ଔଐƁ ƛ ܸߢࢽ ࡦƁ߁ ዖዄ Ž૊ સၫƋࣝެ࠾ኃ ቮƁƊܸެࢲ ଘଐƁ 1ࡴ ߘ༟༎ƄƎ࣠ࠩ࣌ Ƣ߄ၮơ ܻࡺޯࠁ Ƅƣ ࣠ࡓ ‫ݹ‬ƥ P(1′) = 1 – P(1) = 1- 1/6 = 5/6 P(1) + P(1′) = 1/6 +5/6 = 6/6 = 1 Ƅ କଐܸࡷެ߾ କଐƁ݇ ዖወ ƃ߁ ၫܸߕ .2 ‫ۑ‬౰౗ࡁ ƀƥƀ໠ ໄ‫ ߄މ‬ၰ ࡼ ࡷ߄ ၰ ƀƥƖ ࣠ߘƥơ ܻࡺޯࠁ Ƅਔ ৲߄ ႁ ޯࡩƇ ଘଓܻࡺޯࠁ ଘଓƄ݊ ዙዋ Ɔ߄ ၮܻߘƢ߄ ႀ ޯఱ ఠ ࠦ‫ࡷޯެܻߙޢ‬ƈ ƌƄ࣠ஊै Ƅƣ ࣠ࡓ ‫ۑߙߥ ߄ކ‬ƈ ƌƄ࣠ஊै Ƅ໠ ໄ‫ߘࠦހ‬ൎ൉‫ࡩݘ‬Ƅࠦ‫઻ ે ࣍ީޢݘ‬ၮơ ܻࡺޯࠁ ଘଓƄƅ ܻࣀ ࠁ ߎ ߘ࣠୓෺ƥ ܻࡷ߄ ެ‫ࡷޯެܻߙޢࠦݑ‬౰ేၩƥƀ ݊ ࣍ ܻߥ‫ ߄ކ‬ၴ ࣠‫ކ‬ଘଓƧ Ƅ ܻ‫ࡷݘ‬ƈ ƌƄ࣠ஊै ƄƖ ࣠ߘ࣠ࡩࠦ୆คᇴᇍ ࡷƀ߄ ၴ ࣠‫ކ‬ଘଓ‫ࠁޢ ݟ‬ ݊ ࣍ ܻߥ‫ ߄ކ‬ၴ ࣠‫ݹ‬Ƨ ƀ ܻ‫ࠦࡩ࣠ߘ࣠ݑࠩࡺܻࡷݘ‬୆คᇴᇍ ࡷƀƠ ܻެࠧƣ ࣠ࡓ ‫ކ‬ ࠣ‫ެީܸߖޟ‬଍ଉƁƅ Ɓ߁ ၭ ൎᑆᐺ݇ ଵ଩ଠ Ɓƃ߁ ၫܸߕ P ( A ‫ أو‬B ) = P ( A ∪ B ) = P(A) + P(B) P ( A ∪ B ) = P(A) + P(B) ࠣ‫ެީܸߖޟ‬଍ଉƁ౭ౄ‫ݐ‬ ࡦƁƅ Ɓ߁ ၭ ൎᑆᐺ݇ ଵ଩ଠ Ɓƃ߁ ၫܸߕ P ( A ‫ أو‬B ) = P (A) + P(B) – P ( A ‫ و‬B )    P ( A ∪ B ) = P(A) + P (B) – P ( A ∩ B) P ( A ∪ B ) = P(A) + P (B) – P ( A B ) Ƅ କଐܸࡷެ߾ କଐƁƂ ྯ Ⴆ ƃ߁ ၫܸߕ .3 ƄƍƄܻࡺ‫ߙޢ‬ơ ܻࡺޯࠁ ଘଓƄߎ ࡯ޯ༧࿻ ƥ‫ࡷݘܻۑ‬౰౗ࡁ ƀƥƀ໠ ໄ‫ ߄މ‬ၰ Ɩ ࣠ߘƥơ ܻࡺޯࠁ ܻ‫ހ‬ƅ ྲ Ⴎ ࡩƄƆ߄ ၮܻߘள ஭‫୓ݘ‬ข ‫ۑ‬ቿቷߥޯࣀ ࡷ౰ేၩƥƀቿቷߥޯࣀ ࡷƈ ƌƄ࣠ஊै Ƅบ คᇴᆝ ቼቴߢެࢽ ଍ଉƁƅ Ɓ߁ ၭ ൎᑆᐺƂ ྯ Ⴋ ࡦƁƃ߁ ၫܸߕ  ، BƖ ࣠ߘ࣠‫ ࣍࢖ݹߥࠦހ‬Ƨ ܾ‫୍ ࣊ހ‬ง୹ ୪౰ేၩ A Ɩ ࣠ߘƥƣ ᇴᆝ Ƅƍƀ໠ ໄ࡯ߥޯࣀ ࡷ B ‫ و‬A ƣ ܻ‫ ߄މ‬ஊै Ƅ౰౎ޯ‫ݹݘ‬ ࣠ࠧ B ‫ و‬A໠ ໄ‫ॱॶ ߄މ‬५Ơ ౰౔ࣃ ଐଌƄơ ܻࡺޯࠁ ଘଓƄਓ ৯߄ ެ‫ݑ‬ P ( A ‫ و‬B ) = P ( A ∩ B) = P (A)‫ ۑ‬P(B) 68 ܸୃกܸ଍ଉŽ–ໝ ແࡦफउ –ࠣ‫ݶ‬ Ɖܸࢤ ެߕକଐƁƢࠣ‫ࡦ ࢽޟܸ ࢲޟ‬ƁƢࠣ‫ޟ‬පඓ ൷Ɓ౭౑ࢲ କଐƁƄ ܸࢲ ƁƋኃ ᒔᐺ። ጏ࢓‫ݕ‬ ࡦƁ࣬ ࣊ Ɓ࢓ߢೕಹኃ ቮƁ೾೥࢓଍ଉƁƄ ƁƋƁ߁ ࢟ ſ ። ፎକଐ߁ ެࢲ କଐƁƢ। ्࢟ ࣝࡦƁŻܸࢤ ߾ ଔଐƁ ƛ ܸߢࢽ ࡦƁ߁ ዖዄ Ž૊ સၫƋࣝެ࠾ኃ ቮƁƊܸެࢲ ଘଐƁ P ( A ∩ B) = P (A) . P(B) ቼቴߢެࢽ ࡴ౭ౄ‫ݐ‬ ࡦƁƅ Ɓ߁ ၭ ൎᑆᐺƂ ྯ Ⴋ ࡦƁƃ߁ ၫܸߕ  ਓ ৯߄ ެ‫ۑݑ‬༇໩ ଘଓƄƖ ࣠ߘ࣠‫୍ ࣊ހ ࣐࢖ݹߥࠦࡷܻހ‬ง୹ ୪ܻዙ዗߄ ၰ ƀƖ ࣠ߘƥƣ ᇴᆝ Ƅƍƀ໠ ໄ࡯ߥޯࣀ ࡷ౰ేၩƣ ܻ‫ ߄މ‬ஊै Ƅ౰౎ޯ‫ݹݘ‬ P ( A ‫ و‬B ) = P ( A ∩ B) = P (A) . P(B /A ) P ( A ∩ B) = P(A) . P(B / A ) P ( A ∩ B) = P(A) . P(B / A) Ɩ ࣠ߘƥơ ܻࡺޯࠁ ƄƧ ƥܻఱ ఓ B ‫ و‬A໠ ໄ‫ ߄މ‬ஊै Ƅࡼ ࡷൎᇩᆝ Ɩ ࣠ߘƥơ ܻࡺޯࠁ Ƅ፥ ጘଘଓᇴᆝ ƀ࢖ߥ‫ކ‬ቿቱƌܻ‫ݘ‬ଐଌƄƥ ‫ۑ‬ൎ൉‫ߘ ݊ߙݘ‬ƥ߄ ߘ A ƈ ߄ ஊै Ƅƣ ƀࣷࣳၮƄƍƄ B ƈ ߄ ஊै ƄƖ ࣠ߘƥơ ܻࡺޯࠁ Ƅ፥ ጿܻ‫ހ‬ƥྲ Ⴎ ࡷ A ƈ ߄ ஊै Ƅ 2 ،žƄ࢖ߦࢢ 4 ،žܻࢥ ୆ก 3ܻᐱᐟࡷƇ Ƅഃ೨ 10 ે઻ၮƧ ࣠ޯ༨࿻ žܻၮƥƣ ᇴᆝ ƄƍƄ ( ‫ ێ‬ƞ ܸޯࡴ ࠦ‫ࡩޯܻࡩޢ‬ƄƇ ଘଓܻࡺޯࠁ ଘଓƄ፥ ፕ ܻዙኽ‫ۑ‬žƄ࢖ዙ዇ Ɔ߄ ၰ Ƅƥ ‫ و‬،žƄƌ࣠ࢵ žܻࢥ ୆กƆഃ೨ ‫ ݪ‬༨೛ .I žƄ࢖ዙ዇ Ɔഃ೨ ‫ ݪ‬༨೛ .II žƄ࢖ߦࢢ ƥƀžƄƌ࣠ࢵ Ɔഃ೨ ‫ ݪ‬༨೛ .III ࡠ இॅ Ɓ žܻࢥ ୆กƆഃ೨ ‫ ݪ‬༨೛ .I P (W) = nw / N = 3/10 PR = nR / N = 1/10 P(A žƄ࢖ዙ዇ Ɔഃ೨ ‫ ݪ‬༨೛ .II žƄ࢖ߦࢢ ƥƀžƄƌ࣠ࢵ Ɔഃ೨ ‫ ݪ‬༨೛ .III ࠦ‫ࡷޯެܻߙޢ‬ƈ Ƅ߄ ၰ ƀܻᐱ᏶ଘଓ 2Ɔ߄ ၮܻߥࡩƄࢇ ‫୓ ࣐ީޢ‬ก ‫ أو‬B ) = P ( A ∪ B ) = P ( A) + P(B) P ( A ∪ B ) = 2/10 + 4/10 = 6/10 Ƣ߄ ႀ ޯࢵ ƀ ،ࠦ‫ܻࡷݘ‬எै Ƅ፥ ጿơ ƥଘଓƄƦ ࣠ޯࣀ ଐଌƄ፥ ጿƅ ൎ൉࣐ ࡩƄࡼ ࡷࠦެ‫ࡩݘޢ‬Ƥܻ‫ރ‬ƌƀơ ƥ߄ எै Ƅ ( ‫ ێ‬ƞ ܸޯࡴ ࠦ‫ࡩޯܻࡩޢ‬ƄƇ ଘଓܻࡺޯࠁ ଘଓƄਔ ৲߄ ༨ဈ ፥ ጿơ ƥ߄ எै Ƅ 69 ܸୃกܸ଍ଉŽ–ໝ ແࡦफउ –ࠣ‫ݶ‬ Ɖܸࢤ ެߕକଐƁƢࠣ‫ࡦ ࢽޟܸ ࢲޟ‬ƁƢࠣ‫ޟ‬පඓ ൷Ɓ౭౑ࢲ କଐƁƄ ܸࢲ ƁƋኃ ᒔᐺ። ጏ࢓‫ݕ‬ ࡦƁ࣬ ࣊ Ɓ࢓ߢೕಹኃ ቮƁ೾೥࢓଍ଉƁƄ ƁƋƁ߁ ࢟ ſ ። ፎକଐ߁ ެࢲ କଐƁƢ। ्࢟ ࣝࡦƁŻܸࢤ ߾ ଔଐƁ ƛ ܸߢࢽ ࡦƁ߁ ዖዄ Ž૊ સၫƋࣝެ࠾ኃ ቮƁƊܸެࢲ ଘଐƁ 19 ‫ أو‬18Ƥ࢖ዙኸ ‫ࡩ ࣐ܻࡩ ݪ‬Ƅƣ ࣠ࡓ ‫ݹ‬ƣ Ƅơ ܻࡺޯࠁ Ƅ .I ౰౗ࡁ ܾߙࠦެࢵ 19Ƥ࢖ዙኸ ‫ࡩ ࣐ܻࡩ ݪ‬Ƅƣ ƀơ ܻࡺޯࠁ Ƅ .II ࠦެࢵ 19ࡼ ‫ࡩ ࣐ܻࡩ ݪݑ‬Ƅ࢖ዙኸ ࡣ ߥ‫ݹ‬ƣ Ƅơ ܻࡺޯࠁ Ƅ .III ƈ ߄ ஊै Ƅ ࢖ࡺ‫ࡩݘ‬Ƅ B 19 45 21 13 A ƎƄಒ౷ޯࡩƄ 18 C 5 20 D E ‫ ۑ‬ ‫ ۑ‬ 35 22 Ɩ ࣠ࡺ഻വറƄ ơ ܻࡺޯࠁ ଘଓƄ ‫ ۑ‬ ‫ ۑ‬ 2 - ‫ ۑ‬ 100 ‫ ۑ‬ 19 ‫ أو‬18Ƥ࢖ዙኸ ‫ࡩ ࣐ܻࡩ ݪ‬Ƅƣ ࣠ࡓ ‫ݹ‬ƣ Ƅơ ܻࡺޯࠁ Ƅ .I ౰౗ࡁ ܾߙࠦެࢵ 19Ƥ࢖ዙኸ ‫ࡩ ࣐ܻࡩ ݪ‬Ƅƣ ƀơ ܻࡺޯࠁ Ƅ .II P ( a ‫ أو‬b ) = P ( a ∪ b ) = P (a) + P(b) = 0.05 + 0.45 = 0.50 P ( b ‫ أو‬c‫ أو‬d ‫ أو‬e ) = P ( b ∪ c ∪ d ∪ e ) = P (b) + P (c) + P(d) + P (e) = 0.45 + 0.35 + 0.13 + 0.02 = 0.95 ‫ ࢵެࠦ = ۑ‬19ࡼ ‫ࡩ ࣐ܻࡩ ݪݑ‬Ƅ࢖ዙኸ ࡣ ߥ‫ݹ‬ƣ Ƅơ ܻࡺޯࠁ Ƅ 1 – 0.95 = 0.05 .III ơ ܻࡺޯࠁ ଘଓƄࡴ ࡺޯࡷ࣠ࠧƥ ơ ƥଘଓƄ፥ ጹƄ༟༎Ƅࠦ‫ ࢢܻހ‬Ƅơ ܻࡺޯࠁ Ƅƣ ƀܻެ൞൙ၮƄƍܺߙ‫ۑ‬ƞ ߄ ࠧે ઻ၮƣ ܻ‫ࡷޢ‬ƄƎள ஭࣐ ‫ ێ ( ݹ‬ƞ ܸޯࡴ ƅ ॅळ࣐ ଐଌƄ‫ۑ‬2/3 ࣠ࠧƞ ߄ࠩᑃᐽ፥ ጕܻ޲ࡩƄ፥ ጹƄ༟༎Ƅࠦ‫ ࢢܻހ‬Ƃơ ܻࡺޯࠁ Ƅƣ ƀƥ ، 4 /7 ƞ ߄ ࠩᑃᐽ ໠ ໄ‫ ୹ކ‬୪ƞ ߄ ಛಗƄࠦ‫ ࢢܻހ‬Ƅơ ܻࡺޯࠁ Ƅܻ࣠ࠧࡷ .I ƞ ߄ ಛಗƄ໠ ໄ‫ࡷޢ‬Ƅ༟༎Ƅ߄ ၰ ƀ‫ ݪ‬ყࢧ ‫ݹ‬ƣ ƀơ ܻࡺޯࠁ Ƅܻ࣠ࠧࡷ .II ƞ ߄ ಛಗƄܻࡺᐱᐟࡷƧ ƀ‫ ݪ‬ყࢧ ‫ݹ‬ଘଓƣ ƀơ ܻࡺޯࠁ Ƅܻ࣠ࠧࡷ .III 70 ܸୃกܸ଍ଉŽ–ໝ ແࡦफउ –ࠣ‫ݶ‬ Ɖܸࢤ ެߕକଐƁƢࠣ‫ࡦ ࢽޟܸ ࢲޟ‬ƁƢࠣ‫ޟ‬පඓ ൷Ɓ౭౑ࢲ କଐƁƄ ܸࢲ ƁƋኃ ᒔᐺ። ጏ࢓‫ݕ‬ ࡦƁ࣬ ࣊ Ɓ࢓ߢೕಹኃ ቮƁ೾೥࢓଍ଉƁƄ ƁƋƁ߁ ࢟ ſ ። ፎକଐ߁ ެࢲ କଐƁƢ। ्࢟ ࣝࡦƁŻܸࢤ ߾ ଔଐƁ ƛ ܸߢࢽ ࡦƁ߁ ዖዄ Ž૊ સၫƋࣝެ࠾ኃ ቮƁƊܸެࢲ ଘଐƁ ໠ ໄ‫ ୹ކ‬୪ƞ ߄ ಛಗƄࠦ‫ ࢢܻހ‬Ƅơ ܻࡺޯࠁ Ƅܻ࣠ࠧࡷ P ( A ∩ B) = P (A) . P(B) ࡠ இॅ Ɓ ‫ۑ‬I ƣ ܾߙƣ ൎ൉ߥޯࣀ ࡷ໠ ໄ‫ ߄މ‬ஊै Ƅƣ Ƅܻ೘ೃ = 4/7 . 2/3 = 8/21 ƞ ߄ ಛಗƄ໠ ໄ‫ࡷޢ‬Ƅ༟༎Ƅ߄ ၰ ƀ‫ ݪ‬ყࢧ ‫ݹ‬ƣ ƀơ ܻࡺޯࠁ Ƅܻ࣠ࠧࡷ .II P ( A ∪ B ) = P(A) + P (B) - P ( A ∩ B) = 4/7 + 2/3 – 8/21 =( 12 + 14 - 8 ) /21 = 18/21 ƞ ߄ ಛಗƄܻࡺᐱᐟࡷƧ ƀ‫ ݪ‬ყࢧ ‫ݹ‬ଘଓƣ ƀơ ܻࡺޯࠁ Ƅ‫ۑ‬Iii P (Aʹ ∩ Bʹ ) = P ( A U B )ʹ .. de Morgan’s Law P (Aʹ ∩ Bʹ ) = P ( A U B )ʹ = 1 – P ( A U B ) = 1- 18/21 = 3/21 [ Conditional Probability] ࣬ ࣊ ྯ ႞ࡦƁƞ ܸࡷެ߾ କଐƁ .4 ࣩ ࡺఱ ఓൎ൉‫ߙݘ‬ƈ ߄ ၰ ߄ ߘ B ƈ ߄ ஊै Ƅƣ ƀܻެ൞൙ၮƄƍƀ ، Aࡼ ࡓ ‫ࡩޢ‬ƥܻࡷƈ ߄ ၰ ƈ ƥ߄ ၰ ơ ܻࡺޯࠁ Ƅ ፥ ጘଘଓᇴᆝ ܻ‫ݹܻࢠޢ‬Ǝ࣯ ࣍ ྲ ႡࡩƄơ ܻࡺޯࠁ ଘଓƄࠦၩܻ‫ ࡓ ࡼ ࢢޢ‬೘಼ƥ‫ ྲ ࣍ ࣯ۑ‬ႡࡩƄơ ܻࡺޯࠁ ଘଓܻ‫ހ‬ P(‫ܣ‬/B) = Or P(B/A) = ௉(஺∩஻) ………………. P(B) ǂ 0 ௉(஺∩஻) ………………. P(A) ǂ 0 ௉(஻) ௉(஺) ໠ ໄ‫ކ‬ഃ೨ ‫ ݪ‬༨೛ Ꭸ᎙‫ۑ‬žƄƌ࣠ࢵ 5ƥžܻࢥ ୆กƇ Ƅഃ೨ 3ે ઻ၮƧ ࣠ޯ༨࿻ Ɵ ƥ߄ ެࢢ ( ‫ ێ‬ƞ ܸޯࡴ ƣ ܻ୓ก࣠ࠒ ࣀ ଐଌƄƣ ܻ‫ކ‬ಒ౷ᚻࡩƄƣ ࣠ࡓ ‫ކ‬ƣ Ƅơ ܻࡺޯࠁ Ƅ߄ ၴ ƥƀ‫ۑ‬Ɩ ܻၴ Ǝ Ƅƣ ƥਔ ৹Ʀ ༇໩ ଘଓƄॅळ‫ކ‬Ɔ߄ ၰ Ƅƥ ،ܻ୆෷Ƅ࣠ࣃ ‫ݑ‬ ‫ۑ‬໠ ໄ୓෺ƥƄƌ࣠ࢵ P(A) = nA / N ፟ ፑƥଘଓƄቿቱƥܻഷവറƄ፥ ጿžƄƌ࣠ࢵ Ɔഃ೨ ‫ ݪ‬༨೛ ơ ܻࡺޯࠁ Ƅ ࡠ இॅ Ɓ  = 5/ 3+5 = 5/8 71 ܸୃกܸ଍ଉŽ–ໝ ແࡦफउ –ࠣ‫ݶ‬ Ɖܸࢤ ެߕକଐƁƢࠣ‫ࡦ ࢽޟܸ ࢲޟ‬ƁƢࠣ‫ޟ‬පඓ ൷Ɓ౭౑ࢲ କଐƁƄ ܸࢲ ƁƋኃ ᒔᐺ። ጏ࢓‫ݕ‬ ࡦƁ࣬ ࣊ Ɓ࢓ߢೕಹኃ ቮƁ೾೥࢓଍ଉƁƄ ƁƋƁ߁ ࢟ ſ ። ፎକଐ߁ ެࢲ କଐƁƢ। ्࢟ ࣝࡦƁŻܸࢤ ߾ ଔଐƁ ƛ ܸߢࢽ ࡦƁ߁ ዖዄ Ž૊ સၫƋࣝެ࠾ኃ ቮƁƊܸެࢲ ଘଐƁ žƄƌ࣠ࢵ ࠦ‫ ࣀ ࠒ࣠ހ‬ଐଌƄ፟ ፑƥଘଓƄƆಒ౷ᚻࡩƄƣ ƀܻ൞൙ၮ ،žƄƌ࣠ࢵ ࠦ୆คܻ‫މ‬Ɔഃ೨ ‫ ݪ‬༨೛ ơ ܻࡺޯࠁ Ƅ  ƣ ܻ୓෺ƥƄƌ࣠ࢵ ໠ ໄ୓ก࣠ࠒ ࣀ ଐଌƄ໠ ໄ‫ކ‬ಒ౷ᚻࡩƄƣ ƀơ ܻࡺޯࠁ Ƅ  P (B/A) = 4 / 3+4 = 4/7 P(AB) = P(A) . P(B/A) = 5/8 . 4/7 = 20/56 = 5/14 ࡼ ࡷ ‫ ۑ‬،ࠦ‫ݹ‬ƎƄƌଘଓƄƢॅळ‫ࡩݘ‬Ƅࠦ‫ޢ‬ᆿᆝ ፥ ጿƅ ൎ൉࣐ ᑃᐽơ ࣠ީߥࡩƄƣ ܻႁ ޯࡷƄ፥ ጿ ( ‫ ێ‬ƞ ܸޯࡴ ƣ ࣠ީࢵ मअࡴ ᐱᐟࡷ ‫ ۑ‬ƥࠦ‫ݹ‬౮ే࡯༬စ ଘଓƄࠦ‫ݓ‬ ᑃᐽƄƟ ܻࣀ ࡷ፥ ጿ ‫ ۑ‬ƥƇ ܻ‫ݹܻࢠޢ‬༟༎ƄƟ ܻࣀ ࡷ፥ ጿƣ ࣠ީࢵ मअƅ ൎ൉࣐ ࡩƄ ƅ ॅळ࣐ ଐଌƄ‫ۑ‬፥ ገƄ࣠ࣃ ‫ݑ‬ᇯ ᇍ ఴ చ‫ ܻ࣍ࡩ ݪ‬Ǝܻ‫ ࠀޯޢ‬ƄᎨ᎙‫ࡷݘܻۑ‬௪ ௎Ǝ࢖ߥଐଌƄ፥ ጿ ܻީࢵ ƄƎƣ ࣠ࡓ ‫ݹ‬ƣ ƀơ ܻࡺޯࠁ Ƅܻ࣠ࠧዙኽ ،Ƈ ܻ‫ݹܻࢠޢ‬༟༎ƄƟ ܻࣀ ࡷ፥ ጿܻީࢵ ƄƎ‫ࡩ ࣐ܻࡩ ݪ‬Ƅƣ ᇴᆝ Ƅƍƀ  ፥ ጿܻީࢵ ƄƎƣ ࣠ࡓ ‫ݹ‬ƣ Ƅơ ܻࡺޯࠁ Ƅܻ࣠ࠧዙኽ ،ࠦ‫ݹ‬౮ే࡯༬စ ଘଓƄࠦ‫ݓ‬ ᑃᐽƄ፥ ጿܻީࢵ ƄƎ‫ࡩ ࣐ܻࡩ ݪ‬Ƅƣ ᇴᆝ Ƅƍƀ  ࠦ‫ݹ‬౮ే࡯༬စ ଘଓƄࠦ‫ݓ‬ ᑃᐽƄ፥ ጿ ໠ ໄߘܻࣀ ଐଌƄ߄ ၰ ƀ፥ ጿܻީࢵ ƄƎ‫ࡩ ࣐ܻࡩ ݪ‬Ƅƣ ࣠ࡓ ‫ݹ‬ƣ ƀơ ܻࡺޯࠁ Ƅܻ࣠ࠧࡷ Ƈ ܻ‫ݹܻࢠޢ‬༟༎Ƅ  ࡠ இॅ Ɓ Ơ ŽƐ ࢓ߣ‫ހ‬ Ƈ ܻ‫ݹܻࢠޢ‬༟༎ƄƟ ܻࣀ ࡷ፥ ጿƣ ࣠ީࢵ Ƅ༟༎Ƅƅ ൎ൉࣐ ࡩƄ A P(A) = 0.25 ࠦ‫ݹ‬౮ే࡯༬စ ଘଓƄࠦ‫ݓ‬ ᑃᐽƄƟ ܻࣀ ࡷ፥ ጿƣ ࣠ީࢵ Ƅ༟༎Ƅƅ ൎ൉࣐ ࡩƄ B P(B) = 0.15 P(A∩B) = 0.10 ፥ ጿ ‫ ࢵ ݪ‬ƄƎࠢ‫ܻހܾރ‬൞൙ၮ،Ƈ ܻ‫ݹܻࢠޢ‬༟༎ƄƟ ܻࣀ ࡷ፥ ጿܻީࢵ ƄƎ‫ࡩ ࣐ܻࡩ ݪ‬Ƅƣ ࣠ࡓ ‫ݹ‬ƣ Ƅơ ܻࡺޯࠁ Ƅ P ( A/B) = P( A∩B) /P(B)  ࠦ‫ݹ‬౮ే࡯༬စ ଘଓƄࠦ‫ݓ‬ ᑃᐽƄ = 0.10/0.15 = 2/3= 0.66 72 ܸୃกܸ଍ଉŽ–ໝ ແࡦफउ –ࠣ‫ݶ‬ Ɖܸࢤ ެߕକଐƁƢࠣ‫ࡦ ࢽޟܸ ࢲޟ‬ƁƢࠣ‫ޟ‬පඓ ൷Ɓ౭౑ࢲ କଐƁƄ ܸࢲ ƁƋኃ ᒔᐺ። ጏ࢓‫ݕ‬ ࡦƁ࣬ ࣊ Ɓ࢓ߢೕಹኃ ቮƁ೾೥࢓଍ଉƁƄ ƁƋƁ߁ ࢟ ſ ። ፎକଐ߁ ެࢲ କଐƁƢ। ्࢟ ࣝࡦƁŻܸࢤ ߾ ଔଐƁ ƛ ܸߢࢽ ࡦƁ߁ ዖዄ Ž૊ સၫƋࣝެ࠾ኃ ቮƁƊܸެࢲ ଘଐƁ ፥ ጿ‫ ࢵ ݪ‬ƄƎࠢ‫ܻހܾރ‬൞൙ၮ،ࠦ‫ݹ‬౮ే࡯༬စ ଘଓƄࠦ‫ݓ‬ ᑃᐽƄƟ ܻࣀ ࡷ፥ ጿܻީࢵ ƄƎƣ ࣠ࡓ ‫ݹ‬ƣ Ƅơ ܻࡺޯࠁ Ƅ P( B/A) = P( A∩B) /P(A)  Ƈ ܻ‫ݹܻࢠޢ‬༟༎Ƅ = 0.10/0.25 = 2/5 = 0.40 : ໠ ໄߘܻࣀ ଐଌƄ߄ ၰ ƀ፥ ጿܻީࢵ ƄƎƣ ࣠ࡓ ‫ݹ‬ƣ ƀơ ܻࡺޯࠁ Ƅ P(A UB) = P(A) + P(B) – = P( A∩B) = 0.25 + 0.15 – 0.10 = 0.30 [ Bays Theorem] ౫ౄୃ෾ࠣ‫ހ ࣌࢓ݶ‬  .5 : ƣ ƀ߄ ༬စ ،ቿቷߥޯࣀ ଐଌƄ౰ేၩƈ Ƅ߄ ၰ ൑ᑉᐽƅ ྲ Ⴎ ࡩƄƆ߄ ၮܻߘࡼ ࡷ P ( B ∩ A) = P(B) . P(A/ B) P(B∩ A )/ P(B) = ୔(୆). ௉(஺/஻) P(B)ૄ સၫໝ ແߖ࢓࣍ ࡦƁࠣࡷࢽ ߢ‫ݽ‬ ୔(୆) P(B∩ A )/ P(B) = P( A / B) P( A / B) = P(A∩B ) / P(B) ……………..{ P(A∩B ) = P(B∩ A ) } P(A/B) = P(A) . P(B) / P(B) …………{ P(A∩B ) = P(A) . P(B)} P (A/B) = ‫)ۯ(۾‬. ࡼ(࡮) ‫(۾‬۰) ……………{ Bays Theorem }౮ే୆กࠦ‫ۑۑۑۑރ ࣏࢖ݹ‬ ơ ƥ߄ எै Ƅ‫ ࠁ ࣀ ݪ‬໠ ໄၮƏ࣠ࡷ‫ ހܻ ࠁ ݟ‬200ܻᐱᏬኆ ቱࠦ‫ࡩީ ࠒ޲ޢ‬ƄƇ ܻࣀ ࢵ ࣓ଐଌƄƦ ߄ ၰ ƀ ( ‫ ێ‬ƞ ܸޯࡴ ࢓ࡷ‫ݕ‬ ࡦƁ < 30 30- 40 40+ ፥ ፑܻዙዋ ଘଓƄ 73 ࠣၱ ƋƉूरࡴܸၭ ƍ ࣝ‫ݶ‬Ƌࣝࡦᇱᇊ ަࡦƁ 20 45 10 75 ࠣၱ ƋƉूरࡴܸၭ ౭ౄެࢽ ࠂ ܸ଍ଉƁ 5 Ƥܻ‫ރ‬ƌƄ ። ፎܸዖወ Ž 25 5 50 25 100 15 25 ܸୃกܸ଍ଉŽ–ໝ ແࡦफउ –ࠣ‫ݶ‬ Ɖܸࢤ ެߕକଐƁƢࠣ‫ࡦ ࢽޟܸ ࢲޟ‬ƁƢࠣ‫ޟ‬පඓ ൷Ɓ౭౑ࢲ କଐƁƄ ܸࢲ ƁƋኃ ᒔᐺ። ጏ࢓‫ݕ‬ ࡦƁ࣬ ࣊ Ɓ࢓ߢೕಹኃ ቮƁ೾೥࢓଍ଉƁƄ ƁƋƁ߁ ࢟ ſ ። ፎକଐ߁ ެࢲ କଐƁƢ। ्࢟ ࣝࡦƁŻܸࢤ ߾ ଔଐƁ ƛ ܸߢࢽ ࡦƁ߁ ዖዄ Ž૊ સၫƋࣝެ࠾ኃ ቮƁƊܸެࢲ ଘଐƁ ߄ ၴ ƥƀ ،ࠦ‫ࡩީ ࠒ޲ޢ‬Ƅࠦࣀ ࢵ ࣓ଐଌƄࡼ ࡷܻ୆෷Ƅ࣠ࣃ ‫ހܻ ࠁ ݟݑ‬Ǝܻ‫ ࠀޯޢ‬ƄᎨ᎙ƄƍƄ ࣊ ߥߙƐ ࣠‫ݹ‬Ǝ࣠ࡩᇴᇍ ީࡩƄࠦၴ Ǝƌࠢ‫ݹ‬ኆ ቱ‫ࡩީܻ ࠁ ݟ‬Ƅƣ Ƅơ ܻࡺޯࠁ Ƅ .I ܻࡷܻၮ 40ࡼ ࡷ౰౎ࡁ ƄƤ࢖ዙኸ ƥ౰ేޯࣀ ࠅ ܻଐଌƄࠦၴ Ǝƌࡣ ዙ၁ဤ‫ࡩީܻ ࠁ ݟ‬Ƅƣ ƀơ ܻࡺޯࠁ Ƅ .II Ɛ ࣠‫ݹ‬Ǝ࣠ࡩᇴᇍ ީࡩƄࠦၴ Ǝƌࡣ ዙ၁ဤƥܻࡷܻၮ 30ࡼ ࡷࡣ ߘƀ‫ࡩީܻ ࠁ ݟ‬Ƅ࢖ዙኸ ƣ ƀơ ܻࡺޯࠁ Ƅ .III a =࣊ ߥߙƐ ࣠‫ݹ‬Ǝ࣠ࡩᇴᇍ ީࡩƄࠦၴ Ǝƌࡣ ዙ၁ဤ‫ࡩީܻ ࠁ ݟ‬Ƅƣ ƄƓ ࢖ߦ‫ރ‬ ࡠ இॅ Ɓ b ౰ేޯࣀ ࠅ ܻଐଌƄࠦၴ Ǝƌࠢ‫ݹ‬ኆ ቱ‫ࡩީܻ ࠁ ݟ‬Ƅ c ࠦެࢵ 30ࡼ ࡷࡣ ߘƄ‫ࡩީܻ ࠁ ݟ‬Ƅ d ࠦެࢵ 40ࡼ ࡷ౰౎ࡁ ƀ‫ࡩީܻ ࠁ ݟ‬Ƅ ࣊ ߥߙƐ ࣠‫ݹ‬Ǝ࣠ࡩᇴᇍ ީࡩƄࠦၴ Ǝƌࠢ‫ݹ‬ኆ ቱ‫ࡩީܻ ࠁ ݟ‬Ƅƣ Ƅơ ܻࡺޯࠁ Ƅ  ܻࡷܻၮ 40ࡼ ࡷ౰౎ࡁ ƄƤ࢖ዙኸ ƥ౰ేޯࣀ ࠅ ܻଐଌƄࠦၴ Ǝƌࡣ ዙ၁ဤ‫ࡩީܻ ࠁ ݟ‬Ƅƣ ƀơ ܻࡺޯࠁ Ƅ  P(a) = 75/100 = 0.75 P(b/d) = P(b∩ d) / P(d) = (15/100)/(25/100) = 15/25 = 3/5 Ɛ ࣠‫ݹ‬Ǝ࣠ࡩᇴᇍ ީࡩƄࠦၴ Ǝƌࡣ ዙ၁ဤƥܻࡷܻၮ 30ࡼ ࡷࡣ ߘƀ‫ࡩީܻ ࠁ ݟ‬Ƅ࢖ዙኸ ƣ ƀơ ܻࡺޯࠁ Ƅ P(c/a) = P(c ∩ a ) / P(a)  = (20/100)/(75/100) = 20/75 = 4/ 15 74 ܸୃกܸ଍ଉŽ–ໝ ແࡦफउ –ࠣ‫ݶ‬ Ɖܸࢤ ެߕକଐƁƢࠣ‫ࡦ ࢽޟܸ ࢲޟ‬ƁƢࠣ‫ޟ‬පඓ ൷Ɓ౭౑ࢲ କଐƁƄ ܸࢲ ƁƋኃ ᒔᐺ። ጏ࢓‫ݕ‬ ࡦƁ࣬ ࣊ Ɓ࢓ߢೕಹኃ ቮƁ೾೥࢓଍ଉƁƄ ƁƋƁ߁ ࢟ ſ ƛ ܸߢࢽ ࡦƁ߁ ዖዄ Ž૊ સၫƋࣝެ࠾ኃ ቮƁƊܸެࢲ ଘଐƁ ። ፎକଐ߁ ެࢲ କଐƁƢ। ्࢟ ࣝࡦƁŻܸࢤ ߾ ଔଐƁ ࢧ ࡴܸஆॅ Ɓࡠ ࢤ ߣࡦƁ௧ ோƋܸೕೆ ،žܻߘƎƏ 3 ،žƄ࢖ዙ዇ 5ܻᐱᐟࡷƣ ƀžܻެ޲ჴࢵ ܻ‫ހ‬ቿቷ‫ࡷޯࡺܻމ‬Ƈ Ƅഃ೨ 10ે ઻ၮƧ ࣠ޯ༨࿻ žܻၮƥƣ ᇴᆝ Ƅƍƀ‫ۑ‬1 ે ઻ၮơ ࣠ࢧ ஊै Ƅơ ܻࡺޯࠁ Ƅܻ࣠ࠧࡷ ،žƄྲ Ⴎ ࠀ 2 Ɩ ܻၴ ƎƄƣ ƥਔ ৹໠ ໄ‫ ୹ކ‬୪‫ࡩ ࣀ ࠒ ݪ‬Ƅ߄ ެ‫ݑ‬،žܻၮ࣠ࡩƄࡼ ࡷžܻߘƎƏƣ ܻ‫ކ‬ഃ೨ .I Ɩ ܻၴ ƎƄƣ ƥਔ ৹໠ ໄ‫ ୹ކ‬୪‫ࡩ ࣀ ࠒ ݪ‬Ƅ߄ ެ‫ݑ‬،žƄྲ Ⴎ ࠀ ƣ ܻ‫ކ‬ഃ೨ .II Ɩ ܻၴ ƎଘଓƄ݊ ࡷ໠ ໄ‫ ୹ކ‬୪‫ࡩ ࣀ ࠒ ݪ‬Ƅ߄ ެ‫ݑ‬žܻߘƎƏƣ ܻ‫ކ‬ഃ೨ .III Ɔ߄ ၰ Ƅƥƥ ،žƄƌ࣠ࢵ 2 ،žƄ࢖ߦࢢ 3 ،žܻࢥ ୆ก 4Ƈ Ƅഃ೨ ે ઻ၮƧ ࣠ޯ༨࿻ žܻၮƥƣ ᇴᆝ ƄƍƄ .1 Ɩ ܻၴ ƎƄƣ ƥਔ ৹ ࠦ‫ࡩޯܻࡩޢ‬ƄƇ ଘଓܻࡺޯࠁ ଘଓƄ፥ ፕ ܻዙኽ ،žƄ࢖ዙ዇ žƄƌ࣠ࢵ Ꭸ᎜žƄྲ Ⴋ Ɔഃ೨ ‫ ݪ‬༨೛ .I žƄ࢖ߦࢢ Ꭸ᎜žܻࢥ ୆กƆഃ೨ ‫ ݪ‬༨೛ .II žƄ࢖ዙ዇ Ꭸ᎜žƄƌ࣠ࢵ Ꭸ᎜žƄ࢖ߦࢢ Ɔഃ೨ ‫ ݪ‬༨೛ .III ‫ީ ݝ‬༨೛ ،žƄྲ Ⴎ ࠀ 2 ،žܻߘƎƏ 3 ،žƄ࢖ዙ዇ 5 ،ቿቷ‫ࡷޯࡺܻމ‬Ƈ Ƅഃ೨ 10ે ઻ၮžܻၮƥƧ ࣠ޯ༨࿻ .2 žƄ࢖ዙ዇ (‫أ‬ ،ƣ ࣠ࡓ ‫ކ‬ƣ ƀơ ܻࡺޯࠁ Ƅܻ࣠ࠧࡷ ،žܻၮ࣠ࡩƄࡼ ࡷƆഃ೨ žܻߘƎƏ (‫ب‬ žܻߘƎƏ‫ ࣀ ݝ‬ყࡩ (‫د‬ žƄྲ Ⴎ ࠀ žƄྲ Ⴎ ࠀ ‫ ࣀ ݝ‬ყࡩ žƄྲ Ⴎ ࠀ ‫ ࣀ ݝ‬ყࡩƥƄžƄྲ Ⴎ ࠀ ‫ۑ‬Ɩ ܻၴ ƎଘଓƄ݊ ࡷ໠ ໄޯ‫ࡷޯޯܻࡩޢ‬໠ ໄޯީ༨೛ ፥ ጿ௪ ௎ƥܻߘƎƏ໠ ໄ‫ކ‬ഃ೨ ‫ ݪ‬༨೛ ơ ܻࡺޯࠁ Ƅܻ࣠ࠧࡷ ࣊ ‫ހܻࡩޯެߥޢ‬Ƨ ༟༎ƄƢ߄ ႀ ޯఱ ఠƖ ƎƄ࢔ଐଌƄࡼ ࡷ %40ƣ Ƅ߄ ၴ ƥࠦ‫ݑޢ‬ƄƎ༝༎Ƅࢇ ࣍ ܻެଐଌƄƦ ߄ ၰ Ƅ፥ ጿ (‫ج‬ (‫ه‬ (‫و‬ (‫ز‬ .3 Ɩ ƎƄ࢔ଐଌƄƦ ߄ ၰ ƀƇ ౰ేޯࠀ ƄƄƍܻߙ ،ࠦ‫ࡷޯެܻߙޢ‬౰ేၩƇ ܻ࣍ ܻఴ ఝ፥ ፕ ƥ ،ࠦ‫ޢ ࡓޢ‬ჴࢵ ൎ൉ީࡩƄƇ ࣠‫ࡩީޢ‬ƄƢ߄ႀ ޯఱ ఠ %10 ‫و‬ ‫ܻۑ‬ዙ዗ൎᇩᆝ ƥƀࠦ‫ޢ ࡓޢ‬ჴࢵ ൎ൉ީࡩƄƇ ࣠‫ࡩީޢ‬Ƅƥƀ࣊ ‫ހܻࡩޯެߥޢ‬Ƨ ༟༎ƄƢ߄ႀ ޯఱ ఠܻᐱ᏶Ƅơ ܻࡺޯࠁ Ƅܻ࣠ࠧࡷ ، ፥ ጿ 6 ‫ أو‬5 ،4ࡴ ߘ༟༎Ƅે ઻ၮơ ࣠ࢧ ஊै Ƅơ ܻࡺޯࠁ Ƅ߄ ၴ ƥƀ‫ۑ‬໠ ໄ‫ ୹ކ‬୪‫ࡷޢ ݝ‬Ǝƌ౰౑ࡩƄ࢖ᄽ஛ .4 .ࠦ୆คܻ޲ࡩƄࠦ‫ࡷޢ‬༟༎Ƅ፥ ጿ 4 ‫ أو‬،3 ،2 ،1ƥ፟ ፑƥଘଓƄࠦ‫ࡷޢ‬༟༎Ƅ 75 ܸୃกܸ଍ଉŽ–ໝ ແࡦफउ –ࠣ‫ݶ‬ Ɖܸࢤ ެߕକଐƁƢࠣ‫ࡦ ࢽޟܸ ࢲޟ‬ƁƢࠣ‫ޟ‬පඓ ൷Ɓ౭౑ࢲ କଐƁƄ ܸࢲ ƁƋኃ ᒔᐺ። ጏ࢓‫ݕ‬ ࡦƁ࣬ ࣊ Ɓ࢓ߢೕಹኃ ቮƁ೾೥࢓଍ଉƁƄ ƁƋƁ߁ ࢟ ſ ƛ ܸߢࢽ ࡦƁ߁ ዖዄ Ž૊ સၫƋࣝެ࠾ኃ ቮƁƊܸެࢲ ଘଐƁ ። ፎକଐ߁ ެࢲ କଐƁƢ। ्࢟ ࣝࡦƁŻܸࢤ ߾ ଔଐƁ ʫCorrelationŚ ŅŉľɩļŐ ſʆ ń 76 ܸୃกܸ଍ଉŽˍ ໝ ແࡦफउˍ ࠣ‫ݶ‬ Ɖܸࢤ ެߕକଐƁƢࠣ‫ࡦ ࢽޟܸ ࢲޟ‬ƁƢࠣ‫ޟ‬පඓ ൷Ɓ౭౑ࢲ କଐƁƄ ܸࢲ ƁƋኃ ᒔᐺ። ጏ࢓‫ݕ‬ ࡦƁ࣬ ࣊ Ɓ࢓ߢೕಹኃ ቮƁ೾೥࢓଍ଉƁƄ ƁƋƁ߁ ࢟ ſ ƛ ܸߢࢽ ࡦƁ߁ ዖዄ Ž૊ સၫƋࣝެ࠾ኃ ቮƁƊܸެࢲ ଘଐƁ ። ፎକଐ߁ ެࢲ କଐƁƢ। ्࢟ ࣝࡦƁŻܸࢤ ߾ ଔଐƁ ƥ‫ݹݘࠢۑ‬Ə࣠‫ކ‬ƥ [ x ]߄ ၰ Ƅƥ౰ే‫ݓ‬ ޯ೘ೃࠦߥ࡯‫ޯݘ‬ଐଌƄࠦ୆෷ܻࢧ ࠁ ଘଓƄƟ ࢖࣐ ࡩƄᎨᎌ߄ ߥ‫ކ‬Ꭸ᎙ ، ơ ࣠ࢧ ߦࡩƄࡼ ࡷࢇ ީࢵ ܻࡺ‫ߙޢ‬ Ƈ ൎᇩᇍ ࣃ ଐଌƄ࢛ ‫ހݘ‬፥ ጿ‫ۑ‬Ɔƌ߄ ‫ࡷޯݘ‬Ƈ Ƅ౰ే‫ݓ‬ ޯࡷࡼ ࡺࢥ ୓ขžܻࢧ ࠁ ଘଓƄ፥ ጿƇ ൎᇩᇍ ࣃ ଐଌƄࡼ ࡷਔ ৲߄ ‫ࡩݘ‬ƄƠ ܻެࠧ ༇໩ ଘଓƄ࢛ ‫ࡩީݘ‬Ƅ፥ ጿƥ‫ࡺܻۑ‬ᐱᐟყჅࠦߘൎ൉‫ࡩݘ‬Ƅࠦࢵ ƄƎኆ ቱƇ Ƅ౰ే‫ݓ‬ ޯࡷƆ߄ ၮࠦࢵ ƄƎƌযঠ‫ ࠅޯࡺܻݑޢࠦކ‬ଘଓƄƥࠦ‫ݹ‬ƌܻࢧ ޯߘଘଓƄ Ƅ߃ ࠧƞ ࢖‫ݹݘ‬ƥ‫ۑ‬౰ే‫ݓ‬ ޯଐଌƄƄ߃ ᐱᏳܻᐱᐢߘൎ൉‫ހݘ‬Ʀ ༇໩ ଘଓƄƇ Ƅ౰ే‫ݓ‬ ޯଐଌƄࠦࢵ ƄƎƌযঠ‫ކ‬ƥ߄ ၰ Ƅƥ౰ే‫ݓ‬ ޯ೘ೃƢܻࡺޯࠧଘଓƄযঠ‫ݹ‬ ‫ۑ‬Ɣ ୍ܻงƎଘଓܻ‫ހ‬žܻࢧ ࠁ ଘଓƄ፥ ጿƇ ܻࢵ ƄƎኆ ቱƄࡼ ࡷƖ ࣠ެࡩƄ ࡼ ࡷƉ ƥƏ໠ ໄ‫ߘࠦހ‬ൎ൉ၮࠦ‫ݹ‬ƀƠ ܻެࠧบ คᇴᆝ ƄƍƄܻࡺ‫ߙޢ‬ơ Ɓܻࣀ ჴࡩƄযঠ‫ݑެ ߄ࡷܻݹ‬Ɣ ୍ܻงƎଘଓƄƇ ൎᇩᇍ ࣃ ࡷƏ౰౎‫ކ‬ ،࢖‫ࡩ ࣀݘ‬Ƅƥࠦެ‫઻ ે ࢵ࡯ݘࠦࡷݘޢ‬ၮ‫ࡩ ࣐࡯ ݪ‬Ƅ໠ ໄ‫ߘࠦހ‬ൎ൉‫ࡩݘ‬ƄƦ ߄ ࡷơ ࣠ࠁ ơ Ɓܻࣀ ჴࡩƄൎ൉޲ዙኽ‫ ࢵࠦۑ‬ƄƎኆ ቱƄࡣ ༨ீ Ƈ Ƅ౰ే‫ݓ‬ ޯଐଌƄ ƣ ൎ൉ၮଘଓƄƥࠦ‫ܻݹ‬ၮኆ ቱƄ Ƈ ܻߥߦ‫ރ‬໠ ໄ‫ހ‬ƥƀ ،Ƈ ܻ‫ݹܻࢠޢ‬༟༎Ƅƥžܻࢧ ࠁ ଘଓƄƧ Ǝ࢖ߥଐଌ‫ ࣐ܻࡩ ݪ‬ᑃᐽ࣯ ൞൙‫ࡩݘ‬Ƅࡣ ‫ࡩޯ ࠒ ࢧޢ‬Ƅ໠ ໄ‫ހ‬ƥƀ ࡼ ࡷƒ ܻߥ‫ރ‬ଘଓƄƥࠦࢠܻ‫ݹ‬༟༎Ƅࠦࢵ Ǝܻያዪ໠ ໄ‫ހ‬ƥƀƟ ܻߦ‫ރ‬ଘଓƄƥࡣ ၯ ኆ ቱƄ໠ ໄ‫ހ‬ƥƀࠦެ‫ࡷݘޢ‬በሼྲ ႖፥ ጿƇ ܻ‫ީޢݘ‬ଐଌƄࡴ ᄽ஛ ƥ ‫ ࡴۑ‬ᐱᐜභච ൺ཯དྷࡩƄơ ܻߦ࣍ ଘଓƄƌ߄ ၮƥƆƀ࢖ଐଌƄযও࡯‫ކݘ‬໠ ໄ‫ހ‬ƥƀƣ Ə࣠ࡩƄ Ƣܻႅ ఱ ఝƄ፥ ጿ௪ ௎౰ే‫ݓ‬ ޯଐଌƄበሼ༇໪ ƥ‫ۑ‬Ɣ ୍ܻงƎଘଓƄࣩ ࡺఱ ఓ࢛ ‫ࡷ ݊ހݘ‬Ƣܻႅ ఱ ఝƄ፥ ጿ௪ ௎౰ే‫ݓ‬ ޯࡷበሼ༇໪ Ɔ࢖ܻࠧ࣌ ƣ ƀ ࢪ ࡁ ܻ‫ݘ‬ଐଌƄƤܻ༬ဈ ଘଓƄ፥ ጿƥƢܻႅ ఱ ఝܻ‫ހ‬௪ ௎౰ే‫ݓ‬ ޯࡷƠ ࢖ඳඣ ൭ܻࡷ߄ ެ‫ݑ‬ƥ‫ۑ‬፥ ጒܻ༬࿻ ƄƔ ୍ܻงƎƄࣩ ࡺఱ ఠƤܻ༬ဈ ଘଓƄࢪ ߦ‫ރ‬፥ ጿƥ ࠦߘൎ൉ၮ߄ ၴ ࣠‫ކ‬ଘଓƠ ܻެࠧƣ ܾߙ ،࢛ ‫ࡷ ݊ހݘ‬Ƣܻႅ ఱ ఝܻ‫ހ‬ƣ Ƅ౰ే‫ݓ‬ ޯଐଌƄƠ ࢖ඳඣ ൭ଘଓܻࡷ߄ ެ‫ݑ‬ƥ‫཯ۑ‬ཌ࡯ࢵ Ɣ ୍ܻงƎƄࣩ ࡺఱ ఓ ‫ۑ‬Ɣ ୍ܻง‫ار‬ Ƥ Ƌܸࣀ ჱჅକଐƁᇬ ᇊ ࣀ ࡦƁ ે ઻ၮ ܻެ࡯ࢧ ࠁ ‫و‬Ƈ ܻ୆ชƄ߄ ၰ ଗଓƄƦ ࣠ޯࣀ ࡷ፥ ጿܻࠩ࡯‫޲ޢ‬๻๬฽ܻެዙኼƥ௪ ௎౰ే‫ݓ‬ ޯଐଌƆ࢖࣌ ܻެޯࡷযওߥࡁ y ، x ܻ‫ ߃ރ‬ၯ ƀƄƍƂ ƞ ࢖‫ݹݘ‬ơ ƥଛଓƄ౰ే‫ݓ‬ ޯଐଌܻߙ‫ ۑ‬y ౰ే‫ݓ‬ ޯ൞ᒑᐽܻࠩ࡯‫ ߄ߘޢࡺࠦކߥܻހ‬ၴ ࣠‫ ކ‬x ౰ే‫ݓ‬ ޯ൞ᒑᐽࠦࡺ‫ߘޢ‬ᇯ ᇍ ࡯ߙ ،ࠦ‫ݹ‬Ǝܻࣃ ჴ჈ଘଓƄơ ᇴᇍ ࢸ ଛଓƄ Ǝܻࣃ ჴ჈ଘଓƄࠦၰ ॅळ‫ހ‬ƞ ࢖‫࢖ߙ ࢇࠧެܻݹݘ‬ଐଌƄᇯ ᇍ ࣃ ࡩƄū݊ ‫ࡩޯܻހ‬Ƅ౰ే‫ݓ‬ ޯଐଌܻ‫ހ‬ƞ ࢖‫ݹݘ‬༇໩ ଜଓƄ໠ ໄၰ ፥ ጿࡣ ߥޯࣀ ଐଌƄ౰ే‫ݓ‬ ޯଐଌܻ‫ހ‬ ‫ ۑ‬y،x ƆƎ࣠ࢧ ࡩܻ‫ހ‬ว ง୹ ୪Ɖ ƥƏࡣ ޲೘೉ܻެ࣐ࠧࠦ ߥ‫ރ‬ᇯ ᆝƥ 77 ܸୃกܸ଍ଉŽ–ໝ ແࡦफउ –ࠣ‫ݶ‬ Ɖܸࢤ ެߕକଐƁƢࠣ‫ࡦ ࢽޟܸ ࢲޟ‬ƁƢࠣ‫ޟ‬පඓ ൷Ɓ౭౑ࢲ କଐƁƄ ܸࢲ ƁƋኃ ᒔᐺ። ጏ࢓‫ݕ‬ ࡦƁ࣬ ࣊ Ɓ࢓ߢೕಹኃ ቮƁ೾೥࢓଍ଉƁƄ ƁƋƁ߁ ࢟ ſ ƛ ܸߢࢽ ࡦƁ߁ ዖዄ Ž૊ સၫƋࣝެ࠾ኃ ቮƁƊܸެࢲ ଘଐƁ ። ፎକଐ߁ ެࢲ କଐƁƢ। ्࢟ ࣝࡦƁŻܸࢤ ߾ ଔଐƁ Y ‫ و‬X௧ ோ౭ౄ‫ݐ‬ ެ଍ଉƁໝ ແ‫ݽ‬ Ƥ Ƌܸࣀ ჱჅକଐƁᇬ ᇊ ࣀ ࡦƁ6.1 ᇬ ᇊ ࢵ Y Ƒ ܸ୊คƋକଐƁࡠ ࡴܸ‫ݕ‬ ࡴ  Ɣ ୍ܻงƎଘଓƄƖ ࣠‫ރ‬ƥƆ࣠ߘࢪ ყߥ‫ ࣯ݹ‬ዙኤ Ɛ ܻ‫ [ހܾރࠢࡷߥޢ‬r ]୷ ୪༟༎ܻ‫ހ‬ቻቱ୷ ୪मअƧ ኅ ቱƄƥƔ ୍ܻงƎଘଓƄࡣ ࡷܻ‫ࡷݘ‬ƞ ࢖‫ݹݘ‬ ࠦ‫ݹ‬ƌ࢖࣐ ࡩƄࠦߘൎ൉‫ࡩݘ‬Ƅે ઻ၮࠦީࠅ ࣠ଐଌƄࡣ ࡷܻ‫ݘ‬ଐଌƄƆƎܻࢸ Ƃơ ਔ ৿ƥ‫ ۑ‬ƥ ໠ ໄ‫ߘޢࡺޯࠢހ‬Ɗ ƥƄ౰౔‫ކ‬ƥ‫ۑ‬௪ ௎౰ే‫ݓ‬ ޯଐଌƄ໠ ໄ‫ހ‬ ໠ ໄ‫ހ‬Ɗ ƥƄ౰౔‫ކ‬Ɣ ୍ܻงƎଘଓƄࡣ ࡷܻ‫ߘޢࡺࠦࡷݘ‬ƣ ܾߙࠢ‫࡯ޢ‬ၮƥ ūࠦ‫ࡩݘ ࡓ ࣀޢ‬Ƅࠦߘൎ൉‫ࡩݘ‬Ƅે ઻ၮࠦީࡩܻࣀ ࡩƄƆƎܻࢸ ଘଓƄܻࡺެყჅū –1≤r≤+1 ፥ ጘଘଓᇴᆝ ፥ ፕ Ƒ ܸ୊คƋକଐƁࡠ ࡴܸ‫ݕ‬ ଍ଉ[ Karl Pearson] Ơ ࣝࢲ ౭ౄ‫ݽ‬ ƞ Ƌᇱᆚ ቼቮƉܸ‫ݕ‬ ࡴ ‫ێ‬ r= ഥ )(‫ࢅି܇‬ ഥ) ∑(‫ࢄି܆‬ ഥ )૛∑(‫ࢅି܇‬ ഥ)૛ ඥ∑(‫ࢄି܆‬ X -X̅ = ‫ݔ‬ Y – Y̅ = ‫ݕ‬ 78 ܸୃกܸ଍ଉŽˍ ໝ ແࡦफउˍ ࠣ‫ݶ‬ Ɖܸࢤ ެߕକଐƁƢࠣ‫ࡦ ࢽޟܸ ࢲޟ‬ƁƢࠣ‫ޟ‬පඓ ൷Ɓ౭౑ࢲ କଐƁƄ ܸࢲ ƁƋኃ ᒔᐺ። ጏ࢓‫ݕ‬ ࡦƁ࣬ ࣊ Ɓ࢓ߢೕಹኃ ቮƁ೾೥࢓଍ଉƁƄ ƁƋƁ߁ ࢟ ſ ። ፎକଐ߁ ެࢲ କଐƁƢ। ्࢟ ࣝࡦƁŻܸࢤ ߾ ଔଐƁ ƛ ܸߢࢽ ࡦƁ߁ ዖዄ Ž૊ સၫƋࣝެ࠾ኃ ቮƁƊܸެࢲ ଘଐƁ r= ∑࢞‫ܡ‬ ඥ∑࢞૛ ∑࢟૛ Ơ Ž‫߾ޟݜ‬ ௪ ௎౰ే‫ݓ‬ ޯଐଌƄযওߘ Y ، X ƣ ࣠ࢵ ౰ే‫ހ‬Ɣ ୍ܻงƎƄࡣ ࡷܻ‫ ࡷݘ‬r ፥ ጘଘଓƄƣ ࣠‫ࡩߥܻރ‬ƄƢƄ߄ ႀ ޯࢵ ܻ‫ހ‬ᑑ ᐽƍƥ‫ۑۑ‬Ɔྲ ႖ܻީଐଌƄࠦߥ‫ࡩ ࣐࢖ݹ‬ƄƢƄ߄ ႀ ޯࢵ ܻ‫ހ‬Ɣ ୍ܻงƎଘଓƄࡣ ࡷܻ‫ࡷݘ‬ƅ ܻࣀ ࠁ ࡼ ࡓ ೘಼ƥ r= ඥࡺ ‫) ܇∑ () ܆∑ (ି܇܆∑ۼ‬ ∑ࢄ૛ି (∑‫)܆‬૛ ඥࡺ ∑ ࢅ૛ି (∑‫)܇‬૛ ࡼ ࡓ ೘಼Ɣ ୍ܻงƎଘଓƄࡣ ࡷܻ‫ࡷݘ‬ƣ ܾߙƇ Ƅ߄ ܻࠧࣃ ଐଌƄƌ߄ ၮ n บ คᇴᆝƥ௪ ௎౰ే‫ݓ‬ ޯ൞ᒑᐽƧ Ǝܻ‫ݘޢ‬ଐଌƄƞ Ƅ࢖༨စ ଘଓƄࣷࣳၮƄƍƄ r = ࠦ୆งଛଓƄࠦ‫ݓ‬ ‫୓ࠢހܻࡩ ࢧޢ‬กܻૻ૓ ૚ ഥ )(‫࢟ିܡ‬ ഥ) ∑(‫࢞ିܠ‬ ‫ܖ‬ ࣌࢞ ࣌࢟ ࢖࣏ࠩ ‫ݹ‬Ɣ ୍ܻงƎଘଓƄࡣ ࡷܻ‫ࡷݘ‬ƣ ܾߙ፥ ጒܻࣀ ஊै Ƅ࣊ ࢵ ࣠ࡩƄࡼ ‫ݑ‬যওߥࡩƄƇ ܻߙƄ࢖༨စ ƄƢƄ߄ ႀ ޯࢵ ƄƢ߄ ၮࡼ ࡓ ೘಼ƣ ᇴᆝ ƄƍƄƥ ‫=ݎ‬ ∑xy − ፥ ጘଘଓᇴᆝ ∑x∑y n ∑‫ݕ‬ଶ ∑‫ݔ‬ଶ ∑‫ݕ‬ଶ − ൠ൜ ݊ ݊ ൠ [ Rank Correlation Coefficient ] ฤ ค༜་ƁƑ ܸ୊คƋƁࡠ ࡴܸ‫ݕ‬ ࡴ ‫ێ‬ ට ൜∑‫ݔ‬ଶ − ‫ީޢࠦۑ‬ყ჋मऒ௪ ௎౰ే‫ݓ‬ ޯଐଌƄƐ ܻ‫ߘޢ‬ƣ ᇴᆝ ƄƍƄ ƣ ܻࡷ౰౎‫ ࢵޢ‬ว ง༟༎ƄƔ ୍ܻงƎƄࡣ ࡷܻ‫ࡷݘ‬Ƣ߄ ႀ ޯఱ ఓ ໠ ໄ‫ހ‬Ɵ ࢖ߦᑃᐽ୷ ୪मऒ d ‫ و ان‬، Ry ว ง༟༎Ƅቻቱ y ౰ే‫ݓ‬ ޯଐଌƄƥ Rxว ง༟༎Ƅቻቱ x౰ే‫ݓ‬ ޯଐଌƄƣ Ƅܻެࢠ࢖ߙƄƍƀ ࠦ୆งଛଓƄࠦߘൎ൉‫ݹݘ ࣐ ࣩހܻࡩݘ‬ว ง࢖ᑃᐽƣ ܻࡷ౰౎‫ ࢵޢ‬Ɣ ୍ܻงƎƄࡣ ࡷܻ‫ࡷݘ‬ƣ ܾߙ [ d = Ry – Rx ]ƣ ƀƧ ƀ໠ ໄޯხ჋༟༎Ƅ rs = ૚ − 39 32 79 ૟∑ࢊ૛ ࢔(࢔૛ି૚) 36 30 ୍ࠦง࢖ଐଌƄƉ ƄƥƏଘଓƄƌ߄ ၮ፥ ፕ n ƣ ƀ‫ࠁޢ ݟ‬ 35 29 Ƥܻ‫ރ‬ƌƄƇ ܻ‫ࡩީޢܻރ‬Ƅࡼ ࡷƔ ୍ܻงƎൎᑉᐽƣ ࣠ࢵ ౰ే‫ ߄ࡷݘܻࡷ ࡣހ‬ၴ ƥƀ ( ‫ ێ‬ƞ ܸޯࡴ 33 28 31 26 29 24 28 23 27 22 23 (XƯ Ɖ ƥ༝༎Ƅ࢖ዙኸ 18 ( Y ࠦၴ ƥ༝༎Ƅ࢖ዙኸ ܸୃกܸ଍ଉŽ–ໝ ແࡦफउ –ࠣ‫ݶ‬ Ɖܸࢤ ެߕକଐƁƢࠣ‫ࡦ ࢽޟܸ ࢲޟ‬ƁƢࠣ‫ޟ‬පඓ ൷Ɓ౭౑ࢲ କଐƁƄ ܸࢲ ƁƋኃ ᒔᐺ። ጏ࢓‫ݕ‬ ࡦƁ࣬ ࣊ Ɓ࢓ߢೕಹኃ ቮƁ೾೥࢓଍ଉƁƄ ƁƋƁ߁ ࢟ ſ ። ፎକଐ߁ ެࢲ କଐƁƢ। ्࢟ ࣝࡦƁŻܸࢤ ߾ ଔଐƁ ƛ ܸߢࢽ ࡦƁ߁ ዖዄ Ž૊ સၫƋࣝެ࠾ኃ ቮƁƊܸެࢲ ଘଐƁ X Y ࡠ இॅ Ɓ Ơ ࣝࢲ ౭ౄ‫ݽ‬ Ƒ ܸ୊คƋƁࡠ ࡴܸ‫ݕ‬ ࡴƂ ܸࢽ ߾ 6.1ƞ Ƣ߁ ၱ (X- ̅X) (Y - ̅Y) ‫ݔ‬ ‫ݕ‬ (X- ̅X)2 (Y- Y)2 ‫ݕݔ‬ ‫ݔ‬2 23 18 -8.2 -7.7 63.14 67.24 28 23 -3.2 -2.7 8.64 10.24 27 22 29 24 31 26 33 28 35 29 36 30 39 32 ∑x ∑Y =281 =232 -4.2 -3.7 -2.2 -1.7 -0.2 1.8 4.8 - 17.64 3.74 13.69 7.29 4.84 2.89 -0.06 3.3 12.54 14.44 10.89 6.3 49.14 60.84 39.69 4.3 7.8 59.29 0.3 2.3 3.8 15.54 ‫ݕ‬2 - 0.04 4.14 0.09 3.24 20.64 5.29 23.04 177.46 18.49 201.56 ‫ ۑ‬ X̅ = ∑X /n = 281/9 = 31.2 Y̅ = ∑Y /n = 232/9 = 25.7 r= ത) ∑(ଡ଼ି௑ത )(ଢ଼ି௒ ത)మ ඥ∑(ଡ଼ି௑ത )మ∑(ଢ଼ି௒ X- X̅ = ‫ݔ‬ Y - Y̅ = ‫ݕ‬ r= r= ∑௫୷ ඥ∑௫మ ∑௬మ ଵ଻଻.ହ଼ ඥ(ଶ଴ଵ.ହ଺)(ଵ଺ଷ.ଽଵ) 0.98 == ଵ଻଻.ହ଼ √ଷଷ଴ଷ଻.଻ === ଵ଻଻.ସ଺ √ଷଷ଴ଷ଻.଻ = ଵ଻଻.ସ଺ ଵ଼ଵ.଻଺ଶ଻ = 0.9763 ≈ ࡼ ࡓ ೘಼Ɣ ୍ܻงƎଘଓƄࡣ ࡷܻ‫ࡷݘ‬ƣ ܾߙƇ Ƅ߄ ܻࠧࣃ ଐଌƄƌ߄ ၮ n บ คᇴᆝƥ௪ ௎౰ే‫ݓ‬ ޯ൞ᒑᐽƧ Ǝܻ‫ݘޢ‬ଐଌƄƞ Ƅ࢖༨စ ଘଓƄࣷࣳၮƄƍƄ ፥ ጘଘଓᇴᆝࠢ‫ࠁ ࣀܻހ‬ 80 ܸୃกܸ଍ଉŽ–ໝ ແࡦफउ –ࠣ‫ݶ‬ Ɖܸࢤ ެߕକଐƁƢࠣ‫ࡦ ࢽޟܸ ࢲޟ‬ƁƢࠣ‫ޟ‬පඓ ൷Ɓ౭౑ࢲ କଐƁƄ ܸࢲ ƁƋኃ ᒔᐺ። ጏ࢓‫ݕ‬ ࡦƁ࣬ ࣊ Ɓ࢓ߢೕಹኃ ቮƁ೾೥࢓଍ଉƁƄ ƁƋƁ߁ ࢟ ſ ƛ ܸߢࢽ ࡦƁ߁ ዖዄ Ž૊ સၫƋࣝެ࠾ኃ ቮƁƊܸެࢲ ଘଐƁ Sx = ඥ∑(࢞ − ࢞ ̅ )૛ /࢔ ። ፎକଐ߁ ެࢲ କଐƁƢ। ्࢟ ࣝࡦƁŻܸࢤ ߾ ଔଐƁ r= ૚ ഥ )(‫࢟ିܡ‬ ഥ) ∑(‫࢞ିܠ‬ ‫ܖ‬ ࣌࢞ ࣌࢟ = ඥ૛૙૚. ૞૟/ૢ = = √૛૛. ૜૞ = 4.72 Sy = ඥ∑(࢟ − ࢟ ̅ )૛ /࢔ = ඥ૚૟૜. ૢ૚/ૢ = = √૚ૡ. ૛૚ = 4.26 r= భ ∑(୶ି௫̅ ౤ ఙೣ ఙ೤ = 5 5 r = 0.98 4 12 )(୷ି௬ത) 14 11 భ (ଵ଻଻.ସ଺) వ (ସ.଻ଶ)(ସ.ଶ଺) 6 14 = ଵଽ.଻ଵ଻ ଶ଴.ଵ଴ = 0.98 ࠦ୆งଘଓƄƇ ܻ‫ࡩީޢܻރ‬Ƅࡼ ࡷƔ ୍ܻงƎଘଓƄࡣ ࡷܻ‫ ࠁ ࣀ ݪࡷݘ‬ƀ : ( ‫ ێ‬ƞ ܸޯࡴ 6 11 8 7 10 9 10 11 12 7 13 3 X Y ࡠ இॅ Ɓ ፥ ጘଘଓƄƣ ࣠‫ࡩߥܻރ‬ƄƢƄ߄ ႀ ޯࢵ ܻ‫ހ‬ᑑ ᐽƍƥ‫ۑۑ‬Ɔྲ ႖ܻީଐଌƄࠦߥ‫ࡩ ࣐࢖ݹ‬ƄƢƄ߄ ႀ ޯࢵ ܻ‫ހ‬Ɣ ୍ܻงƎଘଓƄࡣ ࡷܻ‫ࡷݘ‬ƅ ܻࣀ ࠁ r= X Y 12 7 13 ඥே ∑௑ మି (∑ଡ଼)మ ඥே ∑ ௒మି (∑ଢ଼)మ Ɔྲ ႖ܻީଐଌƄࠦߥ‫ࡩ ࣐࢖ݹ‬ƄƢƄ߄ ႀ ޯࢵ ܻ‫ހ‬Ɣ ୍ܻงƎଘଓƄࡣ ࡷܻ‫ࡷݘ‬ƅ ܻࣀ ࠁ 6.2ơ ƥ߄ ၴ XY 64 49 7 56 5 11 6 5 4 14 14 12 49 121 8 11 144 100 84 110 6 Y2 169 11 9 X2 39 10 10 81 3 ୒ ∑ଡ଼ଢ଼ି( ∑ଡ଼ )( ∑ଢ଼ ) 90 66 84 55 70 48 100 9 81 36 121 25 121 36 25 16 196 196 144 ܸୃกܸ଍ଉŽ–ໝ ແࡦफउ –ࠣ‫ݶ‬ Ɖܸࢤ ެߕକଐƁƢࠣ‫ࡦ ࢽޟܸ ࢲޟ‬ƁƢࠣ‫ޟ‬පඓ ൷Ɓ౭౑ࢲ କଐƁƄ ܸࢲ ƁƋኃ ᒔᐺ። ጏ࢓‫ݕ‬ ࡦƁ࣬ ࣊ Ɓ࢓ߢೕಹኃ ቮƁ೾೥࢓଍ଉƁƄ ƁƋƁ߁ ࢟ ſ ። ፎକଐ߁ ެࢲ କଐƁƢ। ्࢟ ࣝࡦƁŻܸࢤ ߾ ଔଐƁ ƛ ܸߢࢽ ࡦƁ߁ ዖዄ Ž૊ સၫƋࣝެ࠾ኃ ቮƁƊܸެࢲ ଘଐƁ ∑X=79 r= = ∑XY =702 ∑Y=99 ୒ ∑ଡ଼ଢ଼ି( ∑ଡ଼ )( ∑ଢ଼ ) ඥே ∑௑ మି (∑ଡ଼)మ ඥே ∑ ௒మି (∑ଢ଼)మ ଵ଴(଻଴ଶ)ି( ଻ଽ )(ଽଽ ) = ඥଵ଴( ଻ଵହ)ି (଻ଽ)మ ඥଵ଴(ଵ଴଼଻)ି (ଽଽ)మ ଻଴ଶ଴ି଻଼ଶଵ r= √଻ଵହ଴ ି ଺ଶସଵ √ଵ଴଼଻଴ିଽ଼଴ଵ ି ଼଴ଵ r== ଽ଼ହ.ଶ଻ = - 0.813 = ି ଼଴ଵ ∑X2 =715 √ଽ଴ଽ √ଵ଴଺ଽ = ∑Y2 =1087 ି ଼଴ଵ (ଷ଴.ଵସ )(ଷଶ.଺ଽ) บ คᇴᆝ ƥƅ ൎ൉࣍ 10ࡼ ࡷࠦ‫ࡷ ࡓ࣠ރ‬ƌܻࢧ ޯߘଘଓƄࠦ‫ޢ‬ᆿᆝ ፥ ጿƅ ൎ൉࣐ ࡩƄࡼ ࡷࠦެ‫ݑޢ‬Ƈ ౰ేޯࠀ Ƅ ( ‫ ێ‬ƞ ܸޯࡴ ፥ ጘଘଓƄ࣠ࠒ ެࡩƄે ઻ၮƇ ܻ‫ݹܻࢠޢ‬༟༎Ƅƥƌܻࢧ ޯߘଘଓƄ፥ ጘƌܻࡷƣ ܻႁ ޯࡷƄ፥ ጿࡴ ࠩ༬࿸ ܻ୓ค 78 84 36 51 98 91 25 60 75 68 82 62 90 86 62 58 ‫ۑ‬ว ง༟༎ƄƔ ୍ܻงƎƄࡣ ࡷܻ‫ ߄ࡷݘ‬ၴ ƥƀ 65 53 39 47 ƌܻࢧ ޯߘଘଓƄ Ƈ ܻ‫ݹܻࢠޢ‬༟༎Ƅ ࡠ இॅ Ɓ ƌܻࢧ ޯߘଘଓƄƆƌܻࡷ፥ ጿࠦၴ Ǝƌ፟ ጕƌଛଓ 1 ୍ࠦง༟༎Ƅžܻ࣐ ‫ܺݑ‬ᄣ໠ ໄ‫ކ‬ƌܻଐଌƄൎᇩᇍ ࡩܻ‫ ߄ݹ‬ၮܻࢧ ‫ކ‬ƅ ൎ൉࣐ ࡩƄƇ ܻၴ Ǝƌว งमए Ɔƌܻଐଌࠦߥ‫ࡩ ࣐࢖ݹ‬Ƅࢪ ߦ୐กƥ‫ۑ‬10୍ࠦง༟༎Ƅ፥ ፕ ƥ 98 ࠦၴ Ǝኆ ᒗᐽ୍ࠦงƎે ઻ၮƀƥ 25 ࠦၴ Ǝኆ ቱƄ፥ ፕ ƥ ࠦၴ Ǝƌે ઻ၮଛଓ 10 ୍ࠦงƎે ઻ၮƄƥ 47 ፥ ፕ ƥࠦၴ Ǝƌ፟ ጕƌଘଓ 1 ୍ࠦง༟༎Ƅࣩ ࣐ ‫ ࠁޢ ݟކݘ‬Ƈ ܻ‫ݹܻࢠޢ‬༟༎Ƅ (91 ፥ ፕ ƥ 82 ܸୃกܸ଍ଉŽ–ໝ ແࡦफउ –ࠣ‫ݶ‬ Ɖܸࢤ ެߕକଐƁƢࠣ‫ࡦ ࢽޟܸ ࢲޟ‬ƁƢࠣ‫ޟ‬පඓ ൷Ɓ౭౑ࢲ କଐƁƄ ܸࢲ ƁƋኃ ᒔᐺ። ጏ࢓‫ݕ‬ ࡦƁ࣬ ࣊ Ɓ࢓ߢೕಹኃ ቮƁ೾೥࢓଍ଉƁƄ ƁƋƁ߁ ࢟ ſ ƛ ܸߢࢽ ࡦƁ߁ ዖዄ Ž૊ સၫƋࣝެ࠾ኃ ቮƁƊܸެࢲ ଘଐƁ Ɖܸࢤ ެߕକଐƁ Ƅ ܸ‫ݶܸ ࢝ޟ‬༜་Ɓ X Y 39 47 62 58 65 53 90 86 82 62 75 68 25 60 98 36 - rs =૚ − rs =૚ − 3 5 4 9 8 6 1 ࠣ୊คƋ ( Y ࠣ୊คƋ – ( x ࠣ୊คƋ 1 2 Y 3 4 9 6 0 2 0 5 84 7 8 - 0 -1 10 - 2 7 10 2 D 2 - d2 4 4 0 0 4 1 -4 16 0 0 -1 ∑d =0 0 1 ∑d2 =30 ଺∑ୢమ r = ૚− 83 X Ơ ܸࡴ౭ౄަࢲ ฤ ค༜་Ɓࣇ ୊คƋƁࡠ ࡴܸ‫ݕ‬ ࡴƂ ܸࢽ ߾ ‫ ێ‬ƞ Ƣ߁ ၱ 91 51 78 ࠣ୊คƋ ። ፎକଐ߁ ެࢲ କଐƁƢ। ्࢟ ࣝࡦƁŻܸࢤ ߾ ଔଐƁ ୬(୬మିଵ) ଺(ଷ଴) ଵ଴(ଵ଴଴ିଵ) ଵ଼଴ ଽଽ଴ = 1 – 0.1818 = 0.818 ܸୃกܸ଍ଉŽ–ໝ ແࡦफउ –ࠣ‫ݶ‬ Ɖܸࢤ ެߕକଐƁƢࠣ‫ࡦ ࢽޟܸ ࢲޟ‬ƁƢࠣ‫ޟ‬පඓ ൷Ɓ౭౑ࢲ କଐƁƄ ܸࢲ ƁƋኃ ᒔᐺ። ጏ࢓‫ݕ‬ ࡦƁ࣬ ࣊ Ɓ࢓ߢೕಹኃ ቮƁ೾೥࢓଍ଉƁƄ ƁƋƁ߁ ࢟ ſ ። ፎକଐ߁ ެࢲ କଐƁƢ। ्࢟ ࣝࡦƁŻܸࢤ ߾ ଔଐƁ ƛ ܸߢࢽ ࡦƁ߁ ዖዄ Ž૊ સၫƋࣝެ࠾ኃ ቮƁƊܸެࢲ ଘଐƁ ƍ Ɖܸࢽ ࡦƁࡠ ࢤ ߣࡦƁ௧ ோƋܸೕೆ X Y 77 35 54 58 27 60 Ƥܻ‫ރ‬ƌƀƇ ܻ‫ࡩީޢܻރ‬Ƅࡼ ࡷƔ ୍ܻงƎൎᑉᐽƣ ࣠ࢵ ౰ే‫ ࠁ ࣀ ݪࡷݘܻࡷ ࡣހ‬ƀ‫ۑ‬1 52 40 14 50 35 40 90 35 25 56 56 34 60 42 ࡣ ࡷܻ‫ ࠁ ࣀ ݪࡷݘ‬ƀ‫࢖ۑ‬ᐳ႖ơ ൎ൉ၯ ቿቷႅ ࣀ ଐଌƄƅ ܻ‫ݓޢ‬ ࡩƄƢܻ‫ݹ‬Ƅƌ߄ ၮƥ໠ ໄߦ࣌ ࣠ଐଌƄƎܻዙኸ Ƅ໠ ໄ୍งࠦ‫ࡩޯܻࡩޢ‬ƄƇ ܻ‫ࡩީޢܻރ‬Ƅ‫ۑ‬2 X Y 30 1 32 0 35 2 40 5 48 2 50 4 ࠦႅ ‫ޢ‬ჴެࡩƄྲ ႞ߙƥƔ ୍ܻงƎൎᑉᐽƣ ࣠ࢵ ౰ే‫ހ‬ 52 6 55 5 57 7 61 8 บ คᇴᆝ ƥܻީࡩܻ࣍ 12ࡼ ࡷࠦ‫ ࠅޯࡺܻݑޢࠦࡷ ࡓ࣠ރ‬ଘଓƄࠦࡷ߄ உै Ƅࠦ‫ޢ‬ᆿᆝ ፥ ጿƅ ൎ൉࣐ ࡩƄࡼ ࡷࠦެ‫ݑޢ‬Ƈ ౰ేޯࠀ Ƅ : (3) ፥ ጘଘଓƄ࣠ࠒ ެࡩƄે ઻ၮ፥ ጕƄ߄ ‫ޢ‬ଐଌƄࡣ ࡺ‫ࡩݘ‬Ƅƥ፥ ፎ ܻࡺޯࠅ ଘଓƄࡣ ࡺ‫ࡩݘ‬Ƅ፥ ጿࡴ ࠩ༬࿸ ܻ୓ค ‫ێ‬ว ง༟༎ƄƔ ୍ܻงƎƄࡣ ࡷܻ‫ ߄ࡷݘ‬ၴ ƥƀ 41 54 51 48 55 49 34 42 62 52 55 45 47 60 47 50 59 52 40 47 65 55 62 52 84 ࡣ ዙኸ ፥ ፎ ܻࡺޯࠅ Ƅ ࡣ ዙኸ ፥ ጕƄ߄ ‫ࡷޢ‬ ܸୃกܸ଍ଉŽ–ໝ ແࡦफउ –ࠣ‫ݶ‬ Ɖܸࢤ ެߕକଐƁƢࠣ‫ࡦ ࢽޟܸ ࢲޟ‬ƁƢࠣ‫ޟ‬පඓ ൷Ɓ౭౑ࢲ କଐƁƄ ܸࢲ ƁƋኃ ᒔᐺ። ጏ࢓‫ݕ‬ ࡦƁ࣬ ࣊ Ɓ࢓ߢೕಹኃ ቮƁ೾೥࢓଍ଉƁƄ ƁƋƁ߁ ࢟ ſ ƛ ܸߢࢽ ࡦƁ߁ ዖዄ Ž૊ સၫƋࣝެ࠾ኃ ቮƁƊܸެࢲ ଘଐƁ ። ፎକଐ߁ ެࢲ କଐƁƢ। ्࢟ ࣝࡦƁŻܸࢤ ߾ ଔଐƁ Ļ Ƌ [ Regression] Ő ńŏư ſʆ ń 85 ܸୃกܸ଍ଉŽˍ ໝ ແࡦफउˍ ࠣ‫ݶ‬ Ɖܸࢤ ެߕକଐƁƢࠣ‫ࡦ ࢽޟܸ ࢲޟ‬ƁƢࠣ‫ޟ‬පඓ ൷Ɓ౭౑ࢲ କଐƁƄ ܸࢲ ƁƋኃ ᒔᐺ። ጏ࢓‫ݕ‬ ࡦƁ࣬ ࣊ Ɓ࢓ߢೕಹኃ ቮƁ೾೥࢓଍ଉƁƄ ƁƋƁ߁ ࢟ ſ ። ፎକଐ߁ ެࢲ କଐƁƢ। ्࢟ ࣝࡦƁŻܸࢤ ߾ ଔଐƁ ƛ ܸߢࢽ ࡦƁ߁ ዖዄ Ž૊ સၫƋࣝެ࠾ኃ ቮƁƊܸެࢲ ଘଐƁ ࠦߘൎ൉ၮࠦ‫ݹ‬ƀƠ ܻެࠧบ คᇴᆝ ƄƍƄܻࡺ‫ߙޢ‬ơ Ɓܻࣀ ჴࡩƄযঠ‫ݑެ ߄ࡷܻݹ‬Ə౰౎‫ކ‬Ɣ ୍ܻงƎଘଓƄƇ ൎᇩᇍ ࣃ ࡷƣ ܾ‫ހ‬Ƣ߄ ߥ‫ ྲރܻߙޢࡺܻކ‬႖Ƅ Ɔ࣠ߘࢪ ყߥ‫ݹ‬Ƨ ኅ ቱƄƔ ୍ܻงƎଘଓƄࡣ ࡷܻ‫ߘࠦݑ ࡼ ࣍࢖ݹ ࢇࡷݘ‬ൎ൉‫ࡩݘ‬Ƅमअ߄ ߥ‫ކ‬যঠ‫ݹ‬ƥ‫ۑ‬ଘଓƢƀƇ Ƅ౰ే‫ݓ‬ ޯଐଌƄࡼ ࡷƉ ƥƏ໠ ໄ‫ހ‬ ࣓ީჱޯࡩƄƇ ൎ ᇩᇍ ࣃ ࡷࡣ ၰ ፥ ጿƆ߄ ၮܻࣀ ଐଌƄࡼ ࡷࡼ ࡓ ๻๬ึଘଓƔ ୍ܻงƎଘଓƄࡼ ࡓ ᚻࡩ‫ۑ‬౰౗ࡁ ƀƥƀ௪ ௎౰ే‫ݓ‬ ޯࡷ໠ ໄ‫ހ‬Ɣ ୍ܻงƎଘଓƄ ‫ۑ‬मअ߄ ߥޯࡩƄƥƀ ፥ ጿࠦႅ ޯެჴࣀ ଐଌƄࠦߘൎ൉‫ࡩݘ‬ƄƢƄ߄ ႀ ޯࢵ Ƅࡣ ࡷƀે ઻ၮ౰౗ࡁ Ƅƥƀ௪ ௎౰ే‫ݓ‬ ޯࡷ໠ ໄ‫ߘࠦހ‬ൎ൉‫ࡩݘ‬Ƅࠦࢵ ƄƎƌযঠ‫ݹ‬ƎƄ߄ ༨စ ଘଓƄ፥ ጿ ‫ۑ‬Ƈ Ƅ౰ే‫ݓ‬ ޯଐଌƄƤ߃ ࠧ߄ ၰ ƀযওߥ‫ހ‬मअ߄ ߥޯࡩƄƥƀ࣓ީჱޯࡩƄ፥ ጿƆ߄ ၮܻࣀ ଐଌƄ ᑑ ᐽƍƥ y ‫ و‬Xযওߥࡩࠦ୆งଛଓƄƇ ܻ‫ࡩީޢܻރ‬ƄƓ ౰౔ߦ‫ީ࣓ރ‬ჱޯࡩƄƥ࣊ ‫ ࣀޢ‬ხࡩƄ࣯ ࣐ உै ƄƎƄ߄ ༨စ ଘଓƄࠦߥ‫ࡩޯ࣠ࢠޢ ߰ ࣍࢖ݹ‬ƥ X 8 Y 3 2 11 6 10 3 6 5 8 4 12 12 1 9 4 Ƈ Ƅƌ࢖ߦࡷ ࡼ ࡷࠦެ‫ࡩݘޢ‬ 6 9 1 14 ‫ ۑ‬ᇯ ᇍ ࣃ ࡩƄ፥ ጿ໠ ໄ୍෺Y ‫ و‬X ௪ ௎౰ే‫ݓ‬ ޯଐଌƄ໠ ໄ‫ߘࠦހ‬ൎ ൉‫ࡩݘ‬Ƅࢪ ࡓ ‫ݹݘ‬Ƨ ኅ ቱƄƎܻࣃ ჴ჈ଘଓƄᇯ ᇍ ࢸ Y ، X ௧ ோ౭ౄ‫ݐ‬ ެ଍ଉƁໝ ແ‫ݽ‬ Ƌܸࣀ ჱჅକଐƁᇬ ᇊ ࢵ ‫ ێ‬ᇬ ᇊ ࢵ (x ໠ ໄ‫ߘࠦހ‬ൎ൉‫ࡩݘ‬Ƅƣ Ƅᇯ ᇍ ࣃ ࡩƄࡼ ࡷ߰ ࢥ ୓෺ƥ‫ۑ‬Ƈ ܻ‫ࡩީޢܻރ‬ƄƤ߃ ಛಗƧ Ǝܻࣃ ჴ჈ଘଓƄᇯ ᇍ ࣃ ࡩƄၒল ࣠‫ ۑ ݹ‬ᇯ ᇍ ࢸ ᑑ ᐽƍƥƎܻࣃ ჴ჈ଘଓƄƔ ܻߥ‫ރ‬໠ ໄ‫ݹ࣠ߙ ࢇހ‬ƣ Ƅࡼ ࡓ ೘಼যওߥޯࣀ ଐଌƄ࣊ உै ܻߙᑑ ᐽኅ ቱƥ‫୍ܻۑ‬෺࢖ߥ‫ߘࠦ ࠀ ࣐ޢࠦކ‬ൎ൉ၮ y) ‫و‬ 86 ܸୃกܸ଍ଉŽˍ ໝ ແࡦफउˍ ࠣ‫ݶ‬ Ɖܸࢤ ެߕକଐƁƢࠣ‫ࡦ ࢽޟܸ ࢲޟ‬ƁƢࠣ‫ޟ‬පඓ ൷Ɓ౭౑ࢲ କଐƁƄ ܸࢲ ƁƋኃ ᒔᐺ። ጏ࢓‫ݕ‬ ࡦƁ࣬ ࣊ Ɓ࢓ߢೕಹኃ ቮƁ೾೥࢓଍ଉƁƄ ƁƋƁ߁ ࢟ ſ ƛ ܸߢࢽ ࡦƁ߁ ዖዄ Ž૊ સၫƋࣝެ࠾ኃ ቮƁƊܸެࢲ ଘଐƁ ። ፎକଐ߁ ެࢲ କଐƁƢ। ्࢟ ࣝࡦƁŻܸࢤ ߾ ଔଐƁ ࠦ‫ޯ࣠ߘݘ‬ଐଌƄ Y ࠦࡺ‫ߘޢ‬ƣ ܾߙ 4 = x ൎ൉޲ࡷࠦࡺ‫ߘޢ‬Ƨ ଛଓƥ‫ ۑ‬X)যওߥࡩቿቷ‫ߥܻހ‬ଐଌƄ Y)যওߥ‫ީ࣓ހ‬ჱޯࡩƄƓ ࢖‫ݓ‬ ‫ހ‬ ࠦ‫ޯ࣠ߘݘ‬ଐଌƄ y ࠦࡺ‫ߘޢ‬ƆžƄ࢖ߥ‫ހ‬ƥ Xࠦࡺ‫ߘޢ‬Ɵ ࣠ߙযওߥޯࣀ ଐଌƄ࣊ உै Ƅે ઻ၮቿቷ޲ࡺଐଌƄࠦ‫ ࢵޢ‬Ƅ༟༎Ƅࠦߙܻࣀ ଐଌƄ፥ ፕ ƣ ࣠ࡓ ޯࢵ ‫୍ܻۑ‬෺࢖ߥ‫ ކ‬12፥ ፕ 4 = xࠦࡺ‫ࡩߥޢ‬Ɔ࢖࣌ ܻެଐଌƄƥ ࠦ‫ࡩޯ ࠖ࢖ހ‬ƄƤ߃ ࠧƇ Ǝಒ౷‫ކ‬ƄƍƄࠢ‫ރ‬Ƅᑑ ᐽƍ཯ཏ‫ݹݘ‬ƥ‫ߘࠦ ࠀ ࣐ޢࠦۑ‬ൎ൉ၮ፥ ፕ Y ‫ و‬X໠ ໄ‫ߘࠦހ‬ൎ൉‫ࡩݘ‬Ƅƣ ƄƓ ࢖ߦެࡩƥ Ɔ࢖࣌ ܻެଐଌƄ y যওߘ࣊ ࢵ ࣠ޯࡷƅ ܻࣀ ࠁ Ꭸ᎙ࠢ‫ރ‬ƄƥƆ߄ ၰ Ƅƥƞ ƥ࢖࣌ ‫ ݝ‬༨ဈ N Ƈ Ƅ࢖ଐଌƄࡼ ࡷƄ౰ేީࡁ Ƅƌ߄ ၮ ‫ۑ‬যওߥޯࣀ ࡷ࣊ ࠀ ે ઻ၮ୍ܻ෺࢖ߥ‫ކߥ ݊ކ‬Ɣ ܻߥ‫ࠦ ރ‬ၮ࣠ዙಣಝે ઻ၮࡣ ࢧ ༨စ ܻ୐คܾߙ xࠦࡺ‫ߘޢ‬ᇯ ᇍ ࡩ [ Least Square Method ] ƣ ࢓‫ݐ‬ ࢤ ࡦƁƄ ܸ‫ݕ‬ ‫ݽ‬ ࢓଍ଉƁࠣߢ‫ ࣊࢓ݶ‬ ࠦߘൎ൉ၮƥƀࠦ‫ ࠀ ࣐ޢ‬ƣ ࣠ࡓ ‫ކ‬ƣ Ƅࡼ ࡓ ೘಼ ‫ ۑ‬ᇯ ᇍ ࣃ ࡩƄ፥ ጿ Y ‫ و‬، x ໠ ໄ‫ߘࠦހ‬ൎ൉‫ࡩݘ‬Ƅƣ ܻ‫ހ‬ƣ ܻߦ࡯ࢵ ܻ‫ ྲރ‬႖Ƅሯሡሁ ‫ۑ‬Ɣ ܻߥެࡩƄࡼ ࡷࠦၮ࣠ࡺ഻വറযওߥޯࣀ ࡷ࣊ ࠀ ࢇ ‫ކ࣠ߙޢ‬ቿᇝᇍ ࣃ ࡷ፟ ፑƄơ ƥ࣓‫ ࣐ ࣯ކ‬உै Ƅ࣓ީჱޯࡩƄቿᇝᇍ ࣃ ࡷƥ‫ ࠀ ࣐ޢࠦۑ‬౰ేၩ ، Ʀ ࢖‫ݓ‬ ࢧ ࡩƄ Ƈ ܻ‫࢖ހݘ‬ଐଌƄ ࠦߥ‫ ࣍࢖ݹ‬፥ ፕ যওߥޯࣀ ଐଌƄ ࣊ உै Ƅ ᑑ ᐽƍ ƌܻ༬࿻ ଗଓ ࡣ ࡺ‫ޯݘ‬ఱ ఠ ཯དྷࡩƄ Ɵ ࢖࣐ ࡩƄ Ʀ ߄ ၰ Ƅ، ࢖‫ݓ‬ ࢢ ƀžܻ࣐ ࠀ ଘଓƄ Ƈ ܻ‫ ୹ހݘ‬୪Ɩ ࣠༬ீ ࡣ ‫ݘ‬༬ဈ ཯དྷࡩƄቿቱƌܻ‫ݘ‬ଐଌƄƇ ൎ൉ࡷܻ‫ࡷݘ‬যওߘƌܻ༬࿻ ƀ፥ ጿࠦߥ‫ࡩ ࣐࢖ݹ‬ƄƤ߃ ࠧ࢝ ࠑ ࡯୓ขƥ ፥ ጿܻᐱᐢ᠊ܻૻ૓ࡼ ࡓ ೘಼ 7.1 Ǝܻࣃ ჴ჈ଘଓƄᇯ ᇍ ࢸ ፥ ጿƇ ܻ‫ࡩީޢܻރ‬Ƅࢇ ߙ࣠‫ݹ‬Ƨ ኅ ቱƄযওߥޯࣀ ଐଌƄ࣊ ࠀ ቿቱƌܻ‫ࡷݘ‬ƥ‫ ࡓ ࡼۑ‬೘಼ܻࡷ :ƆƎ࣠ࢧ ࡩƄ Y = a + bx ‫ۑ‬যওߥޯࣀ ଐଌƄ࣊ உै Ƅܻޯ൞൙‫ ࡷݘ‬b ، a ‫ࠁޢ ݟ‬ ࠦၮ࣠ዙಣಝࢇ ߙ࣠‫ݹ‬ƣ ƀযওߥޯࣀ ଐଌƄ࣊ ॵॱ५ࡼ ࡓ ೘಼ཀྵདྷࠁ ، b ، a ໠ ໄޯ൞൙‫ݘ‬ଐଌƄযওߘƅ ܻࣀ ࠁ ፥ ፕ ቿᇝᇍ ࣃ ଐଌƄƣ ƀ‫ ࠁޢ ݟ‬ƥ ‫ܻࡩޢࠦۑ‬ၮƆžܻߦࡁ Ƈ Ƅƍࠦߘ࢖࣐ ‫ހ‬Ƈ ܻ൞൙‫ݘ‬ଐଌƄযওߘƅ ܻࣀ ஊै ࡣ ‫ݶ‬ ܻࣀ ଐଌƄƦ ߄ ၰ ƀ፥ ፕ ቿቱܻࣀ ଐଌƄƣ ܾߙᑑ ᐽኅ ቱ ،Ɣ ܻߥެࡩƄ Ɵ ࢖࣐ ࡩƄƤ߃ ࠧࡣ ࢥ ߙƄƣ ƀଘଓƄमअ߄ ߥޯࡩƄƄ߃ ࠧƅ ܻࣀ ჴၰ ଘଓƟ ࢖࣐ ࡩƄࡼ ࡷਔ ৲߄‫ࡩݘ‬Ƅƌ࣠ࠅ ƥࡼ ࡷࡴ ‫ ݌‬༟༎Ƅે ઻ၮƥ Ƣ߄ ႀ ޯࣀ ყࢵ ƅ ॅळ࣐ ଐଌƄ࣊ உै Ƅƣ Ƅ‫ ࠁޢ ݟ‬ƥ‫ۑ‬Ʀ ࢖‫ݓ‬ ࢧ ࡩƄƇ ܻ‫࢖ހݘ‬ଐଌƄࠦߥ‫ ࣍࢖ݹ‬፥ ፕ ƎƄ߄ ༨စ ଘଓƄƇ ൎᇩᇍ ࣃ ଐଌ ࣓ީჱޯࡩƄžܻ࣐ ࠀ Ƅƣ ࣠ࡓ ‫ޢ ݟކ‬༨ဂ ࠦߘኆ ቱƄࡼ ࡷ࣊ உै Ƅᑑ ᐽƍƣ ࣠ࡓ ‫ݹ‬ƣ ƀ‫ެܻ ࢵ ݪ‬ଐଌƄࡼ ዙኽᑑ ᐽኅ ቱ ،࣓ީჱޯࡩƄƓ Ƅ୹ ‫ ݌‬ଛଓ ‫ܻۑ‬ಛಗƆ࢖࣌ ܻެଐଌƄࠦࡺ ‫ࡩߥޢ‬ƄƥƆ߄ ܻࠧࣃ ଐଌƄ ࠦࡺ‫ࡩߥޢ‬Ƅ໠ ໄ‫ހ‬Ɵ ƥ࢖ߦࡩƄ࣠ࠧमअ߄ ߥޯࡩƄžܻ࣐ ࠀ ܾ‫ࠧެܻހ‬ƌ࣠ࢧ ߥଐଌƄƥ‫ۑ‬Ƅ߄ ၴ Ɔ౰ే‫ݓ‬ ࢢ ࢇ ߙ࣠‫཯ކ‬དྷࡩƄࠦࡺ‫ ࣀޯߥޢ‬ଐଌƄƔ ࣐࣠ உै Ƅࡣ ࢥ ߙƄે ઻ၮơ ࣠ࢧ ஊै Ƅ݊ ‫ޯ ࣐ޢ‬ఱ ఝƦ ࢖‫ݓ‬ ࢧ ࡩƄƇ ܻ‫࢖ހݘ‬ଐଌƄࠦߥ‫ ࣍࢖ݹ‬ƢƄ߄ ႀ ޯࢵ ܻ‫ހ‬ƥ ƥƆ߄ ‫ ࢸޢ‬Ǝƥࠦࡺ࣏ ެࡷࠦߥ‫ހ ࣐࢖ݹ‬Ɣ ࣐࣠ உै Ƅࡣ ࢥ ߙƄે ઻ၮࡣ ࢧ ༨စ ƣ Ƅ፥ ፕ ቿᇝᇍ ࣃ ଐଌƄƥ ، ( ‫ ۑ‬ᇯ ᇍ ࢸ Ɣ ܻߥެࡩƄ ‫ۑ‬Ʀ ࢖‫ݓ‬ ࢧ ࡩƄƇ ܻ‫࢖ހݘ‬ଐଌƄƀ߄ ީࡷࠢࡷ߄ ߥ‫ࡷܻݹ‬Ƅ߃ ࠧ 87 ܸୃกܸ଍ଉŽ–ໝ ແࡦफउ –ࠣ‫ݶ‬ Ɖܸࢤ ެߕକଐƁƢࠣ‫ࡦ ࢽޟܸ ࢲޟ‬ƁƢࠣ‫ޟ‬පඓ ൷Ɓ౭౑ࢲ କଐƁƄ ܸࢲ ƁƋኃ ᒔᐺ። ጏ࢓‫ݕ‬ ࡦƁ࣬ ࣊ Ɓ࢓ߢೕಹኃ ቮƁ೾೥࢓଍ଉƁƄ ƁƋƁ߁ ࢟ ſ 7.1 ። ፎକଐ߁ ެࢲ କଐƁƢ। ्࢟ ࣝࡦƁŻܸࢤ ߾ ଔଐƁ ƛ ܸߢࢽ ࡦƁ߁ ዖዄ Ž૊ સၫƋࣝެ࠾ኃ ቮƁƊܸެࢲ ଘଐƁ ࣊ ‫ ࣀޢ‬ხࡩƄƎƄ߄ ༨စ ଘଓƄ࣊ ࠀ ቿቱƌܻ‫ࡷݘ‬Ꭸ፻ܻ‫ݘ‬ଐଌࠦ‫ݹ‬मअ߄ ߥ‫ކ‬যওߘƅ ܻࣀ ჴၰ Ƅ፥ ጿƦ ࢖‫ݓ‬ ࢧ ࡩƄƇ ܻ‫࢖ހݘ‬ଐଌƄࠦߥ‫୓࡯ ࠑ ࢝ ࣍࢖ݹ‬ขƥ ‫ۑ‬፥ ጒܻࣀ ஊै Ƅܻ࣐ࠩ ࢵ ƥࡼ ‫ݑ‬যওߥࡩƄƇ ܻߙƄ࢖༨စ ƄƇ ܻ‫ ୹ހݘ‬୪Ɩ ࣠ዙಣಝ౰ే‫ݓ‬ ࢧ ‫ކ‬Ɛ ܻࢵ ƀે ઻ၮ b ، a) ፥ ፕ ࣊ ‫ ࣀޢ‬ხࡩƄƎƄ߄ ༨စ ൎᑉᐽࠦ‫ ࣐ޢ‬உै Ƅቿቱƌܻ‫ݘ‬ଐଌƄ Y = a + bx b= ፥ ፕ ቿቱƌܻ‫ݘ‬ଐଌƄ཯དྷ൞൙‫ࡷݘ‬मअ߄ ߥ‫ކ‬ƣ ܾߙ …..(1) ∑௫௬ ∑௫మ a = y̅ - b x̅ …..(2) ƎƄ߄ ༨စ ଘଓƄቿቱƌܻ‫ ߄ࡷݘ‬ၴ ƥƀ ‫ ۑ‬Ƨ Ǝܻࣃ ჴ჈ଘଓƄᇯ ᇍ ࣃ ࡩƄ፥ ጿࣩ ࣐ ‫ݘ‬ଐଌƄơ ܻ޲ଐଌƄ፟ ፑƄƆƎܻࢸ ଗଓܻ‫ ێ ( ހ‬ƞ ܸޯࡴ X ે ઻ၮ Y ࡠ இॅ Ɓ i xi yi 2 2 10 1 3 4 5 6 7 8 9 8 3 11 6 5 6 8 4 12 9 6 10 1 3 12 1 4 9 ∑ 64 14 70 ‫ۑ‬ƣ ࢓‫ݐ‬ ࢤ ࡦƁƄ ܸ‫ݕ‬ ‫ݽ‬ ࢓଍ଉƁࠣߢ‫ ࣊࢓ݶ‬። ጼƄ ܸ‫ݽ‬ ܸࢽ இॅ ƁƄ Ɓࣝ࣍ ߽ ၏য ࣝ‫ݶ‬7.1 ƞ Ƣ߁ ஋ॅ Ɓ X − X̅ Y −Y¯ −4.4 3 1.6 4.6 −4 −0.4 −1 −2.4 5 −1.4 5.6 2.6 1 −0.4 2 - - −5.4 −4 7 (x−X¯)(y−Y¯) (x−X¯) ‫ݔ‬2 ‫ݕݔ‬ 2 −6.4 2.56 −18.4 21.16 −13.2 0.4 −1.4 −6 −3 −12 −33.6 −7.8 −0.8 −37.8 −131 19.36 0.16 1.96 5.76 31.36 6.76 0.16 29.16 118.4 x̅ = ∑x / n = 64/10 = 6.4 88 ܸୃกܸ଍ଉŽ–ໝ ແࡦफउ –ࠣ‫ݶ‬ Ɖܸࢤ ެߕକଐƁƢࠣ‫ࡦ ࢽޟܸ ࢲޟ‬ƁƢࠣ‫ޟ‬පඓ ൷Ɓ౭౑ࢲ କଐƁƄ ܸࢲ ƁƋኃ ᒔᐺ። ጏ࢓‫ݕ‬ ࡦƁ࣬ ࣊ Ɓ࢓ߢೕಹኃ ቮƁ೾೥࢓଍ଉƁƄ ƁƋƁ߁ ࢟ ſ ƛ ܸߢࢽ ࡦƁ߁ ዖዄ Ž૊ સၫƋࣝެ࠾ኃ ቮƁƊܸެࢲ ଘଐƁ b= b= σ ௫௬ σ ௫మ ିଵଷଵ ଵଵ଼Ǥସ = - 1.1 a = y̅ - b x̅ a = 7 – (- 1.1)(6.4) a = 7 + 7 = 14. y = a + bx …..(3) ። ፎକଐ߁ ެࢲ କଐƁƢ। ्࢟ ࣝࡦƁŻܸࢤ ߾ ଔଐƁ y̅ = ∑y /n = 70 /10 = 7 …..(4) ƋƁ߁ ༥ဂ କଐƁࣇ ߽ ቼቮƉܸ‫ݕ‬ ࡴ y = 14 – 1.1 x X ે ઻ၮYƎƄ߄ ༨စ ଘଓƄ࣊ உै Ƅቿቱƌܻ‫ݘ‬ଐଌࡣ ࢥ ߙƄࢇ ‫ކ࣠ߙޢ‬ၒল ࣠‫ݹ‬፥ ጕܻ‫ࡩީޢ‬Ƅᇯ ᇍ ࣃ ࡩƄƥ ࣇ ‫ ࢽޟ‬ძࡦƁƋƁ߁ ༥ဂ କଐƁࣇ ߽ ቼቮƉܸ‫ݕ‬ ࡴ ‫ ێ‬ᇬ ᇊ ࢵ 89 ܸୃกܸ଍ଉŽˍ ໝ ແࡦफउˍ ࠣ‫ݶ‬ Ɖܸࢤ ެߕକଐƁƢࠣ‫ࡦ ࢽޟܸ ࢲޟ‬ƁƢࠣ‫ޟ‬පඓ ൷Ɓ౭౑ࢲ କଐƁƄ ܸࢲ ƁƋኃ ᒔᐺ። ጏ࢓‫ݕ‬ ࡦƁ࣬ ࣊ Ɓ࢓ߢೕಹኃ ቮƁ೾೥࢓଍ଉƁƄ ƁƋƁ߁ ࢟ ſ ƛ ܸߢࢽ ࡦƁ߁ ዖዄ Ž૊ સၫƋࣝެ࠾ኃ ቮƁƊܸެࢲ ଘଐƁ Yi = a + bxi + ei ። ፎକଐ߁ ެࢲ କଐƁƢ। ्࢟ ࣝࡦƁŻܸࢤ ߾ ଔଐƁ :ࠦߘൎ൉‫཯ކݘ ࣐ ࣩހܻࡩݘ‬དྷࡩƄ࣊ ‫ ࣀޢ‬ხࡩƄƎƄ߄ ༨စ ଘଓƄ࣊ ࠀ ቿቱƌܻ‫ࡷݘ‬፥ ጿ Ơ Ž‫߾ޟݜ‬ บ กܻ‫ މ‬b ‫ و‬a ࡣ ߥޯࣀ ଐଌƄ౰ే‫ݓ‬ ޯଐଌƄ X ݊ ‫ࡩޯܻހ‬Ƅ౰ే‫ݓ‬ ޯଐଌƄ Y ܾ࣐ உै Ƅ߄ ၰ ፥ ገƄ࣠ࣃ ‫ࡩݘ‬Ƅ౰ే‫ݓ‬ ޯଐଌƄ ei ( 0 = x ƣ ࣠ࡓ ‫ݑެ ߄ࡷܻކ‬y ࠦࡺ‫ ߘޢ‬Ƈ Ƅƌܻࢧ ࡩƄƎ࣠༨ீ ࡼ ࡷƖ ࣐࣠ ߥଐଌƄž࢔எै Ƅ aࡣ ޲೘೉ƥ ‫ۑ‬Ɔ߄ ၰ ƄƥƆ߄ ၰ ƥƎƄ߄ ߥ೘ೃ x ‫ اذا زادت‬y ፥ ጿƆƌܻ‫ݹ‬༝༎ƄƧ ƀ যওߥޯࣀ ଐଌƄ࣊ உै Ƅࡣ ‫ ࡷޢ‬b ࡣ ޲೘೉ƥ ໠ ໄ‫ࡩߥܻރ࣠ރ‬Ƅ‫ ࠁ ࣀ ݪ‬a ‫ و‬b ཯དྷࡺ‫ߘޢ‬ƅ ܻࣀ ࠁ ࡼ ࡓ ೘಼ƥ ܾ = ே ∑௑௒ି∑௑∑௒ ே ∑௑ మି(∑௑)మ a = y̅ - b ‫ݔ‬ҧ ….. (3) ….. (4) ơ ൎ൉ၯ ࡼ ࡷ a & b ཯དྷࡺ‫ߘޢ‬मअ߄ ߥ‫ކ‬ƣ ܻߙ፥ ጒܻࣀ ஊै Ƅܻ࣐ࠩ ࢵ ƥࡼ ‫ݑ‬যওߥࡩƄƇ ܻߙƄ࢖༨စ ƄƢƄ߄ ႀ ޯࢵ ƄᎨ᎙ƄƍƄƥ ܾ= ∑௫௬ ∑௫మ a = y̅ - b ‫ݔ‬ҧ ……(5) …….(6) ໠ ໄޯ‫ࡩޯܻࡩޢ‬Ƅ໠ ໄޯࡩƌܻ‫ݘ‬ଐଌƄ Ơ Ž‫߾ޟݜ‬ ‫ = ݔ‬X̅ - X ‫ = ݕ‬Y̅ - Y 90 ܸୃกܸ଍ଉŽ–ໝ ແࡦफउ –ࠣ‫ݶ‬ Ɖܸࢤ ެߕକଐƁƢࠣ‫ࡦ ࢽޟܸ ࢲޟ‬ƁƢࠣ‫ޟ‬පඓ ൷Ɓ౭౑ࢲ କଐƁƄ ܸࢲ ƁƋኃ ᒔᐺ። ጏ࢓‫ݕ‬ ࡦƁ࣬ ࣊ Ɓ࢓ߢೕಹኃ ቮƁ೾೥࢓଍ଉƁƄ ƁƋƁ߁ ࢟ ſ ። ፎକଐ߁ ެࢲ କଐƁƢ। ्࢟ ࣝࡦƁŻܸࢤ ߾ ଔଐƁ ƛ ܸߢࢽ ࡦƁ߁ ዖዄ Ž૊ સၫƋࣝެ࠾ኃ ቮƁƊܸެࢲ ଘଐƁ Yે ઻ၮ X ‫ و‬X ે ઻ၮ Y ƎƄ߄ ༨စ ଘଓƄቿቱƌܻ‫ࡷݘ‬Ǝ߄ ߘ ،Ƥܻ‫ރ‬ƌƀƇ ܻ‫ࡩީޢܻރ‬Ƅࡼ ࡷ ( 7.2 )ƞ ܸޯࡴ X Y 27 31 26 27 21 24 24 20 Y2 XY 25 27 625 729 675 27 24 33 36 ∑X = 240 23 1225 26 529 841 21 ∑Y= 200 900 X2 = ∑ 7342 ଼(଻ଷସଶ)ି(ଶସ଴)మ a = y̅ - b ‫ݔ‬ҧ = 957 Y2 5092 ܰ∑ܻܺ − ∑ܺ∑ܻ ܰ ∑ܺ ଶ − (∑ܺ)ଶ ଼(଺଴ହ଴)ି(ଶସ଴ ୶ ଶ଴଴) 480 841 1296 ∑ 648 400 1089 30 ࡠ இॅ Ɓ 651 576 576 29 30 754 441 729 20 29 36 805 676 961 24 33 ƋƁ߁ ༥ဂ କଐƁཬཏࡦƉܸ‫ݕ‬ ࡴफं߁ ߢ‫ ێ ރ‬ƞ Ƣ߁ ၱ X2 31 91 23 29 Y 29 b= 25 X 35 ܾ= 35 1080 = ∑XY= 6050 x̅ = ∑X /n = 240/8 = 30 Y̅= ∑Y / n = 200/8 = 25 ସ଼ସ଴଴ି ସ଼଴଴଴ ହ଼଻ଷ଺ିହ଻଺଴଴ = ସ଴଴ ଵଵଷ଺ = 0.352 ܸୃกܸ଍ଉŽ–ໝ ແࡦफउ –ࠣ‫ݶ‬ Ɖܸࢤ ެߕକଐƁƢࠣ‫ࡦ ࢽޟܸ ࢲޟ‬ƁƢࠣ‫ޟ‬පඓ ൷Ɓ౭౑ࢲ କଐƁƄ ܸࢲ ƁƋኃ ᒔᐺ። ጏ࢓‫ݕ‬ ࡦƁ࣬ ࣊ Ɓ࢓ߢೕಹኃ ቮƁ೾೥࢓଍ଉƁƄ ƁƋƁ߁ ࢟ ſ ። ፎକଐ߁ ެࢲ କଐƁƢ। ्࢟ ࣝࡦƁŻܸࢤ ߾ ଔଐƁ ƛ ܸߢࢽ ࡦƁ߁ ዖዄ Ž૊ સၫƋࣝެ࠾ኃ ቮƁƊܸެࢲ ଘଐƁ a = 25 – (0.352)(30) a = 25 – 10.56 a = 14.44 Y = a + bX Y = 14.44 + 0.352 X ܾ= b= ܰ∑ܻܺ − ∑ܺ∑ܻ ܰ ∑ܻଶ − (∑ܻ)ଶ ଼(଺଴ହ଴)ି(ଶସ଴ ୶ ଶ଴଴) ଼(ହ଴ଽଶ)ି(ଶ଴଴)మ a = X̅ - b ‫ݕ‬ҧ = x̅ = ∑X /n = 240/8 = 30 Y̅= ∑Y / n = 200/8 = 25 ସ଼ସ଴଴ି ସ଼଴଴଴ ସ଴଻ଷ଺ିସ଴଴଴଴ = ସ଴଴ ଻ଷ଺ = 0.543 a = X̅ – (0.543)(25) a = 30 – 13.575 a = 16.425 X= a + bY X= 16.425+ 0.543 Y Ƣॅळ‫ࡩݘ‬Ƅࠦ‫ޢ‬ᆿᆝ ơ ƥଘଓƄƦ ࣠ޯࣀ ଐଌƄ፥ ጿƅ ൎ൉࣍ Ɔྲ Ⴁ‫ݑ‬Ƈ ܻၴ Ǝƌơ ࣠ࠁ Ƥܻ‫ރ‬ƌƀƇ ܻ‫ࡩީޢܻރ‬Ƅࡼ ࡷ ‫ ۑ‬ơ ܻ޲ࡷ ߄ ၴ ƥƀ 50ࡼ ࡷࠦၴ Ǝኆ ቱƄ žܻࢧ ࠁ ଘଓƄƥƌܻࢧ ޯߘଘଓƄ፥ ጾܻࣀ ࡷ፥ ጿࠦ‫ݹ‬ƎƄƌଘଓƄ Y ે ઻ၮ X ‫ و‬X ે ઻ၮ Y ƎƄ߄ ༨စ ଘଓƄ཯དྷࡩƌܻ‫ࡷݘ‬ žܻࢧ ࠁ ଘଓƄƇ ܻၴ Ǝƌƥƌܻࢧ ޯߘଘଓƄƇ ܻၴ Ǝƌ໠ ໄ‫ހ‬Ɣ ୍ܻงƎଘଓƄࡣ ࡷܻ‫ࡷݘ‬ 30 ƌܻࢧ ޯߘଘଓƄ፥ ጿࠦၴ Ǝኆ ቱƄบ คᇴᆝ ƄƍƄžܻࢧ ࠁ ଘଓƄ፥ ጿࠦ‫ޯ࣠ߘݘ‬ଐଌƄࠦၴ Ǝኆ ቱƄ 92 ܸୃกܸ଍ଉŽ–ໝ ແࡦफउ –ࠣ‫ݶ‬ Ɖܸࢤ ެߕକଐƁƢࠣ‫ࡦ ࢽޟܸ ࢲޟ‬ƁƢࠣ‫ޟ‬පඓ ൷Ɓ౭౑ࢲ କଐƁƄ ܸࢲ ƁƋኃ ᒔᐺ። ጏ࢓‫ݕ‬ ࡦƁ࣬ ࣊ Ɓ࢓ߢೕಹኃ ቮƁ೾೥࢓଍ଉƁƄ ƁƋƁ߁ ࢟ ſ .I .II .III ። ፎକଐ߁ ެࢲ କଐƁƢ। ्࢟ ࣝࡦƁŻܸࢤ ߾ ଔଐƁ ƛ ܸߢࢽ ࡦƁ߁ ዖዄ Ž૊ સၫƋࣝެ࠾ኃ ቮƁƊܸެࢲ ଘଐƁ 32 34 38 29 36 31 32 35 28 25 39 33 30 31 32 36 41 49 46 43 Ƈ ܻၴ Ǝƌ ƌܻࢧ ޯߘଘଓƄ Ƈ ܻၴ Ǝƌ žܻࢧ ࠁ ଘଓƄ ࡠ இॅ Ɓ ƋƁ߁ ༥ဂ କଐƁܸެࡦƉܸ‫ݕ‬ ࡴ Y = a + bx X = a + by ܾ= a = y̅ - b ‫ݔ‬ҧ ∑‫ݕݔ‬ ∑‫ݔ‬ଶ Żܸࢤ ߾ କଐƁƢƉܸࢤ ެߕକଐƁ። ጻܸࢽ ࡴ። ጼƂ ോെ࣍ ࡦƁƄ ܸၱ Ƌኃ ቮƋƁ߁ ༥ဂ କଐƁཬཏࡦƉܸ‫ݕ‬ ࡴफं߁ ߢ‫ ێ ރ‬ƞ Ƣ߁ ၱ (Y-Y̅) 43 -7 5 49 3 Y 28 46 25 35 32 31 36 29 38 34 32 ∑X 320 93 (X-X̅) X 41 36 32 31 30 33 39 = ∑Y =380 -4 0 -1 4 -3 6 2 0 0 ‫ݔ‬ 8 11 3 -2 -6 -7 -8 -5 1 0 ‫ݕ‬ 49 16 ‫ݔ‬ଶ 9 0 1 16 9 36 4 0 ∑‫ݔ‬ଶ = 140 25 64 ‫ݕ‬ଶ 121 9 4 36 49 64 25 1 ∑‫ݕ‬ଶ=398 -35 -32 ‫ݕݔ‬ 33 0 2 -24 21 -48 -10 0 ∑‫ =ݕݔ‬-93 ܸୃกܸ଍ଉŽ–ໝ ແࡦफउ –ࠣ‫ݶ‬ Ɖܸࢤ ެߕକଐƁƢࠣ‫ࡦ ࢽޟܸ ࢲޟ‬ƁƢࠣ‫ޟ‬පඓ ൷Ɓ౭౑ࢲ କଐƁƄ ܸࢲ ƁƋኃ ᒔᐺ። ጏ࢓‫ݕ‬ ࡦƁ࣬ ࣊ Ɓ࢓ߢೕಹኃ ቮƁ೾೥࢓଍ଉƁƄ ƁƋƁ߁ ࢟ ſ .IV ። ፎକଐ߁ ެࢲ କଐƁƢ। ्࢟ ࣝࡦƁŻܸࢤ ߾ ଔଐƁ ƛ ܸߢࢽ ࡦƁ߁ ዖዄ Ž૊ સၫƋࣝެ࠾ኃ ቮƁƊܸެࢲ ଘଐƁ X̅ = ∑X / n = 320/10 = 32 Y̅ = ∑Y / n = 380/10 = 38 ܾ= ∑‫ݕݔ‬ ∑‫ݔ‬ଶ a = y̅ - b ‫ݔ‬ҧ ܾ= ିଽଷ ଵସ଴ = -0.664 a = 38 – (-0.664)(32) a = 38 + 21.248 = 59.248 Y = a + bX Y = 59.248 - 0.664 X X = a + by ܾ= a = X̅ - ܾ‫ݕ‬ҧ ∑‫ݕݔ‬ ∑‫ݕ‬ଶ ܾ= ିଽଷ∑ ଷଽ଼ = - 0.234 a = 32 – (- 0.234)(38) a = 32 + 8.892 = 40.892 X = a + bY X = 40.892 – 0.234 Y r= 94 ∑௫୷ ඥ∑௫మ ∑௬మ ܸୃกܸ଍ଉŽ–ໝ ແࡦफउ –ࠣ‫ݶ‬ Ɖܸࢤ ެߕକଐƁƢࠣ‫ࡦ ࢽޟܸ ࢲޟ‬ƁƢࠣ‫ޟ‬පඓ ൷Ɓ౭౑ࢲ କଐƁƄ ܸࢲ ƁƋኃ ᒔᐺ። ጏ࢓‫ݕ‬ ࡦƁ࣬ ࣊ Ɓ࢓ߢೕಹኃ ቮƁ೾೥࢓଍ଉƁƄ ƁƋƁ߁ ࢟ ſ ። ፎକଐ߁ ެࢲ କଐƁƢ। ्࢟ ࣝࡦƁŻܸࢤ ߾ ଔଐƁ ƛ ܸߢࢽ ࡦƁ߁ ዖዄ Ž૊ સၫƋࣝެ࠾ኃ ቮƁƊܸެࢲ ଘଐƁ r= ିଽଷ ඥ(ଵସ଴)(ଷଽ଼) = ିଽଷ √ହହ଻ଶ଴ Y = 59.248 - 0.664 X = ିଽଷ ଶଷ଺.଴ହ = - 0.394 Y = 59.248 - 0.664 (30) Y = 59.248 – 19.92 = 39 በሼྲ Ⴁࡩ Y Ƈ ܻ‫ީޢݘ‬ଐଌƄࡴ ᄽ஛ ƥ X ƣ ൎ൉ၮଘଓƄƥࠦ‫ܻݹ‬ၮኆ ቱƄƇ ܻߥߦ‫޲ ࡣރ‬೘೉ࠦ‫ࡩޯܻࡩޢ‬ƄƇ ܻ‫ࡩީޢܻރ‬Ƅ ‫ ۑ‬ơ ܻ޲ࡷ บ คᇴᆝ ƄƍƄƇ ܻ‫ީޢݘ‬ଐଌƄࡴ ᄽ஛ Ǝ߄ߘᎨ᎜ X ે ઻ၮ Y ƎƄ߄ ༨စ ଘଓƄቿቱƌܻ‫ ߄ࡷݘ‬ၴ ƥƀ‫ۑ‬मंܻࢧ ‫ࡩݘ‬ƄƥƇ ܻީ࣍ ࢖ଐଌƄ ƣ ൎ൉ၮଘଓƄƥࠦ‫ܻݹ‬ၮኆ ቱƄƇ ܻߥߦ‫ރ‬໠ ໄ‫ހ‬ƣ ࣠ࢵ ౰ే‫ހ‬Ɣ ୍ܻงƎଘଓƄࡣ ࡷܻ‫ࡷݘ‬ᑑ ᐽቦሼƥ‫ۑ‬20ƣ ൎ൉ၮଘଓƄƥࠦ‫ܻݹ‬ၮኆ ቱƄƇ ܻߥߦ‫ރ‬ ‫ۑ‬Ƈ ܻ‫ީޢݘ‬ଐଌƄࡴ ᄽ஛ ‫و‬ 900 500 600 1200 900 300 500 X 1100 ($) Y 15000 10000 12000 25000 20000 8000 13000 25000 ($) Ơ ോെၫକଐƁƢࠣ‫ݶ‬ ܸၫኃ ቮƁƄ ܸߢߣ‫ ૄހ‬સၫ Y Ƅ ܸ‫ݕ‬ ‫ަޟ‬଍ଉƁࡱ ᄺ஘ ƋƁ߁ ༥ဂ ƁቼቮƉܸ‫ݕ‬ ࡴफं߁ ߢ‫ ێ ރ‬ƞ Ƣ߁ ၱ ࡠ இॅ Ɓ X Y $ (100) $ (1000) (X) (X-X̅) (Y-Y̅) ‫ݔ‬ଶ 11 25 ‫ݔ‬ 3.5 ‫ݕ‬ 9 12.25 81 3 8 -4.5 -8 20.25 64 5 9 12 6 5 9 13 20 25 12 10 15 ∑X = 60 ∑Y =128 95 -2.5 1.5 4.5 -1.5 -2.5 1.5 0 -3 4 9 -4 -6 -1 0 6.25 2.25 20.25 2.25 6.25 2.25 ∑‫ݔ‬ଶ =72 ‫ݕ‬ଶ 9 16 81 16 36 1 ∑‫ݕ‬ଶ=304 ‫ݕݔ‬ 31.5 7.5 36 6 40.5 6 15 -1.5 ∑‫=ݕݔ‬14 1 ܸୃกܸ଍ଉŽ–ໝ ແࡦफउ –ࠣ‫ݶ‬ Ɖܸࢤ ެߕକଐƁƢࠣ‫ࡦ ࢽޟܸ ࢲޟ‬ƁƢࠣ‫ޟ‬පඓ ൷Ɓ౭౑ࢲ କଐƁƄ ܸࢲ ƁƋኃ ᒔᐺ። ጏ࢓‫ݕ‬ ࡦƁ࣬ ࣊ Ɓ࢓ߢೕಹኃ ቮƁ೾೥࢓଍ଉƁƄ ƁƋƁ߁ ࢟ ſ ። ፎକଐ߁ ެࢲ କଐƁƢ। ्࢟ ࣝࡦƁŻܸࢤ ߾ ଔଐƁ ƛ ܸߢࢽ ࡦƁ߁ ዖዄ Ž૊ સၫƋࣝެ࠾ኃ ቮƁƊܸެࢲ ଘଐƁ X̅ = ∑X / n = 60/8 = 7.5 Y̅ = ∑Y / n = 128/8 = 16 ܾ= ∑‫ݕݔ‬ ∑‫ݔ‬ଶ a = y̅ - b ‫ݔ‬ҧ ܾ= ଵସଵ ଻ଶ = 1.958 a = 16 – (1.958)(7.5) a = 16 – 14.685 = 1.315 Y = a + bX Y = 1.315 + 1.958 X X = 20 Ơ ോെၫକଐƁƢࠣ‫ݶ‬ ܸၫኃ ቮƁƄ ܸߢߣ‫ހ‬ท กᇱᆚ ƁƊƁ Y = 1.315 + 1.958 X Y = 1.315 + 1.958(20) Y = 40.475 = $ 40475 r= r= 96 ∑௫୷ ඥ∑௫మ ∑௬మ ଵସଵ ඥ(଻ଶ)(ଷ଴ସ) = ଵସଵ √ଶଵ଼଼଼ Ơ ࣝࢲ ౭ౄ‫ݽ‬ Ƒ ܸ୊คƋକଐƁࡠ ࡴܸ‫ݕ‬ ࡴ = ଵସଵ ଵସ଻. = 0.959 [ Coefficient of Determination ] ਑ ৯߁ ၾ ެࡦƁࡠ ࡴܸ‫ݕ‬ ࡴ ܸୃกܸ଍ଉŽ–ໝ ແࡦफउ –ࠣ‫ݶ‬ Ɖܸࢤ ެߕକଐƁƢࠣ‫ࡦ ࢽޟܸ ࢲޟ‬ƁƢࠣ‫ޟ‬පඓ ൷Ɓ౭౑ࢲ କଐƁƄ ܸࢲ ƁƋኃ ᒔᐺ። ጏ࢓‫ݕ‬ ࡦƁ࣬ ࣊ Ɓ࢓ߢೕಹኃ ቮƁ೾೥࢓଍ଉƁƄ ƁƋƁ߁ ࢟ ſ 7.2 ። ፎକଐ߁ ެࢲ କଐƁƢ। ्࢟ ࣝࡦƁŻܸࢤ ߾ ଔଐƁ ƛ ܸߢࢽ ࡦƁ߁ ዖዄ Ž૊ સၫƋࣝެ࠾ኃ ቮƁƊܸެࢲ ଘଐƁ ୷ ୪༟༎ܻ‫ހ‬ቻቱ୷ ୪मअƥ‫ ્ۑ‬ᇣᇍ ࡩƄ௪ ௎ܻީޯࡩƄࡼ ࡷྲ ႞ߦଐଌƄ௪ ௎ܻީޯࡩƄ ࠦީఱ ఝࡼ ‫ݑ‬ƆƎܻީ‫ހܾރࠢݑ‬ਔ ৲߄ ႁ ޯࡩƄࡣ ࡷܻ‫ࡷݘ‬ƞ ࢖‫ݹݘ‬ ƎƄ߄ ༨စ ଘଓƄ࣊ உै ࢇ ‫ࡩޯ࣠ߙޢ‬ƄƆƌ࣠ࠅ ਔ ৲߄ ႁ ޯࡩƄࡣ ࡷܻ‫ ࢪࡷݘ‬ყߥ‫ݹ‬ƥ ‫ۑ‬r2 ∑(Yi - Y̅)2 ૊ᇠᇊ ࡦƁ௧ ோܸަެࡦƁ = ∑(Ŷ - Y̅)2 ྯ ႛߣ଍ଉƁ௧ ோܸަެࡦƁ + + Goodness of fit ∑ ei2 ྯ ႛߣ଍ଉƁ౭ౄၦ௧ ோܸަެࡦƁ [Total variation] = [ Explained Variation] + [Unexplained Variation] SST ∑(Yi - Y̅)2 ത)మ ∑(ଢ଼୧ି ௒ ത)మ ∑(ଢ଼୧ି ௒ 1 = = ∑(Ŷ୧ି ଢ଼ጟ )మ ∑(ଢ଼୧ି ଢ଼ጟ )మ = R2 R2 = 1 - + ∑ୣ୧మ + = SSR ∑(Ŷi - Y̅)2 + + SSE ∑ ei2 ∑ ୣ୧మ ∑(ଢ଼୧ି ଢ଼ጟ )మ ∑ୣ୧మ ∑୷మ ∑୷మ ࡴ ᄽ஛ ƥ X ƣ ൎ൉ၮଘଓƄƥࠦ‫ܻݹ‬ၮኆ ቱƄƇ ܻߥߦ‫ރ‬ơ ࣠ࠁ ‫ߘ ࡴ ۑ‬Ǝơ ܻ޲ଐଌƄ Ƈ ܻ‫ܻރ‬୆กࡼ ࡷ (7.5 ƞ ܸޯࡴ ‫ۑ‬ਔ ৲߄ ႁ ޯࡩƄࡣ ࡷܻ‫ ߄ࡷݘ‬ၴ ƥƀ Y Ƈ ܻ‫ީޢݘ‬ଐଌƄ ࡱ ᄺ஘ Ƣ X Ơ ോെၫକଐƁƢࠣ‫ݶ‬ ܸၫኃ ቮƁƄ ܸߢߣ‫ހ‬Ƅ ܸ‫ࡦަޟܸހ‬Ƥ ߁ ၾ ެࡦƁࡠ ࡴܸ‫ݕ‬ ࡴफं߁ ߢ‫ ێ ރ‬ƞ Ƣ߁ ၱ : ࡠ இॅ Ɓ Ƅ ܸ‫ݕ‬ ‫ަޟ‬଍ଉƁ X $ (100) 11 97 Y $(10 00) 25 (X-X̅) 3.5 ‫ݔ‬ (YY̅) 9 ‫ݕ‬ ‫ݔ‬ଶ 12.25 ‫ݕ‬ଶ 81 ‫ݕݔ‬ 31.5 Ŷ 22.85 3 E Y- Ŷ 2.147 (Y ) e2 4.60 9 ܸୃกܸ଍ଉŽ–ໝ ແࡦफउ –ࠣ‫ݶ‬ Ɖܸࢤ ެߕକଐƁƢࠣ‫ࡦ ࢽޟܸ ࢲޟ‬ƁƢࠣ‫ޟ‬පඓ ൷Ɓ౭౑ࢲ କଐƁƄ ܸࢲ ƁƋኃ ᒔᐺ። ጏ࢓‫ݕ‬ ࡦƁ࣬ ࣊ Ɓ࢓ߢೕಹኃ ቮƁ೾೥࢓଍ଉƁƄ ƁƋƁ߁ ࢟ ſ ƛ ܸߢࢽ ࡦƁ߁ ዖዄ Ž૊ સၫƋࣝެ࠾ኃ ቮƁƊܸެࢲ ଘଐƁ ። ፎକଐ߁ ެࢲ କଐƁƢ। ्࢟ ࣝࡦƁŻܸࢤ ߾ ଔଐƁ 5 13 -2.5 -3 6.25 9 7.5 3 8 -4.5 -8 20.25 64 36 9 20 1.5 4 2.25 16 6 12 25 4.5 9 20.25 81 40.5 6 12 -1.5 -4 2.25 16 6 5 10 -2.5 -6 6.25 36 15 9 15 1.5 -1 2.25 1 -1.5 0 0 ∑Y ∑X = 60 =12 8 ଶ ∑‫ݔ‬ =72 ∑‫ݕ‬ଶ =30 4 ∑‫ݕݔ‬ =141 11.10 5 7.189 18.93 7 24.81 1 13.06 3 11.10 5 18.93 7 128 1.895 0.811 1.063 0.189 - 1.063 - 1.105 - 3.937 3.59 1 0.65 7 1.12 9 0.03 5 1.12 9 1.22 1 15.4 9 27.8 61 Ŷ = 1.315 + 1.958 X Ŷ = 1.315 + 1.958(11) = 1.315 + 21.538 = 22.853 Ŷ = 1.315 + 1.958(5) = 1.315 + 9.79 = 11.105 Ŷ = 1.315 + 1.958 (3) = 1.315 + 5.874 = 7.189 Ŷ = 1.315 + 1.958 (9) = 1.315 + 17.622= 18.937 Ŷ = 1.315 + 1.958 (12) = 1.315 + 23.496 = 24.811 Ŷ = 1.315 + 1.958 (6) = 1.315 + 11.748 = 13.063 R2 = 1 - 98 ∑ୣ୧మ ∑୷మ ܸୃกܸ଍ଉŽ–ໝ ແࡦफउ –ࠣ‫ݶ‬ Ɖܸࢤ ެߕକଐƁƢࠣ‫ࡦ ࢽޟܸ ࢲޟ‬ƁƢࠣ‫ޟ‬පඓ ൷Ɓ౭౑ࢲ କଐƁƄ ܸࢲ ƁƋኃ ᒔᐺ። ጏ࢓‫ݕ‬ ࡦƁ࣬ ࣊ Ɓ࢓ߢೕಹኃ ቮƁ೾೥࢓଍ଉƁƄ ƁƋƁ߁ ࢟ ſ ። ፎକଐ߁ ެࢲ କଐƁƢ। ्࢟ ࣝࡦƁŻܸࢤ ߾ ଔଐƁ ƛ ܸߢࢽ ࡦƁ߁ ዖዄ Ž૊ સၫƋࣝެ࠾ኃ ቮƁƊܸެࢲ ଘଐƁ R2 = 1 R2 = 99 72 ଶ଻.଼଺ଵ ଷ଴ସ ∑(Ŷ୧ି ଢ଼ጟ )మ ∑(ଢ଼୧ି ଢ଼ጟ )మ = = 1 – 0.0916 = 0.908 ா௫௣௟௔௜௡௘ௗ ௏௥௜௔௧௜௢௡ ்௢௧௔௟௏௔௥௜௔௧௜௢௡ = ௌௌோ ௌௌ் ݇‫ݽ‬ ܸࢽ ࡦƁࡠ ࢤ ߣࡦƁ௧ ோƋܸೕೆ 75 = X ƣ ࣠ࡓ ‫ݑެ ߄ࡷܻކ‬Y ࠦࡺ‫ߘޢ‬Ǝ߄ ߘƥ ،Ƥܻ‫ރ‬ƌƀƇ ܻ‫ࡩީޢܻރ‬Ƅࡼ ࡷƎƄ߄ ༨စ ଘଓƄቿቱƌܻ‫ ߄ࡷݘ‬ၴ ƥƀ .1 96 70 90 73 81 78 78 79 72 82 69 80 66 84 63 87 60 85 X Y ፥ ጿ‫ࡩ ࣐ܻࡩ ݪ‬ƄƇ ܻၴ Ǝƌ໠ ໄ୍ง཯དྷࡩƄ Y ે ઻ၮ X ‫ و‬، Xે ઻ၮ Y ƎƄ߄ ༨စ ଘଓƄቿቱƌܻ‫ ߄ࡷݘ‬ၴ ƥƀ .2 Ǝ߄ ߘƥ ،Ƥܻ‫ރ‬ƌƀƇ ܻ‫ࡩީޢܻރ‬Ƅࡼ ࡷ X žܻࢧ ࠁ ଘଓƄƟ ܻࣀ ࡷ፥ ጿ‫ࡩ ࣐ܻࡩ ݪ‬ƄƇ ܻၴ Ǝƌƥ Y ƌܻࢧ ޯߘଘଓƄƟ ܻࣀ ࡷ 32 39 34 33 38 30 29 32 36 32 31 36 32 41 75 = X ƣ ࣠ࡓ ‫ݑެ ߄ࡷܻކ‬Y ࠦࡺ‫ߘޢ‬ 35 49 28 46 25 43 X Y Y Ƈ ܻ‫ݹܻࢠޢ‬༟༎Ƅƥ X žܻࢧ ࠁ ଘଓƄ፥ ጿƅ ൎ൉࣍ 10Ƈ ܻၴ Ǝƌࢪ ࡓ ‫ކݘ‬Ƥܻ‫ރ‬ƌƀƇ ܻ‫ࡩީޢܻރ‬Ƅ .3 Ɣ ୍ܻงƎଘଓƄࡣ ࡷܻ‫ ࠁ ࣀ ݪࡷݘ‬ƀ 57 66 žܻࢧ ࠁ ଘଓƄ፥ ጿࠦၴ Ǝƌ 62ે ઻ၮࡣ ࢧ ࠁ Ƨ ኅ ቱƄ‫ ࣐ܻࡩ ݪ‬ᑃᐽƇ ܻ‫ݹܻࢠޢ‬༟༎Ƅ፥ ጿƇ ܻၴ Ǝኆ ቱƄƎ߄ ߘ 69 68 64 66 60 70 56 68 57 65 58 70 58 67 55 67 56 68 X Y .I .II Ƈ ܻ‫ࡩީޢܻރ‬Ƅࡼ ࡷ R2 ਔ ৲߄ ႁ ޯࡩƄࡣ ࡷܻ‫ࡷݘ‬Ǝ߄ ߘᎨ᎜ X ે ઻ၮ Y ƎƄ߄ ༨စ ଘଓƦ ࢖‫ݓ‬ ࢧ ࡩƄƇ ܻ‫࢖ހݘ‬ଐଌƄቿቱƌܻ‫ ࠁ ࣀ ݪࡷݘ‬ƀ .4 79 84 99 72 78 67 75 66 71 63 72 64 73 65 75 74 84 86 91 89 92 Ƥܻ‫ރ‬ƌƀ X Y ܸୃกܸ଍ଉŽ–ໝ ແࡦफउ –ࠣ‫ݶ‬ Ɖܸࢤ ެߕକଐƁƢࠣ‫ࡦ ࢽޟܸ ࢲޟ‬ƁƢࠣ‫ޟ‬පඓ ൷Ɓ౭౑ࢲ କଐƁƄ ܸࢲ ƁƋኃ ᒔᐺ። ጏ࢓‫ݕ‬ ࡦƁ࣬ ࣊ Ɓ࢓ߢೕಹኃ ቮƁ೾೥࢓଍ଉƁƄ ƁƋƁ߁ ࢟ ſ ƛ ܸߢࢽ ࡦƁ߁ ዖዄ Ž૊ સၫƋࣝެ࠾ኃ ቮƁƊܸެࢲ ଘଐƁ ። ፎକଐ߁ ެࢲ କଐƁƢ। ्࢟ ࣝࡦƁŻܸࢤ ߾ ଔଐƁ ļ ȧļŌ ʆ ļ Ļ ŉŮļ ŷŉĿŮŅų ſ ńņ ŅšǫĿŐ ż ń [ Probability Distributions ] 100 ܸୃกܸ଍ଉŽˍ ໝ ແࡦफउˍ ࠣ‫ݶ‬ Ɖܸࢤ ެߕକଐƁƢࠣ‫ࡦ ࢽޟܸ ࢲޟ‬ƁƢࠣ‫ޟ‬පඓ ൷Ɓ౭౑ࢲ କଐƁƄ ܸࢲ ƁƋኃ ᒔᐺ። ጏ࢓‫ݕ‬ ࡦƁ࣬ ࣊ Ɓ࢓ߢೕಹኃ ቮƁ೾೥࢓଍ଉƁƄ ƁƋƁ߁ ࢟ ſ ። ፎକଐ߁ ެࢲ କଐƁƢ। ्࢟ ࣝࡦƁŻܸࢤ ߾ ଔଐƁ ƛ ܸߢࢽ ࡦƁ߁ ዖዄ Ž૊ સၫƋࣝެ࠾ኃ ቮƁƊܸެࢲ ଘଐƁ Ƈ ܻ‫ݹݘ‬Ə࣠ޯࡩƄ࢛ ‫ހݘ‬ள ஭‫୓ݘ‬෺ƥ‫ۑ‬፥ ገܻࢧ ࠁ ଘଓƄơ ଘଓ߄ ޯࢵ ଘଓƄ፥ ጿܻࡺᐱᚑƄƎƥƌ ࠦ‫ ࠁޯࡺܻࡩޢ‬ଘଓƄƇ ܻ‫ݹݘ‬Ə࣠ޯࡩƄ‫ކ࡯ݘ ݪ‬ ‫ۑ‬ቿቷࢧ ޯࡷࠦ୆෷Ƅ࣠ࣃ ‫ݑ‬Ƈ Ƅ౰ే‫ݓ‬ ޯ೘ೃ༇໩ ଘଓƄ࢛ ‫ࡩީݘ‬Ƅƥቿቷࢧ ߦެࡷࠦ୆෷Ƅ࣠ࣃ ‫ݑ‬Ƈ Ƅ౰ే‫ݓ‬ ޯ೘ೃࠦ‫ ࠁޯࡺܻࡩޢ‬ଘଓƄ ቻᇝᇍ ఴ ఓ Ƨ ኅ ቱƄࠦެ‫ࡩݘޢ‬ƄƗ Ƅ࢖ߙƣ ᇴᆝ Ƅƍƀࡣ ࢧ ߦެࡷࠢ‫ރ‬ƀ፥ ገƄ࣠ࣃ ‫ݑ‬౰ే‫ݓ‬ ޯࡷࡼ ‫ݑ‬ơ ࣠ߥ‫ީߣ ࢤ ࡠރ‬଍ଉƁ። ጅƁࣝࣀ ‫ݕ‬ ࡦƁ౭ౄ‫ݐ‬ ެ଍ଉƁ  ‫ۑ‬Ɣ ܻߥެࡩƄࡼ ࡷ߄ ‫ݘ‬ᑃᐽൎ൉‫ޢܻߘܻހ‬ᐱᐢެࡷƄƌ߄ ၮƧ ࣠༨࿻ ౰ే‫ݓ‬ ޯଐଌƄ ፥ ጿƆƌ࣠ࠅ ࣠ଐଌƄܻ‫ݹ‬౰ేޯࡓ ީࡩƄƌ߄ ၮે ઻ၮơ ਔ ৿཯དྷࡩƄࠦ୆෷Ƅ࣠ࣃ ‫ࡩݘ‬ƄƇ Ƅ౰ే‫ݓ‬ ޯଐଌƄ ƣ ܾߙ ،ྲ Ⴐ ஊै Ƅଘଓơ ܻ޲ଐଌƄࡣ ‫ޢ‬ხࢵ ે ઻ၮ ơ ܻߦ࣍ ƀࠦ‫މ‬ൎ൉‫މ‬Ƨ ࣠༨ဈ ཯དྷࡩƄƇ ൎ൉‫ݶ‬ ܻ‫ࡩݘ‬ƄƦ ኆ ቱƎ࣠ࡁኅ ቱƄơ ܻߦ࣍ ଘଓƄƌ߄ၮે ઻ၮƥžܻଐଌƄࡼ ࡷ‫ࡷ ࡓݘ ݪ‬౰౔ࡺ‫ ߄ࡷ࡯ޢ‬ၰ Ƅƥ ƌ߄ၮે ઻ၮơ ਔ ৲ơ ƥଘଓƄ፥ ገƄ࣠ࣃ ‫ࡩݘ‬Ƅ౰ే‫ݓ‬ ޯଐଌƄযওߘࠦၮ࣠ዙಣಝƣ ଘଓᑑ ᐽƍƥቿቷࢧ ߦެࡷࠦ୆෷Ƅ࣠ࣃ ‫ݑ‬Ƈ Ƅ౰ే‫ݓ‬ ޯࡷ፥ ፕ ، 1،0ࠦၮ࣠ࡺ഻വറƄƣ ࣠ࡓ ‫ކ‬ƣ Ƅࡼ ࡓ ೘಼ƥƆƌƥ߄ ༨ீ žܻଐଌƄࡼ ࡷ‫ࡷ ࡓݘ ݪ‬౰౔ࡺ ‫ ߄ࡷ࡯ޢ‬ၰ Ƅƥ፥ ጿƆƌ࣠ࠅ ࣠ଐଌƄܻ‫ݹ‬౰ేޯࡓ ީࡩƄ ે઻ၮơ ਔ ৲Ƨ ኅ ቱƄ፥ ጕܻ޲ࡩƄ፥ ገƄ࣠ࣃ ‫ࡩݘ‬Ƅ౰ే‫ݓ‬ ޯଐଌƄযওߘࠦၮ࣠ዙಣಝƣ ƀƥ‫ۑ‬Ƅ౰ేީࡁ Ƅƌ߄ ၮ n ‫ [ ࠁޢ ݟ‬n ]،…….،2 ، 1 ،0 ࠦ‫ޢ‬ᐱᐢެଐଌƄࠦၮ࣠ࡺ഻വറƄ፥ ፕ ơ ܻߦ࣍ ƀࠦ‫މ‬ൎ൉‫މ‬Ƨ ࣠༨ဈ ཯དྷࡩƄƇ ൎ൉‫ݶ‬ ܻ‫ࡩݘ‬ƄƦ ኆ ቱƎ࣠ࡁኅ ቱƄơ ܻߦ࣍ ଘଓƄƌ߄ ၮ [ 3]،2 ơ ਔ ৲Ƨ ኅ ቱƄ Xࡣ ࢧ ߦެଐଌƄ፥ ገƄ࣠ࣃ ‫ࡩݘ‬Ƅ౰ే‫ݓ‬ ޯ൞ᒑᐽ፥ ፑܻࡺޯࠁ ଘଓƄ݊ ‫ݹ‬Ə࣠ޯࡩƄƣ ࣠‫ߘܻރ‬ơ ƥ߄ எै Ƅ໠ ໄ୍෺ ‫ ۑ‬ơ ܻ޲ࡷ ‫ࡩޯ ࣐ީޢ ࢇۑ‬ƄƆƌܻࡷ౰ేࢥ ༨ဈ ƣ ƥƌƅ ࣠ࢵ ܻஊै Ƅ౰౎ޯ༧ீ ƣ ॅळၯ ਔ ৲௪ ௎ኅ ቱƄƅ ൎ൉࣐ ࡩƄƌ߄ၮે ઻ၮ 5 0.09 4 ? 3 0.4 2 0.3 1 0.1 P(x ˂ 3) ، 0 0.01 x F(x) f(4)߄ ၴ ƥƀ .I P( x ≤ 3) ߄ ၴ ƥƀ .II P( x ≥ 4) ߄ ၴ ƥƀ .III ࡠ இॅ Ɓ Ɩ ࣠ዙಣಝƥ࢖ߦࢧ ࡩƄƧ ƥܻఱ ఓƥƀ౰౎ࡁ ƀযওߥࡩƄƤ߃ ࠧࡼ ࡷࠦࡺ‫ߘޢ‬ᇯ ᆝ ơ ܻࡺޯࠁ Ƅƣ ܾߙ ،ࡣ ࢧ ߦެࡷ౰ే‫ݓ‬ ޯଐଌƄƣ ƀܻ೘ೃ  f(4) = ? ‫ࡩ ࢧ ࠒޢ ߰ۑ‬Ƅ߄ ၰ Ƅ࣠ࡩƄƧ ƥܻఱ ఓƇ ଘଓܻࡺޯࠁ ଘଓƄ f(0) + f(1) + f(2) + …….+ f(5) = 1 0.01 + 0.1 + 0.3 + 0.4 + f(4) + 0.09 =1 f(4) = 1.0 – 0.9 f(4) = 0.1 101 ܸୃกܸ଍ଉŽ–ໝ ແࡦफउ –ࠣ‫ݶ‬ Ɖܸࢤ ެߕକଐƁƢࠣ‫ࡦ ࢽޟܸ ࢲޟ‬ƁƢࠣ‫ޟ‬පඓ ൷Ɓ౭౑ࢲ କଐƁƄ ܸࢲ ƁƋኃ ᒔᐺ። ጏ࢓‫ݕ‬ ࡦƁ࣬ ࣊ Ɓ࢓ߢೕಹኃ ቮƁ೾೥࢓଍ଉƁƄ ƁƋƁ߁ ࢟ ſ ። ፎକଐ߁ ެࢲ କଐƁƢ। ्࢟ ࣝࡦƁŻܸࢤ ߾ ଔଐƁ ƛ ܸߢࢽ ࡦƁ߁ ዖዄ Ž૊ સၫƋࣝެ࠾ኃ ቮƁƊܸެࢲ ଘଐƁ  P(x ≤ 3) = P(x= 0) + P(x = 1 ) + P(x = 2 ) + P(x = 3) = f(0) + f(1) + f(2) + f(3) = 0.01 + 0.1 + 0.3 + 0.4  P(x ˂ 3) = 0.81 P(x ˂ 3) = P(x = 0 ) + P(x = 1) + P(x = 2) = 0.01 + 0.1 + 0.3 = 0.41  P(x ≥ 4) = P(x = 4 ) + P (x = 5) = 0.1 + 0.09 = 0.19 ơ ܻ༬ீ ፥ ጿࠦࡺ‫ݹࠦߘޢ‬ƀƓ ౰౔ߦ‫ݹ‬ƣ ᇴᆝ ƄƍƄࡣ ࢧ ޯࡷࠢ‫ރ‬ƀ፥ ገƄ࣠ࣃ ‫ݑ‬౰ే‫ݓ‬ ޯࡷࡼ ‫ݑ‬ơ ࣠ߥ‫ެ ࢤ ࡠ ރ‬଍ଉƁ። ጅƁࣝࣀ ‫ݕ‬ ࡦƁ౭ౄ‫ݐ‬ ެ଍ଉƁ  Ƅ߃ ࠧƥ ‫ ࢢߦ࢖ۑ‬Ƨ ƥܻఱ ఓࠦެ‫ߘޢࡺࠦࡷݘޢ‬౰ే‫ݓ‬ ޯଐଌƄ߃ ၯ ܾ‫ݹ‬ƣ ƀơ ܻࡺޯࠁ Ƅƣ ƀƥࠦ‫ߥޢߥޢ‬ஊै ƄƌƄ߄ ၮ൑ᑉᐽƇ ଘଓܻ༬ீ ፥ ጿƥƀܻࡷ ‫ݘ ߄ۑ‬ᑃᐽቿቷ‫ߘܻހ‬౰ేၩࠦၮ࣠ዙಣಝ፥ ፕ ࠢࡺ‫ࠦߘޢ‬ၮ࣠ዙಣಝƣ Ƅ཯ཏ‫ݹݘ‬ ƣ ܾߙࠦ‫ߥޢߥޢ‬ஊै ƄƌƄ߄ ၮଘଓƄࡼ ࡷƇ ଘଓܻ༬ீ ƥƄơ ܻ༬ீ ࡼ ‫ݑ‬ƆƎܻީ‫ݑ‬፥ ፕ ࡣ ࢧ ޯଐଌƄ፥ ገƄ࣠ࣃ ‫ࡩݘ‬Ƅ౰ే‫ݓ‬ ޯଐଌƄࠦၮ࣠ዙಣಝƣ ƀܻ೘ೃ ݊ ‫ޯ ࣐ޢ‬ఱ ఝƧ ƀ ،ࡣ ࢧ ޯଐଌƄ፥ ገƄ࣠ࣃ ‫ࡩݘ‬Ƅ౰ే‫ݓ‬ ޯଐଌƄƄ߃ ࠧƠ ॅळࢵ ߎ ࢧ ‫ ࠁޯࡺܻࡩޢࠦކ‬ƄቿቱƄƌᇯ ᇍ ఴ ఠƤ߃ ࠧযওߥࡩƄࠦၮ࣠ዙಣಝ ፥ ጘଘଓƄ࣠ࠒ ެࡩƄે ઻ၮܻ‫ݹܻࢠޢ‬Ǝࡣ ࢧ ޯଐଌƄ፥ ገƄ࣠ࣃ ‫ࡩݘ‬Ƅ౰ే‫ݓ‬ ޯଐଌƄƞ ࢖‫ރݘ‬ƣ Ƅ ે઻ၮቿቷࢧ ޯࡷƥࠦߙ࢖‫ ࢵܻࡩީࠦࡷݘ‬౰ేၩ [ f x ] ቿቱƄƌƇ ߄ ၴ ƥƄƍƄࡣ ࢧ ޯࡷࠢ‫ ހܾރ‬X ፥ ገƄ࣠ࣃ ‫ࡩݘ‬Ƅ౰ే‫ݓ‬ ޯଐଌƄࡼ ‫ݑ‬ơ ࣠ߥ‫ރ‬ P( α ≤ X ≤ ᵝ ) = ∫ࢻ ࢌ (࢞) ࢊ࢞ ᵝ α ≤ ᵝ ‫ࠁޢ ݟ‬ ‫ޢ ݟ‬༨ဂ ࠦ‫ߥޢߥޢ‬ஊै ƄƌƄ߄ ၮଘଓƄࠦၮ࣠ዙಣಝ ، ᵝ ،α ໠ ໄ‫ߥޢߥޢ‬ஊै Ƅ௪ ௎ƌ߄ ‫ࡩݘ‬Ƅࡣ ၴ ƀࡼ ࡷᑑ ᐽƍƥ ܻಛಗ୷ ୪मएࠦ‫ܻހ‬ຨ຀ࡩƄቿቱ࣠ᐳ႞ࡩƥࡣ ࢧ ޯଐଌƄ፥ ገƄ࣠ࣃ ‫ࡩݘ‬Ƅ౰ే‫ݓ‬ ޯ൞ᒑᐽࠦ‫ ࠁޯࡺܻࡩޢ‬ଘଓƄࠦߙܻຫ຀ࡩƄቿቱƄਔ ৹ ݂ (‫)ݔ‬ቿቱƄኆ ቱƄࣩ ࡺఱ ఠ ࠦ‫ࡩޯܻࡩޢ‬Ƅƒ Ƅ࣠உै ܻ‫୓ ࢧ ߎހ‬ขቿቱƄኆ ቱƄƤ߃ ࠧƣ Ƅƥ ،ࠦߙܻ૾૓ቿቱƄƌ፥ ፕ f (x)ƣ ƀơ ࣠ߥ‫ރ‬ƥ f (x) ୷ ୪༟༎ܻ‫ހ‬ ∫௔ ݂ (‫ = ݔ݀)ݔ‬1 .II ࣩ ࣐ ‫ ކݘ‬x ౰ే‫ݓ‬ ޯ൞ᒑᐽࠦ‫ ࠁޯࡺܻࡩޢ‬ଘଓƄࠦߙܻຫ຀ࡩƄቿቱƄƌ ( ‫ ) ێ‬ƞ ܸޯࡴ ௕ (݂(‫ = )ݔ‬0 ) 102 f(x) ≥ 0 .I ‫ ˂ݔ‬1 ܸୃกܸ଍ଉŽ–ໝ ແࡦफउ –ࠣ‫ݶ‬ Ɖܸࢤ ެߕକଐƁƢࠣ‫ࡦ ࢽޟܸ ࢲޟ‬ƁƢࠣ‫ޟ‬පඓ ൷Ɓ౭౑ࢲ କଐƁƄ ܸࢲ ƁƋኃ ᒔᐺ። ጏ࢓‫ݕ‬ ࡦƁ࣬ ࣊ Ɓ࢓ߢೕಹኃ ቮƁ೾೥࢓଍ଉƁƄ ƁƋƁ߁ ࢟ ſ ። ፎକଐ߁ ެࢲ କଐƁƢ। ्࢟ ࣝࡦƁŻܸࢤ ߾ ଔଐƁ ƛ ܸߢࢽ ࡦƁ߁ ዖዄ Ž૊ સၫƋࣝެ࠾ኃ ቮƁƊܸެࢲ ଘଐƁ ିଷ௫య ቀ݂(‫ ଼ = )ݔ‬+ (݂(‫ = )ݔ‬0 ) න ଷ௫మ ଶ − ଽ௫ ଼ 1≤‫≤ ݔ‬3 ‫ ˃ݔ‬3 ߰ ‫ࡩ ࢧ ࠒޢ‬Ƅ߄ ၰ Ƅ࣠ࡩƄƧ ƥܻఱ ఓཀྵཏࠒ ެଐଌƄ‫ ݝ‬༨ဈ ࠦၰ ܻࣀ ଐଌƄƣ ƀ‫ ݝ‬ხ჎ƀ ቁ ࡠ இॅ Ɓ ࠢ‫୍ܻކ‬ชƄƅ ॅळ࣐ ଐଌƄ ஶ ݂ (‫ = ݔ݀)ݔ‬1 ିஶ ஶ ଵ ଷ ஶ ݂ (‫ݔ݀)ݔ‬ ∫ିஶ ݂ (‫ି∫ = ݔ݀)ݔ‬ஶ ݂ (‫ ݔ݀)ݔ‬+ ∫ଵ ݂ (‫ ݔ݀)ݔ‬+ ∫ଷ =∫ିஶ 0 ݀‫ ݔ‬+ ∫ଵ ( ଵ = ଷ ∫ଵ ( ିଷ௫య = │( = ቀ− ଷ ଼ ିଷ௫ర ଷଶ ଷ.଼ଵ ଷଶ ିଷ௫య ଼ ଷ௫మ + ଶ + + ௫య ଶ ଷ + − − ଷ௫మ ଶ ଽ௫ ଼ − ଽ௫ ଼ )݀‫ݔ‬ ଽ௫మ ) │ଵଷ ଵ଺ ଶ଻ ଵ )݀‫ ݔ‬+ ஶ ∫ଷ 0 ݀‫ݔ‬ ଼ଵ ଽ ቁ+ ቀ ଶ − ଶ ቁ+ ቀ − ଵ଺ + ଵ଺ቁ ଷଶ = - 7.5 + 13 – 4.5 = 13-12 = 1 ܻ‫ ߄ރ‬ၮܻఱ ఠܻಛಗࠦީࠁ ܻࢧ ଐଌƄࠦ‫ ࠁޯࡺܻࡩޢ‬ଘଓƄƇ ܻ‫ݹݘ‬Ə࣠ޯࡩƄƥቿቷࢧ ߦެଐଌƄƥቿቷࢧ ޯଐଌƄࠦ୆෷Ƅ࣠ࣃ ‫ࡩݘ‬ƄƇ Ƅ౰ే‫ݓ‬ ޯଐଌƄࠦࢵ ƄƎƌƣ ƀ ࠦߥ࡯‫ޯݘ‬ଐଌƄƓ ƥ࢖ߦࡩƄƇ ƄƎܻީޯࠀ Ƅᑑ ᐽቦሼ ݊ ࡺޯ഻വറƄᎨ፻ܻ‫ࡷݘ‬मअ߄ ߥ‫ކ‬፥ ጿܻᐱᚑƄ߄ ႀ ޯࢵ Ƅࡼ ࡓ ೘಼ၖ࿏ ܻ୓คે ઻ၮơ ࣠ࢧ ஊै Ƅ፥ ጿ ‫ۑ‬Ƈ ƄƎƄ࢖ߥࡩƄƍܻ༧ဈ ܻ‫ހ‬ ࡣ ޲ࡷƇ ܻߥ‫ࡩޯ ࣐ީޢ‬Ƅࡼ ࡷਔ ৲߄ ‫ࡩݘ‬Ƅܻಛಗ཯དྷࡩƄࠦࡺࠩଐଌƄࠦ‫ ࠁޯࡺܻࡩޢ‬ଘଓƄƇ ܻ‫ݹݘ‬Ə࣠ޯࡩƄࡼ ࡷƄƌ߄ ၮ્ ઻‫ߙޢࡺܻݹ‬Ɠ ࢖‫ޯݘ‬ఱ ఝƥ ݊ ‫ݹ‬Ə࣠‫ކ‬ƥഭ ୨‫ࡩ ࣐ީޢ‬Ƅ݊ ‫ݹ‬Ə࣠ޯࡩƄƥࡣ ࢧ ߦެଐଌƄ፥ ገƄ࣠ࣃ ‫ࡩݘ‬Ƅ౰ే‫ݓ‬ ޯ൞ᒑᐽᑑ ᐽƍƥƣ ࣠ࢵ Ƅ࣠‫ݹ ݊ހ‬Ə࣠‫ކ‬ƥ ،௪ ௎߄ ஊै ƄƧ ƍ݊ ‫ݹ‬Ə࣠‫ކ‬ ‫ޯ ࢧ ࡣۑ‬ଐଌƄ፥ ገƄ࣠ࣃ ‫ࡩݘ‬Ƅ౰ే‫ݓ‬ ޯ൞ᒑᐽƞ ݊ ‫ݹ‬Ə࣠‫ކ‬ƥƇ [ Binomial Distribution ]:௧ ோ߁ இॅ ƁƤ Ɗ݇ ‫ێ ރࣝݶ‬ Ɗ ܻ༬စ ܻዙ዗߄ ၰ ƀ࣊ ߥߙ໠ ໄ༬ဈ ܻ‫ܻރ‬ಛಗ ࠦ୆෷Ƅ࣠ࣃ ‫ݹ ݊ࡩޯ ࠖ࢖ހࠦݑ‬Ə࣠‫ ࠧ࣠ކ‬௧ ோ߁ இॅ ƁƢƊ ‫ أو‬። ጅܸީޯࡦƁ። ፎܸࡷެ߾ କଐƁ݇ ‫ݶ‬ƌࣝެࡦƁ ơ ܻ޲ࡷ‫ࡩޯ ࠖ࢖ހࠦۑ‬ƄƎƄಒ౷୓กमकܾ୓෺ଘଓƊ ܻႅ ެࡩƄ ơ ܻࡺޯࠁ Ƅƣ ƀ࿃ ႓ܻࢵ ଛଓƄƔ ྲ ႡࡩƄƣ ࣠ࡓ ‫ݹ‬ƥܻࠩ࡯ࣃ ߙ༇໩ ଜଓƄƥࠦ‫ࡩޯ ࠖ࢖ހ‬Ƅ .ࡴ ‫ރݘ‬ƥƀଘଓࠦ‫ܻހ‬ၴ ଗଓƄ߄ ࡺޯ‫཯ކݘ‬དྷࡩƄቿቷ‫ ࢵޟ‬ଛଓƄƥƀƇ Ƅžܻࢧ ࠁ ଗଓƄūƌ࣠ߥ‫ߘ ࣐ݘࠦރ‬፥ ጹƎ ᑑ ᐽƍે ઻ၮ : ௧ ோ߁ இॅ ƁƤ Ɗ። ጅܸީޯࡦƁ݇ ‫ݶ‬ƌࣝެࡦƁ࢚ ‫߽ ࢤܸݳ‬ 103 ܸୃกܸ଍ଉŽ–ໝ ແࡦफउ –ࠣ‫ݶ‬ Ɖܸࢤ ެߕକଐƁƢࠣ‫ࡦ ࢽޟܸ ࢲޟ‬ƁƢࠣ‫ޟ‬පඓ ൷Ɓ౭౑ࢲ କଐƁƄ ܸࢲ ƁƋኃ ᒔᐺ። ጏ࢓‫ݕ‬ ࡦƁ࣬ ࣊ Ɓ࢓ߢೕಹኃ ቮƁ೾೥࢓଍ଉƁƄ ƁƋƁ߁ ࢟ ſ ƛ ܸߢࢽ ࡦƁ߁ ዖዄ Ž૊ સၫƋࣝެ࠾ኃ ቮƁƊܸެࢲ ଘଐƁ ። ፎକଐ߁ ެࢲ କଐƁƢ। ्࢟ ࣝࡦƁŻܸࢤ ߾ ଔଐƁ :፥ ፕ ࢝ ‫ݶ‬ ܻࢧ ࠀ Ɔ߄ ‫ހݘ‬፥ ገܻެ޲ࡩƄ݊ ‫ݹ‬Ə࣠ޯࡩƄ౮ే๻๬ึ ࠦ‫࢖ހ‬༬ဈ ፥ ጿܻ୐คܺߙūƆ߄ ၰ Ƅƥቿቱƥܻ༨ீ ࡼ ࡷࠦ‫ࡩޯ ࠖ࢖ހ‬Ƅบ ค࣠ࡓ ‫ކ‬ƄƍƂ‫ۑ‬ቿቱƥܻ༨ீ ࡼ ࡷ౰౗ࡁ ƀࡼ ࡷࠦ‫ࡩޯ ࠖ࢖ހ‬Ƅƣ ࣠ࡓ ୓ข .I .፥ ፑ࣠‫ރ‬मऌ݊ ‫ݹ‬Ə࣠‫ކ‬ q.ࡣ ࣃ ߦࡩƄơ ܻࡺޯࠁ ƄᎨ᎜ࡼ ࡷƥ p Ɗ ܻႅ ެࡩƄơ ܻࡺޯࠁ ƄƇ ୍ܻชƧ ƀ࢛ ‫ࡩީݘ‬ƄܻᐳႮ ‫ݑ ࡼހݘ‬Ƈ ଘଓƥܻഷവറƄơ ൎ൉ߥޯࢵ Ƅ .II .ቿቷߥޯࣀ ࡷƥቿቷ‫ޢݘܻࡷޯࡺܻމ‬ዙዋ Ƈ ଘଓƥܻഷവറƄƤ߃ ࠧ .III nƌ߄ ‫ [ ࡩݘ‬p+q =1 ]߰ ‫ࡩ ࢧ ࠒޢ‬Ƅ߄ ၰ Ƅ࣠ࡩƄƧ ƥܻఱ ఓࡣ ࣃ ߦࡩƄƥƊ ܻႅ ެࡩƄ፥ ፑܻࡺޯࠁ Ƅ݊ ዙዋ ࡣ ࢢ ܻၰ .IV ‫ࡩޯ ࠖ࢖ހࠦۑ‬Ƅࡼ ࡷƇ Ƅ୹ ୪ : ። ጕକଐᇱᆚ ࣦ ࣍ ‫ݕ‬ ‫ރ‬௧ ோ߁ இॅ ƁƤ Ɗ݇ ‫ݶ‬ƌࣝެࡦࠣࡴܸ‫ݕ‬ ࡦƁࠣ‫ݐ‬ ‫ࡦ ࢤޟ‬Ɓ ‫ = }ݔ{݌‬c௫௡ ‫݌‬௫‫ݍ‬௡ି௫ P(x) = nCx Px (1 – P)n – x Ɔྲ ႓ଘଓƄ፥ ጿƣ ᇴᆝ Ƅƍܻߙ ¼࣠ࠧ௪ ௎ƥܻߘƎƏ໠ ໄެ‫ݑޢ‬ƥƍࡣ ߦ࣍ ௪ ௎ኆ ቱƄ࣠ࡩƣ ࣠ࡓ ‫ݹ‬ƣ ƀơ ܻࡺޯࠁ Ƅ ‫ێ‬8 3 )ƞ ܸޯࡴ ‫ۑ‬žܻߘƎƏƣ ࣠‫ߘ ࡣݑޢ‬ଘଓƄે ઻ၮࡴ ࠩߦࢧ ެࡩƣ ࣠ࡓ ‫ݹ‬ƣ Ƅơ ܻࡺޯࠁ Ƅܻዙኽơ ܻߦ࣍ ƀ 8 ࡠ இॅ Ɓ ¼ = {p} Ɗ ܻႅ ެࡩƄơ ܻࡺޯࠁ Ƅ ¾ = ¼ – 1 = p – 1 = { q}ࡣ ࣃ ߦࡩƄơ ܻࡺޯࠁ Ƅ 8= n žܻߘƎƏࡴ ᐱ᏶࣠‫ߘ ࡣݑޢ‬ଘଓƄે ઻ၮơ ܻߦ࣍ ଘଓƄƌ߄ ၮߎ ࢧ ‫ރ‬ƣ ࣠ࡓ ‫ ݹ‬ƣ Ƅơ ܻࡺޯࠁ Ƅે ઻ၮࡣ ࢧ ༨စ ƥ P ≥ 4 = { p(4) + p(5) + p(6) + (7) + p(8) } P(x) = nCx Px (1 – P)n – x P(4) = P(5) = P(6) = P(7) = P(8) = 104 ଼! ସ!ସ! ଼! ( )ସ ( )ସ = ଵ ଷ ସ ସ ଵ ହ ଷ ଷ ( ) ( ) ହ!ଷ! ସ ସ ଼! ଵ ଺ ଷ ଶ ( ) ( ) ଺!ଶ! ସ ସ ଼! ଻! ଵ ! ଼! ଼! ଴ ! = = ( )଻( )ଵ = ଵ ସ ସ ଺ହହଷ଺ ଵହଵଶ ଺ହହଷ଺ ଶହଶ ଺ହହଷ଺ ଷ ଶସ ସ ଺ହହଷ଺ ଷ ଵ ( )଼ ( )଴ = ଵ ହ଺଻଴ ସ ଺ହହଷ଺ ܸୃกܸ଍ଉŽ–ໝ ແࡦफउ –ࠣ‫ݶ‬ Ɖܸࢤ ެߕକଐƁƢࠣ‫ࡦ ࢽޟܸ ࢲޟ‬ƁƢࠣ‫ޟ‬පඓ ൷Ɓ౭౑ࢲ କଐƁƄ ܸࢲ ƁƋኃ ᒔᐺ። ጏ࢓‫ݕ‬ ࡦƁ࣬ ࣊ Ɓ࢓ߢೕಹኃ ቮƁ೾೥࢓଍ଉƁƄ ƁƋƁ߁ ࢟ ſ ። ፎକଐ߁ ެࢲ କଐƁƢ। ्࢟ ࣝࡦƁŻܸࢤ ߾ ଔଐƁ ƛ ܸߢࢽ ࡦƁ߁ ዖዄ Ž૊ સၫƋࣝެ࠾ኃ ቮƁƊܸެࢲ ଘଐƁ P ≥ 4 = { p(4) + p(5) + p(6) + (7) + p(8) = = ହ଺଻଴ ଺ହହଷ଺ ହଽସ଻ + ଺ହହଷ଺ ଶହଶ ଺ହହଷ଺ + = 0.090 ଶସ ଺ହହଷ଺ + ଵ ଺ହହଷ଺ [Poisson Distribution] Ơ ࣝࢲ Ɓࣝ‫ݽ‬ ݇ ‫ݶ‬ƌࣝ‫ ރ‬: ‫ێ‬ ࡼ ࡷ౰ే޲ࡁ ፥ ጿƢ߄ ႀ ޯఱ ఓƥ‫ެߥ ࣐ݘࠦ ۑ‬ଐଌƄ ቿቷࢧ ߦެଐଌƄࠦ‫ ࠁޯࡺܻࡩޢ‬ଘଓƄƇ ܻ‫ݹݘ‬Ə࣠ޯࡩƄࡼ ࡷ ƣ ࣠ࢵ Ƅ࣠‫ݹ ݊ހ‬Ə࣠‫ކ‬౰౎ޯ‫ݹݘ‬ žܻ‫ݹ‬౮ేߦࡩƄƢॅळၮࡣ ࡺఴ ఠࠢ‫ࡷܻކ‬Ƅ߄ ႀ ޯࢵ ܻߙ‫ ࠅޯࡺܻݑޢࠦۑ‬ଘଓƄƥࠦ‫ݹ‬ƌܻࢧ ޯߘଘଓƄƥࠦ‫ࡩ ࣐ީޢݘޢ‬ƄƢॅळ‫ࡩݘ‬Ƅ፥ ጿƇ ܻߥ‫ࡩޯ ࣐ީޢ‬Ƅ ƣ ࣠ࢵ Ƅ࣠‫ݹ ݊ހ‬Ə࣠ޯߙ‫ݹࠦۑ‬ƌܻࢧ ޯߘଘଓƄƥࠦ‫ݹ‬ƎƄƌଘଓƄƢॅळ‫ࡩݘ‬ƄƥƇ ܻ‫ࡩݘࡺ࡯ޢ‬Ƅƈ ࣠༨ဂ ƥࠦߥ‫ߘޢ‬ኆ ቱƄžܻ‫ ࠁޢ‬ଘଓƄࠦࢵ ƄƎƌƥ ࠦީఱ ఝƣ ࣠ࢵ Ƅ࣠‫ݹ ݊ހ‬Ə࣠୓ก࣯ ያዜ ƥ‫ޢߦܻࠩۑ‬ჱࢧ ‫ކ‬ƥࠦ‫ ࢧެݘ‬ଐଌƄƇ ܻၴ ࣠ޯެଐଌƄࠦީߘƄ୹ ୪ƥࢭ ყޯߦ‫ކ‬፥ ጿƢƄ߄ ႀ ޯࢵ ଘଓƄ౰ే޲ࡁ ‫ۑ‬Ơ ࣝࢲ Ɓࣝ‫ݽ‬ Ơ ࣝ෩෍ᚿ ࿃ ྜྈ࢖ߦࡩƄƇ ܻ‫ݹܻࢠޢ‬༟༎ƄᎨ፻ܻၮ፟ ፑƄ ࣐ࠢ ࢵ ƥƣ ࣠ࢵ Ƅ࣠‫ݹ ݊ހ‬Ə࣠ޯࡩ݊ ࢥ ༧࿻ ࠢ‫ރ‬ƀ ، X݊ ࣐ ߥެଐଌƄ፥ ገƄ࣠ࣃ ‫ࡩݘ‬Ƅ౰ే‫ݓ‬ ޯଐଌƄࡼ ‫ݑ‬ơ ࣠ߥ‫ ރ‬Ơ ࣝࢲ Ɓࣝ‫ݽ‬ ݇ ‫ݶ‬ƌࣝ‫ݕ࢓ݶ ߋރ‬ ‫ ރ‬ ࠦߘൎ൉‫ ࠁޯࡺܻࡩޢࠦࡷݘ࢖ߙࠦހܻࡩݘ‬ଘଓƄࠢޯߙܻ૾૓บ คᇴᆝ Ƅƍƀ ‫ג‬ ࢞ ……………..، 2، 1، x = 0،،f(x) = P(X = x ) = ‫גି ג‬ ࡼ(࢞) = ࢋ ࢞! ‫ג‬ ࢞ ࢞! ࢋି‫ג‬ ። ፒ Ơ ࣝࢲ Ɓࣝ‫ݽ‬ ݇ ‫ݶ‬ƌࣝ‫ࡴࠣࡦެ ࣍ަޟ ࢄރ‬ƌോᑆᐺƁƑ Ƣྯ ႞ࡦƁ Ɗ ܻႅ ެࡩƄƥƀࡣ ࣃ ߦࡩƄ ƣ Ƅƌܻࢥ ޯࡷƣ ܻ༬ဈ ܻ‫ރ‬Ơ ܻެࠧƣ ࣠ࡓ ‫ݹ‬ƣ ƀ .I ቿቷߥޯࣀ ࡷƈ Ƅ߄ ၰ ଘଓƄƣ ࣠ࡓ ‫ކ‬ƣ ƀ‫ ݪ‬༬࿻ .II ࡣ ࣃ ߦ‫ݹ‬ƥƀ߰ භච ൭ƈ ƌܻஊै Ƅƣ Ƅܻࡷƀࠦ‫࢖ހ‬༬ဈ ᇯ ᆝࠦႅ ‫ޢ‬ჴ჈.III ౰ే‫ݓ‬ ࢢ [ p ]ơ ܻࡺޯࠁ ଘଓƄƥƄ߄ ၴ ౰ేީࡁ [ n]ࠦެ‫ࡩݘޢ‬Ƅࡴ ᄽ஛ .IV ે ઻ၮƧ ࣠ޯ༨࿻ Ɵ ƥ߄ ެࢢ ፥ ጿ౰౗ࡁ ଘଓƄે ઻ၮࠦީ‫ࡷݘޢ‬Ƈ ƄƏ࣠‫ ߙޢ‬5߄ ၴ ࣠‫ݹ‬ƣ ƀơ ܻࡺޯࠁ Ƅ߄ ၴ ƥƀ ( ‫) ێ‬ƞ ܸޯࡴ 6{ ݁ିସ = 0.0183} ࠦީ‫ࡷݘޢ‬Ƈ ƄƏ࣠‫ࡩߦޢ‬ƄƤ߃ ࠧࡼ ࡷ %2ƣ ƄƢॅळ‫ࡷݘ‬ƅ Ǝܻႅ ޯࡩƄࡼ ࡷƥ‫ۑ‬Ə࣠‫ ߙޢ‬200 ࡠ இॅ Ɓ (n) = 200 Ƈ ƄƏ࣠‫ࡩߦޢ‬Ƅƌ߄ ၮ ‫ = ג‬np 105 ‫ࡷݘޢ ݪ ۑ‬Ə࣠‫઻ ેߙޢ‬ၮơ ࣠ࢧ ஊै Ƅơ ܻࡺޯࠁ Ƅ ܸୃกܸ଍ଉŽ–ໝ ແࡦफउ –ࠣ‫ݶ‬ Ɖܸࢤ ެߕକଐƁƢࠣ‫ࡦ ࢽޟܸ ࢲޟ‬ƁƢࠣ‫ޟ‬පඓ ൷Ɓ౭౑ࢲ କଐƁƄ ܸࢲ ƁƋኃ ᒔᐺ። ጏ࢓‫ݕ‬ ࡦƁ࣬ ࣊ Ɓ࢓ߢೕಹኃ ቮƁ೾೥࢓଍ଉƁƄ ƁƋƁ߁ ࢟ ſ ። ፎକଐ߁ ެࢲ କଐƁƢ। ्࢟ ࣝࡦƁŻܸࢤ ߾ ଔଐƁ ƛ ܸߢࢽ ࡦƁ߁ ዖዄ Ž૊ સၫƋࣝެ࠾ኃ ቮƁƊܸެࢲ ଘଐƁ = 200 x 0.02 = 4 ơ ࣠ࢧ ॶॱ५Ƈ ଘଓܻࡺޯࠁ ଘଓƄ፥ ፕ ƥ‫ۑ‬౰౗ࡁ ଘଓƄે઻ၮࠦީ‫ࡷݘޢ‬Ƈ ƄƏ࣠‫઻ ે ߙޢ‬ၮơ ࣠ࢧ ஊै Ƅơ ܻࡺޯࠁ Ƅࠦߙ࢖‫ࡷݘ‬ਔ ৲मए ‫ࡷݘޢީࠦۑ‬Ƈ ƄƏ࣠‫ ߙޢ‬ƥƄ ū ū ū ū ે ઻ၮ ௫ ‫גି ג‬ ܲ(‫= )ݔ‬ ݁ ‫ݔ‬Ǩ { P(0) + P(1) +P(2) +P(3) +P(4) +P(5)} =࣠ࠧƅ ॅळ࣐ ଐଌƄơ ܻࡺޯࠁ ଘଓƄ P = {P(0) + P(1) +P(2) +P(3) +P(4) +P(5)} = ସబ ଴Ǩ ିସ =݁ ݁ିସ + ସభ ଵǨ ݁ିସ + ସమ ଶǨ ݁ିସ + ସయ ଷǨ ݁ିସ + ସర ସǨ ݁ିସ + { 1 + 4 + 8 + 32/3 + 32/3 + 128/15} = 0.0183 (643/15) = 0.78 ସఱ ହǨ ݁ିସ [ Normal Distribution]പ ୥‫ࡦ ࣍ަޟ‬Ɓ݇ ‫ݶ‬ƌࣝެࡦƁ ‫ ێ‬ ݊ ߘƄ࣠ࡩƄ፥ ጿƄ౰ే޲ࡁ Ƣ߄ ႀ ޯఱ ఓƥ‫ ۑ‬Ɔ࢖ࡺޯࣀ ଐଌƄ ቿቷࢧ ޯଐଌƄࠦ‫ ࠁޯࡺܻࡩޢ‬ଘଓƄƇ ܻ‫ݹݘ‬Ə࣠ޯࡩƄࡼ ࡷഭ ୨‫ࡩ ࣐ީޢ‬Ƅ݊ ‫ݹ‬Ə࣠ޯࡩƄ౰౎ޯ‫ݹݘ‬ ‫ ( ࡷ ࡼ ࠁޢ ݟ‬Symmetric)ࡣ ‫ࠧ࣠ࡷޯࡺܻމ‬ƥ‫ۑ‬Ɛ ࢖எै Ƅᇯ ᇍ ࢸ ࡼ ࡷƅ ౰౔ߥ‫ݹ ݊ݹ‬Ə࣠ޯࡩƄᇯ ᇍ ࢸ ƥ‫ࡩݘࡺ઻ ્ۑ‬Ƅ ‫ۑ‬፥ ጒܻࣀ ஊै Ƅ࣊ ࢵ ࣠ࡩƄơ ࣠ࠁ Ƈ ܻ‫ࡩީޢܻރ‬ƄƢܻ࣏ ୓คƄ പ ୥‫ࡦ ࣍ަޟ‬Ɓ݇ ‫ݶ‬ƌࣝެࡦƁསཌࠏ ީࡴ ‫ ێ‬ᇯ ᇍ ࢸ പ ୥‫ࡦ ࣍ަޟ‬Ɓ݇ ‫ݶ‬ƌࣝެࡦƁ  Ɛ ࢖எै Ƅ Ɛ ࣠ߘܻެࡩƄᇯ ᇍ ࢸ ቻቱࡣ ࢧ ޯࡷ݊ ‫ݹ‬Ə࣠‫ ކ‬.I .ơ Ƅ࣠ެଐଌƄƥ࣊ ‫ࡩ࣠ ࢵޢ‬Ƅƥ࣊ ࢵ ࣠ࡩƄࠦ‫ݹ‬ഁ೨࢖ଐଌƄࠦၮ౮౑ࡩƄࢪ ყႾܻߥࡷࠢ‫ߙޢ‬Ʀ ƥܻࣀ ჴ჆ .II 106 ܸୃกܸ଍ଉŽˍ ໝ ແࡦफउˍ ࠣ‫ݶ‬ Ɖܸࢤ ެߕକଐƁƢࠣ‫ࡦ ࢽޟܸ ࢲޟ‬ƁƢࠣ‫ޟ‬පඓ ൷Ɓ౭౑ࢲ କଐƁƄ ܸࢲ ƁƋኃ ᒔᐺ። ጏ࢓‫ݕ‬ ࡦƁ࣬ ࣊ Ɓ࢓ߢೕಹኃ ቮƁ೾೥࢓଍ଉƁƄ ƁƋƁ߁ ࢟ ſ ƛ ܸߢࢽ ࡦƁ߁ ዖዄ Ž૊ સၫƋࣝެ࠾ኃ ቮƁƊܸެࢲ ଘଐƁ ። ፎକଐ߁ ެࢲ କଐƁƢ। ्࢟ ࣝࡦƁŻܸࢤ ߾ ଔଐƁ ࢖ߦࢢ ࣐ࠢ ࢵ ƥơ ࣠ࠁ ࡣ ‫ ࡷޯࡺܻމ‬.III .߰ ‫ࡩ ࢧ ࠒޢ‬Ƅ߄ ၰ Ƅ࣠ࡩƄƧ ƥܻఱ ఓቻቱƧ Ǝܻ‫ݘޢ‬ଐଌƄƞ Ƅ࢖༨စ ଘଓƄ.IV ໠ ໄ‫ߘݘࠦހ‬Ƅ࣠ࡩƄᑻ ᑧ‫ކ‬፥ ፕ ࠦࡺࠩଐଌƄࠦၰ ܻࣀ ଐଌƄƥ .१ ॏߙଛଓƄƎ࣠ഷവറƄܻ‫ݹ࡯ޯߥޢ‬ƣ ƀƣ ƥƌࠦ‫ܻݹ‬ᐱ᏶ଘଓܻࡷ፟ ፑƂƣ Ƅ߄ ޯ೘಼Ƥܻߙ࢖࣍ .V ࡼ ࡷ % ‫ ۑ‬ƣ ܾ‫཯ހ‬ཏ‫ݹݘ‬Ƅ߃ ࠧƥ‫ޢࠦۑ‬ᆿᇍ ࡩƄࠦၰ ܻࣀ ଐଌƄࡼ ࡷ % ‫ ۑ‬፥ ፕ ƥ ϻ - σ ‫ و‬ϻ + σ 2 ϻ + σ ໠ ໄ‫ߘݘࠦހ‬Ƅ࣠ࡩƄࠦၰ ܻࣀ ଐଌƄƥ‫ۑ‬ϻ - σ ‫ و‬ϻ + σ໠ ໄ‫ހ‬໠ ໄ‫ކߥ ݊ހ‬Ƈ Ƅ߄ ܻࠧࣃ ଐଌƄ ‫ و‬3ϻ + σ໠ ໄ‫ހ‬໠ ໄ‫ߘݘࠦހ‬Ƅ࣠ࡩƄࠦၰ ܻࣀ ଐଌƄܻࡺެყჅ‫ޢࠦۑ‬ᆿᇍ ࡩƄࠦၰ ܻࣀ ଐଌƄࡼ ࡷ % ‫ ۑ‬፥ ፕ 2ϻ - σ ‫و‬ ࠦ‫ޢ‬ᆿᇍ ࡩƄࠦၰ ܻࣀ ଐଌƄࡼ ࡷ % ‫ ۑ‬፥ ፕ 3ϻ - σ .߰ ‫ࡩ ࢧ ࠒޢ‬Ƅ߄ ၰ Ƅ࣠ࡩƄƧ ƥܻఱ ఠഭ ୨‫ࡩ ࣐ީޢ‬Ƅཀྵཏࠒ ެଐଌƄ‫ ݝ‬༨ဈ ࠦၰ ܻࣀ ଐଌƄ.VI ࠢ‫ރ‬ƀ σ2ࠢ୐෺୍ܻงƥ ϻ࣐ࠦ ࢵ ࣠ޯࡷƧ ኅ ቱƄ X ࡣ ࢧ ޯଐଌƄ ࢖ࡺޯࣀ ଐଌƄ፥ ገƄ࣠ࣃ ‫ࡩݘ‬Ƅ౰ే‫ݓ‬ ޯଐଌƄࡼ ‫ݑ‬ơ ࣠ߥ‫ݕ࢓ݶ ߋ ރ‬ ‫ ރ‬ ፥ ጘൎᇩᆝࠦߙ࢖‫ ࠁޯࡺܻࡩޢࠦࡷݘ‬ଘଓƄࠢޯߙܻ૾૓บ คᇴᆝ Ƅƍƀ،ഭ ୨‫ࡩ ࣐ީޢ‬Ƅ݊ ‫ݹ‬Ə࣠ޯᑃᐽ݊ ࢥ ༧࿻ [ Standard Normal Distribution ] : Ƥ Ƌܸ‫ݕޟ‬ ଍ଉƁപ ୥‫ࡦ ࣍ަޟ‬Ɓ݇ ‫ݶ‬ƌࣝެࡦƁ Ɛ ܻ‫ࡩߥޢ‬ƄƇ Ƅ߄ ၰ ƥࡼ ࡷƆƌ࢖༬ீ ࠦ‫ݹ‬Ǝܻ‫ࡷݘޢ‬যওߘ፟ ፑƄƢܻஉै ƄযওߥࡩƄࡣ ‫࣠ݹ‬༨ဈ Ƨ Ǝܻ‫ݘޢ‬ଐଌƄഭ ୨‫ࡩ ࣐ީޢ‬Ƅཀྵཏࠒ ެଐଌܻ‫ݹߥ ࢧ ߄ހ‬ ഭ ୨‫ࡩ ࣐ީޢ‬Ƅཀྵཏࠒ ެଐଌƄ‫ ݝ‬༨ဈ ࠦ‫ߘݘ‬Ƅ࣠ࡩƄࢇ ࣍ ܻެଐଌƄࠦߙ࢖‫ࡷݘ‬ƥࠦ‫ރ‬ƎܻߥଐଌƄƇ ଘଓܻၰ ፥ ጿܻᐱᐟࡷƆƌܻߦޯࢵ ଘଓƄࡼ ࡓ ೘಼཯དྷࡩƄƥ ܼ= ‫ݔ‬௜ି ϻ σ Z = 1.54 ્ ઻‫ݹ‬ሯሡሁ ፥ ፕ Z ࠦ‫ݹ‬Ǝܻ‫ݘޢ‬ଐଌƄࠦࡺ‫ࡩߥޢ‬Ƅƅ ܻࣀ ஊै ࠦࡷƏൎᑉᐽƄࠦ‫ݹܻࢠޢ‬༟༎Ƅࠦ‫ݓ‬ ‫ࡩ ࢧޢ‬Ƅƥ‫ۑ‬Ƨ Ǝܻ‫ݘޢ‬ଐଌƄ : Ƨ Ǝܻ‫ݘޢ‬ଐଌƄഭ ୨‫ࡩ ࣐ީޢ‬Ƅ݊ ‫ݹ‬Ə࣠ޯࡩƄཀྵཏࠒ ެࡷ‫ ݝ‬༨ဈ ࠦၰ ܻࣀ ଐଌƄ߄ ၴ ƥƀ ( ‫ ێ‬ƞ ܸޯࡴ Z = -1.54 Z ƌ࣠ዙኸ ፥ ጿࡣ ߦࢵ Ƅ፟ ፑƄƠ ࢖ඳඣ ൷ Ƅ Ɓࣝ࣍ ஆॅ Ɓ  Z = 1.54) ƌܻ༬࿻ ଘଓƧ Ǝܻ‫ݘޢ‬ଐଌƄഭ ୨‫ࡩ ࣐ީޢ‬Ƅ݊ ‫ݹ‬Ə࣠ޯࡩƄơ ƥ߄ ၴ ࡼ ࡷ ƌ࣠ࡺ‫ࡩݘ‬Ƅƥߎ ࢧ ࡩƄ݊ ࣍ ܻߥ‫ކ‬ƥ 0.04 ƌ࣠ࡺ‫ࡩݘ‬Ƅ፟ ፑƄࡣ ࢧ ‫ރ‬ཀྵདྷࠁ ࠢ‫ߙޢ‬Ơ ࢖ඳඣ ൷ƥ ‫ࡩ ࢧ ߎ ۑ‬Ƅ፟ ፑƄࡣ ࢧ ‫ރ‬ཀྵདྷࠁ Z = -1.54 = ( 0.4382) ( ‫ ۑ‬፥ ፕ ƥࠦ‫ॅހ‬ळ࣐ ଐଌƄࠦࡺ‫ࡩߥޢ‬Ƅܻެ‫ݹݘ ࣐ޢ‬ Z = 0.25 ࠦࡺ‫ࡩߥޢ‬Ƅ໠ ໄ೘಼፟ ፑƄƧ Ǝܻ‫ݘޢ‬ଐଌƄഭ ୨‫ࡩ ࣐ީޢ‬Ƅཀྵཏࠒ ެଐଌƄ‫ ݝ‬༨ဈ ࠦၰ ܻࣀ ଐଌƄ߄ ၴ ƥƀ ( ‫ ێ‬ƞ ܸޯࡴ 107 ܸୃกܸ଍ଉŽ–ໝ ແࡦफउ –ࠣ‫ݶ‬ Ɖܸࢤ ެߕକଐƁƢࠣ‫ࡦ ࢽޟܸ ࢲޟ‬ƁƢࠣ‫ޟ‬පඓ ൷Ɓ౭౑ࢲ କଐƁƄ ܸࢲ ƁƋኃ ᒔᐺ። ጏ࢓‫ݕ‬ ࡦƁ࣬ ࣊ Ɓ࢓ߢೕಹኃ ቮƁ೾೥࢓଍ଉƁƄ ƁƋƁ߁ ࢟ ſ ። ፎକଐ߁ ެࢲ କଐƁƢ। ्࢟ ࣝࡦƁŻܸࢤ ߾ ଔଐƁ ƛ ܸߢࢽ ࡦƁ߁ ዖዄ Ž૊ સၫƋࣝެ࠾ኃ ቮƁƊܸެࢲ ଘଐƁ ࡠ இॅ Ɓ ࠦ༬ဈ ܻެࡩƄࠦࡺ‫ࡩߥޢ‬ƄƊ ࢖࣐ ‫ރ‬Ꭸ᎜‫ ۑ ۑ‬፥ ፕ ƥഭ ୨‫ࡩ ࣐ީޢ‬Ƅ݊ ‫ݹ‬Ə࣠ޯࡩƄơ ƥ߄ ၴ ࡼ ࡷ Z = 0.25ଘଓƥƄ߄ ၴ ࣠‫ރ‬ ‫ ۑ‬ཀྵཏࠒ ެଐଌƄࠦၰ ܻࣀ ࡷߎ ࢧ ‫ރ‬፥ ፕ ، ) ‫ࡷ ࡼ ۑ‬ ፥ ፕ Z = 0.25໠ ໄ೘಼፟ ፑƄࠦၰ ܻࣀ ଐଌƄƣ ƄƧ ƀ ‫ ۑ‬- 0.0987 = 0.4013 -1.60 = Z Ǝܻఱ ఓ፟ ፑƄഭ ୨‫ࡩ ࣐ީޢ‬Ƅཀྵཏࠒ ެଐଌƄ‫ ݝ‬༨ဈ ࠦၰ ܻࣀ ଐଌƄ߄ ၴ ƥƀ :( ‫ ێ‬ƞ ܸޯࡴ ( 0‫ۑ‬4452 ፥ ፕ Z = - ‫ߘޢࡺࠦ ۑ‬ƣ ܾߙഭ ୨‫ࡩ ࣐ީޢ‬Ƅ݊ ‫ݹ‬Ə࣠ޯࡩƄơ ƥ߄ၴ ፥ ጿ‫ހܻࡩީ ࠒ ݟ‬ ‫ ࡷ ࡼ ۑ‬0‫ۑ‬4452 Ɗ ࢖࣐ ‫ܻࠧހ‬ƌܻ༬࿻ Ƅࡼ ࡓ ೘಼ Z = - 1.60Ǝܻఱ ఓ፟ ፑƄࠦၰ ܻࣀ ଐଌƄƥ ‫ ۑ – ۑ = ۑ‬ z= % ‫ ۑ ( أو ۑ‬፥ ፕ z = - ‫ ۑ‬Ǝܻఱ ఓ፟ ፑƄࠦၰ ܻࣀ ଐଌƄƣ ƀƧ ƀ ፟ ፑƄ z = 1.60 ࡼ ࡷƧ Ǝܻ‫ݘޢ‬ଐଌƄഭ ୨‫ࡩ ࣐ީޢ‬Ƅཀྵཏࠒ ެଐଌƄ‫ ݝ‬༨ဈ ࠦၰ ܻࣀ ଐଌƄ ߄ ၴ ƥƀ ( 8.8 )ƞ ܸޯࡴ 2.55 ( ‫ ۑ‬፥ ፕ ƥ z = 1.60ཀྵཏࠒ ެଐଌƄ‫ ݝ‬༨ဈ ࠦၰ ܻࣀ ଐଌƄ߄ ၴ ࣠‫ރ‬ଘଓƥƄ ( 0.4946 ፥ ፕ ƥ z= 2.55 ཀྵཏࠒ ެଐଌƄ‫ ݝ‬༨ဈ ࠦၰ ܻࣀ ଐଌƄ߄ ၴ ࣠‫ܻރ‬୆คܻ‫މ‬ : ፥ ፕ z = 2.55 ፟ ፑƄ z = 1.60 ࡼ ࡷࠦၰ ܻࣀ ଐଌƄƣ ƍƀ % 93.98 = 0.9398 = = ‫ ۑ‬+ 0.4452 ፥ ጒܻࣀ ࠁ ࣊ ࢵ ࣠‫ހ‬ഭ ୨‫ࡩ ࣐ީޢ‬Ƅ݊ ‫ݹ‬Ə࣠ޯࡩƄ݊ ީჴႾ፥ ገܻ‫ ࡓࠩ࢖ހ‬ᚻࡩƄƊ ܻީࢧ ଐଌƄ࢖ዙኸ ƣ ᇴᆝ Ƅƍƀ :( 8.9 ƞ ܸޯࡴ ‫ و‬110໠ ໄ‫࢖ހ‬ዙኸ ቻቱܻ୆෷Ƅ࣠ࣃ ‫ݑ‬౰ేޯࠀ Ƅܻၰ ܻީࢧ ࡷƣ ƀơ ܻࡺޯࠁ Ƅܻࡷ‫ۑ‬8 Ƨ Ǝܻ‫ࡷݘޢ‬ƞ Ƅ࢖༨စ Ƅƥ 100 = ‫ۑ‬Ɵ Ƅ౰౔ၰ Ƅࠦၮܻࢵ 120 ࡠ இॅ Ɓ ƌܻ༬࿻ Ƅ࣠ࠧƅ ॅळ࣐ ଐଌƄ P ( 110 ˂ X ˂ 120) X1 = 110 ، X2 = 120 ( ϻ = 100 ، σ = 8) ‫ݔ‬௜ି ϻ ܼ= σ Z1 = 110 – 100/ 8 = 1.25 Ơ Ž‫߾ޟݜ‬ Z2 = 120 – 100/ 8 = 2.50 108 ܸୃกܸ଍ଉŽ–ໝ ແࡦफउ –ࠣ‫ݶ‬ Ɖܸࢤ ެߕକଐƁƢࠣ‫ࡦ ࢽޟܸ ࢲޟ‬ƁƢࠣ‫ޟ‬පඓ ൷Ɓ౭౑ࢲ କଐƁƄ ܸࢲ ƁƋኃ ᒔᐺ። ጏ࢓‫ݕ‬ ࡦƁ࣬ ࣊ Ɓ࢓ߢೕಹኃ ቮƁ೾೥࢓଍ଉƁƄ ƁƋƁ߁ ࢟ ſ ። ፎକଐ߁ ެࢲ କଐƁƢ। ्࢟ ࣝࡦƁŻܸࢤ ߾ ଔଐƁ ƛ ܸߢࢽ ࡦƁ߁ ዖዄ Ž૊ સၫƋࣝެ࠾ኃ ቮƁƊܸެࢲ ଘଐƁ Z2 = ‫ و ۑ‬Z1 = 1.25໠ ໄ‫ࠦހ‬ၰ ܻࣀ ଐଌƄ࣠ࠧƅ ॅळ࣐ ଐଌƄơ ܻࡺޯࠁ ଘଓƄƣ ƀƧ ƀ ƣ ƀ߄ ༬စ Ƨ Ǝܻ‫ݘޢ‬ଐଌƄഭ ୨‫ࡩ ࣐ީޢ‬Ƅ݊ ‫ݹ‬Ə࣠ޯࡩƄơ ƥ߄ ၴ ፥ ጿ‫ހܻࡩީ ࠒ ݟ‬ƥ Z1 = 1.25 =( 0.3944) Z2 = 2.50 = ( 0.4938) Z2 – Z1 = 0.4938 – 0.3944 = 0.0994 = 9.94% ‫ۑ‬0.0994 = P ( 110 ˂ X ˂ 120 ) ƅ ॅळ࣐ ଐଌƄơ ܻࡺޯࠁ ଘଓƄ࣠ࠧƥ [ t]݇ ‫ݶ‬ƌࣝ‫ێ ێރ‬ ഃ೨ƍܻ޲༨ဂ ྲ ྟྈ‫ۑ ࠁޢ ݟ‬1908Ƣܻၮࠢߦࣃ ჴࡁ ƄƧ ኅ ቱƄƥ‫ ࠅ࣠ ࢵ ݝ‬Ƨ ౮ే࡯༬စ ଘଓƄᎨ፻ܻ‫ࡩݘ‬Ƅ፟ ፑƄ t݊ ‫ݹ‬Ə࣠‫ ࣀ ݪކ‬ჱႾ ܻᐱᐟ‫ݹ ݊ࡷ ࡼ ࠁޢ‬Ə࣠ޯࡩƄƄ߃ ࠧ .ࠢ‫࢖ހ‬ᐱᐢࢸ Ƅƥ Student Ǝܻ‫ࡷ ࣀޯݘ‬ᄎ ႓ܻ‫ހ‬Ƥྲ ྟྈƥ݊ ‫ݹ‬Ə࣠ޯࡩƄƄ߃ ࠧቿቱƌܻ‫ߙޢࠢࡷݘ‬ [ student distribution ]บ คƌ࣠‫ޢ‬ჴࢵ ݊ ‫ݹ‬Ə࣠‫ ࣯ކ‬ያዜ Ɔ࢖ࡺޯࣀ ଐଌƄࠦ‫ ࠁޯࡺܻࡩޢ‬ଘଓƄƇ ܻ‫ݹݘ‬Ə࣠ޯࡩƄࡼ ࡷ࣠ࠧ t݊ ‫ݹ‬Ə࣠‫࡯ޢࠢކ‬ၮள ஭࣐ ‫ݹ‬Ƨ ኅ ቱƄ፥ ፑܻࡺޯࠁ ଘଓƄ݊ ‫ݹ‬Ə࣠ޯࡩƄ ߋ ‫ݕ࢓ݶ‬ ‫ރ‬ ቿቷࢧ ޯଐଌƄ ݂(‫ܿ = )ݐ‬ቀ1 + భ ௧ ି௩ା మ ቁ ௩ ࠦ୆งଛଓƄࠦ‫ݓ‬ ‫ࡩ ࢧޢ‬Ƅ߃ ၯ ܾ‫ݹ ݊ކ‬Ə࣠ޯࡩƄƄ߃ ࠧࠢޯߙܻ૾૓ቿቱƄƌƥ ∞ ˂ t ˂ ∞ ƣ ƀ‫ࠁޢ ݟ‬ ࠦ‫࢖ݹ‬ஊै Ƅ Ƈ ܻၴ Ǝƌ = ‫ݒ‬ บ กܻ‫ܿ މ‬ ፥ ጘଘଓƄƣ ࣠‫ ݹݘ ࣐ ࣩހܻࡩߥܻރ‬t ݊ ‫ݹ‬Ə࣠ޯࡩࠦ‫ݹ‬Ǝܻ‫ݘޢ‬ଐଌƄࠦ‫ݓ‬ ‫ࡩ ࢧޢ‬Ƅƥ ࢞Ǉ− ϻ ࢚= ඥ࢙/࢔ 109 ܸୃกܸ଍ଉŽ–ໝ ແࡦफउ –ࠣ‫ݶ‬ Ɖܸࢤ ެߕକଐƁƢࠣ‫ࡦ ࢽޟܸ ࢲޟ‬ƁƢࠣ‫ޟ‬පඓ ൷Ɓ౭౑ࢲ କଐƁƄ ܸࢲ ƁƋኃ ᒔᐺ። ጏ࢓‫ݕ‬ ࡦƁ࣬ ࣊ Ɓ࢓ߢೕಹኃ ቮƁ೾೥࢓଍ଉƁƄ ƁƋƁ߁ ࢟ ſ ƛ ܸߢࢽ ࡦƁ߁ ዖዄ Ž૊ સၫƋࣝެ࠾ኃ ቮƁƊܸެࢲ ଘଐƁ ። ፎକଐ߁ ެࢲ କଐƁƢ। ्࢟ ࣝࡦƁŻܸࢤ ߾ ଔଐƁ ે ઻ၮt݊ ‫ݹ‬Ə࣠‫ކ‬ཀྵཏࠒ ެࡷ߄ ࡺޯ‫ݹݘ‬ƥ‫ߦܻࢠܻࡷެࠢۑ‬༧စ Ƅ౰౗ࡁ Ƅࠢ‫ރ‬ƄଘଓƄƧ Ǝܻ‫ݘޢ‬ଐଌƄഭ ୨‫ࡩ ࣐ީޢ‬Ƅ݊ ‫ݹ‬Ə࣠ޯࡩƄt ݊ ‫ݹ‬Ə࣠‫ީࠢކ‬ఴ ఓ ƅ ౰౔ߥ‫࢖ݹࠦݹ‬ஊै ƄƇ ܻၴ Ǝƌƌ߄ ၮƌƄƌबअܻࡷ߄ ެ‫ߙݘ‬ūࠦ‫࢖ݹ‬ஊै ƄƇ ܻၴ Ǝƌ፥ ፕ ƥཀྵཏࠒ ެଐଌƄᇯ ᇍ ࢸ ƌ߄ ༨ဈ ࠦࡷܻࠧࠦ൞൙‫ࡷݘ‬ ‫ ۑ‬ᇯ ᇍ ࢸ Ƨ Ǝܻ‫ݘޢ‬ଐଌƄഭ ୨‫ࡩ ࣐ީޢ‬Ƅ݊ ‫ݹ‬Ə࣠ޯࡩƄࡼ ࡷt݊ ‫ݹ‬Ə࣠‫ކ‬ t݇ ‫ݶ‬ƌࣝ‫ރ‬སཌࠏ ީࡴ ‫ ێ‬ᇬ ᇊ ࢵ t ݇ ‫ݶ‬ƌࣝ‫ރ‬Ə Ɓ߽ࣝ ߰ ‫ࡩ ࢧ ࠒޢ‬Ƅ߄ ၰ Ƅ࣠ࡩƄƧ ƥܻఱ ఠt݊ ‫ݹ‬Ə࣠‫ކ‬ཀྵཏࠒ ެࡷ‫ ݝ‬༨ဈ ࠦ‫ޢ‬ᆿᇍ ࡩƄࠦၰ ܻࣀ ଐଌƄ १ ॏߙଘଓƄƎ࣠ഷവറƄࢪ ࡷൎ൉‫ݹ‬ƣ Ƅƣ ƥƌ໠ ໄܻࠧ༬ဈ ଘଓƄ፥ ጿࠦ‫ܻݹ‬ᐱ᏶ଘଓܻࡷ፟ ፑƄཀྵཏࠒ ެଐଌƄܻߙ࢖࣍ ߄ ޯ೘಼ ơ ࣠ࠁ ࡣ ‫ࡷޯࡺܻމ‬ƥ Ɛ ࢖எै Ƅ Ɛ ࣠ߘܻެࡩƄࠢީఴ ఓ࣠ࠩߙƧ Ǝܻ‫ݘޢ‬ଐଌƄഭ ୨‫ࡩ ࣐ީޢ‬Ƅ݊ ‫ݹ‬Ə࣠ޯࡩƄࠢީఴ ఓt݊ ‫ݹ‬Ə࣠‫ކ‬ཀྵཏࠒ ެࡷ ‫ۑ‬፥ ጒܻࣀ ஊै Ƅ࣊ ࢵ ࣠ࡩƄ ( )Ƥ ᇱᆚ ݇ ‫ݽ‬ ୶ ୧݇ ‫ݶ‬ƌࣝ‫ێ ێރ‬ ݊ ‫ݹ‬Ə࣠ޯࡩƄࠢޯߙܻ૾૓ቿቱƄƌƥ ቿቷࢧ ޯଐଌƄ Ɔ࢖ࡺޯࣀ ଐଌƄࠦ‫ ࠁޯࡺܻࡩޢ‬ଘଓƄƇ ܻ‫ݹݘ‬Ə࣠ޯࡩƄࡼ ࡷƧ ᇴᆝ ݊ ‫ ୹ހ‬୪፥ ፑܻࡺޯࠁ ଘଓƄ݊ ‫ݹ‬Ə࣠ޯࡩƄ ፥ ፑܻޯࡩƄᇯ ᇍ ࣃ ࡩƄ߃ ၯ ܾ‫ކ‬ F( 110 ) = ܿሺ ሻሺ௩ିଶȀଶ݁௫ /2 మ ܸୃกܸ଍ଉŽˍ ໝ ແࡦफउˍ ࠣ‫ݶ‬ Ɖܸࢤ ެߕକଐƁƢࠣ‫ࡦ ࢽޟܸ ࢲޟ‬ƁƢࠣ‫ޟ‬පඓ ൷Ɓ౭౑ࢲ କଐƁƄ ܸࢲ ƁƋኃ ᒔᐺ። ጏ࢓‫ݕ‬ ࡦƁ࣬ ࣊ Ɓ࢓ߢೕಹኃ ቮƁ೾೥࢓଍ଉƁƄ ƁƋƁ߁ ࢟ ſ .I .II .III ። ፎକଐ߁ ެࢲ କଐƁƢ। ्࢟ ࣝࡦƁŻܸࢤ ߾ ଔଐƁ ƛ ܸߢࢽ ࡦƁ߁ ዖዄ Ž૊ સၫƋࣝެ࠾ኃ ቮƁƊܸެࢲ ଘଐƁ Ƥ ᇱᆚ ݇ ‫ݽ‬ ୶ ୧Ə Ɓ߽ࣝ ‫ࡷޯࡺܻމ ࡣۑ‬౰ేၩ݊ ‫ݹ‬Ə࣠‫ކ‬  ‫ۑ‬Ʀ ࣠ޯࣀ ଐଌƄࡼ ࡷ‫ࡩ ࣀܻࡩ ݪ‬Ƅž࢔எै Ƅ፥ ጿƞ ࢖‫ࡷݘ‬౰ేၩ ‫ܻݹࠦۑ‬ᐱ᏶ଘଓܻࡷ፟ ፑƂ࢖ࡺޯఱ ఓƥ࢖ߦࢧ ࡩƄࡼ ࡷƀ߄ ୍෺ .žƄ࣠ޯࡩଘଓƄ‫ࡷ࣠ ࠅ ݪ‬Ƨ ƀ໠ ໄ৤ৈষƄࠦ‫ރܻ ࠁޢ‬ƪ࣠ޯ࡯ࡷ Ƥ ᇱᆚ ݇ ‫ݽ‬ ୶ ୧݇ ‫ݶ‬ƌࣝ‫ރ‬སཌࠏ ީࡴ ‫ ێ‬ᇬ ᇊ ࢵ F݇ ‫ݶ‬ƌࣝ‫ێ ێރ‬ ‫ۑ‬Ƈ ܻ‫ࡩߦ࢖ࢠޢ‬ƄƎܻީޯࠀ Ƅ፥ ጿƢ߄ ႀ ޯఱ ఓƥࠦࡷܻಛಗƄࠦ‫ ࠁޯࡺܻࡩޢ‬ଘଓƄƇ ܻ‫ݹݘ‬Ə࣠ޯࡩƄࡼ ࡷ ྲ Ⴁყߙ F ݊ ‫ݹ‬Ə࣠‫ކ‬ ፥ ጘଘଓƄƣ ࣠‫ހܻࡩߥܻރ‬F ፥ ገƄ࣠ࣃ ‫ࡩݘ‬Ƅ౰ే‫ݓ‬ ޯ൞ᒑᐽࠦ‫ ࠁޯࡺܻࡩޢ‬ଘଓƄࠦߙܻຫ຀ࡩƄ݊ ‫ݹ‬Ə࣠‫ݹݘ ࣐ ࣩކ‬ F˃0،‫ݔ‬ሺ‫ܨ‬ሻൌ ௖ிሺೇభషమሻȀమ ሺ௩ଶା௩ଵிሻೡభశೡమȀమ F݇ ‫ݶ‬ƌࣝ‫ރ‬Ə Ɓ߽ࣝ ߰ ‫ࡩ ࢧ ࠒޢ‬Ƅ߄ ၰ Ƅ࣠ࡩƄƧ ƥܻఱ ఠཀྵཏࠒ ެଐଌƄ‫ ݝ‬༨ဈ ࠦ‫ޢ‬ᆿᇍ ࡩƄࠦၰ ܻࣀ ଐଌƄ .I v2 ƢܻߥଐଌƄࠦ‫ݹ‬༇໪ ࠦၴ Ǝƌƥ v1࣊ ࣀ ხࡩƄࠦ‫ݹ‬༇໪ ࠦၴ Ǝƌܻዙ዗ƥ໠ ໄޯ൞൙‫઻ ેࡷݘ‬ၮF݊ ‫ݹ‬Ə࣠‫ ݹݘޯࡺ ߄ކ‬.II 111 ܸୃกܸ଍ଉŽˍ ໝ ແࡦफउˍ ࠣ‫ݶ‬ Ɖܸࢤ ެߕକଐƁƢࠣ‫ࡦ ࢽޟܸ ࢲޟ‬ƁƢࠣ‫ޟ‬පඓ ൷Ɓ౭౑ࢲ କଐƁƄ ܸࢲ ƁƋኃ ᒔᐺ። ጏ࢓‫ݕ‬ ࡦƁ࣬ ࣊ Ɓ࢓ߢೕಹኃ ቮƁ೾೥࢓଍ଉƁƄ ƁƋƁ߁ ࢟ ſ .I .II .III .IV ƛ ܸߢࢽ ࡦƁ߁ ዖዄ Ž૊ સၫƋࣝެ࠾ኃ ቮƁƊܸެࢲ ଘଐƁ ። ፎକଐ߁ ެࢲ କଐƁƢ। ्࢟ ࣝࡦƁŻܸࢤ ߾ ଔଐƁ Ǝ࣠ഷവറƄે ઻ၮ࢖ߦࢧ ࡩƄࡼ ࡷཀྵཏࠒ ެଐଌƄƀ߄ ୍෺ƥ‫ [ۑ‬F≥0]ࠦީࡩܻࢵ ƣ ࣠ࡓ ‫ކ‬ƣ Ƅࡼ ࡓ ೘಼ଘଓܻ೘ಹƄƌࠦީࠅ ࣠ࡷFযওߘ .III ‫ۑ‬žƄ࣠ޯࡩଘଓƄ໠ ໄ೘಼१ ॏߙଘଓƄ ཀྵཏࠒ ެ೘ೃࠢ‫ޢ‬ხࢸ ࡣ ‫ࡷޯࡺܻމ‬౰ేၩf݊ ‫ݹ‬Ə࣠‫ކ‬ཀྵཏࠒ ެࡷ .IV F݊ ‫ݹ‬Ə࣠‫ކ‬ཀྵཏࠒ ެࡷ 8.4)ᇯ ᇍ ࢸ ࡹ ࡴܸޯࡦƁࡠ ࢤ ߣࡦƁ௧ ோƋܸೕೆ a) Z = ໠ ໄ‫ ހ‬± 1.64 b) Z =໠ ໄ‫ ހ‬± Ƨ Ǝܻ‫ݘޢ‬ଐଌƄഭ ୨‫ࡩ ࣐ީޢ‬Ƅ݊ ‫ݹ‬Ə࣠ޯࡩƄཀྵཏࠒ ެࡷ‫ ݝ‬༨ဈ ࠦၰ ܻࣀ ଐଌƄ߄ ၴ ƥƀ .1 1.96 c) Z = ໠ ໄ‫ ހ‬±2.58 d) Z ໠ ໄ‫ ހ‬0.90 ‫ و‬2.10 e) Z = Ǝܻఱ ఓ፟ ፑƄ 0.90 f) Z = ໠ ໄ೘಼፟ ፑƄ 2.10 ౰ే‫ݓ‬ ޯଐଌƄƄ߃ ࠧƣ Ƅơ ܻࡺޯࠁ ƀܻ࣠ࠧࡷσ = 3‫ و‬μ = 67 ݊ ࡷഭ ୨‫ࡩ ࣐ީޢ‬Ƅ݊ ‫ݹ‬Ə࣠ޯࡩƄ݊ ީჴႾ፥ ገƄ࣠ࣃ ‫ݑ‬౰ే‫ݓ‬ ޯࡷ‫ۑ‬ ࠦࡺ‫ࡩߥޢ‬Ƅ߃ ၯ ܾ‫ݹ‬ ƥ ໠ ໄ‫ހ‬ 112 ƥ ໠ ໄ‫أ( ހ‬ (‫ب‬ ܸୃกܸ଍ଉŽˍ ໝ ແࡦफउˍ ࠣ‫ݶ‬ Ɖܸࢤ ެߕକଐƁƢࠣ‫ࡦ ࢽޟܸ ࢲޟ‬ƁƢࠣ‫ޟ‬පඓ ൷Ɓ౭౑ࢲ କଐƁƄ ܸࢲ ƁƋኃ ᒔᐺ። ጏ࢓‫ݕ‬ ࡦƁ࣬ ࣊ Ɓ࢓ߢೕಹኃ ቮƁ೾೥࢓଍ଉƁƄ ƁƋƁ߁ ࢟ ſ ƛ ܸߢࢽ ࡦƁ߁ ዖዄ Ž૊ સၫƋࣝެ࠾ኃ ቮƁƊܸެࢲ ଘଐƁ ። ፎକଐ߁ ެࢲ କଐƁƢ। ्࢟ ࣝࡦƁŻܸࢤ ߾ ଔଐƁ 60ࡼ ࡷࡣ ߘƀ(‫ج‬ 65ࡼ ࡷ౰౎ࡁ ƀ (‫د‬ Ƅƍƀ‫ۑ‬Ƨ Ǝܻ‫ݘޢ‬ଐଌƄƞ Ƅ࢖༨စ ଘଓƄƥൎ൉࣍ Ǝ 180࣠ࠧƐ ܻެࡩƄࡼ ࡷƆ౰ేީࡁࠦၮ࣠ዙಣಝƣ ƄƏƥଗଓ፥ ጒܻࣀ ஊै Ƅ࣊ ࢵ ࣠ࡩƄ‫ۑ‬3 ࠦၮ࣠ࡺ ഻വറƄࡼ ࡷܻ୆෷Ƅ࣠ࣃ ‫ݑ‬ƤƎܻ‫ ࠀޯޢ‬ƄᎨ᎙ܻࢧ ༧ೞ ƣ ƀơ ܻࡺޯࠁ Ƅ߄ ၴ ƥƀ ،ഭ ୨‫ࡩ ࣐ީޢ‬Ƅ݊ ‫ݹ‬Ə࣠ޯࡩƄ݊ ީჴ჆ƣ ƄƏƥଘଓƄƣ ᇴᆝ ƣ बअƞ ࣠ࢵ ൎ൉࣍ Ǝ 180 ‫ و‬160໠ ໄ‫أ( ހ‬ ࡣ ࣍ Ǝ 200ࡼ ࡷે ઻ၮƄ (‫ب‬ ࡣ ࣍ Ǝ 150ࡼ ࡷࡣ ߘƄ(‫ج‬ 113 ܸୃกܸ଍ଉŽ–ໝ ແࡦफउ –ࠣ‫ݶ‬ Ɖܸࢤ ެߕକଐƁƢࠣ‫ࡦ ࢽޟܸ ࢲޟ‬ƁƢࠣ‫ޟ‬පඓ ൷Ɓ౭౑ࢲ କଐƁƄ ܸࢲ ƁƋኃ ᒔᐺ። ጏ࢓‫ݕ‬ ࡦƁ࣬ ࣊ Ɓ࢓ߢೕಹኃ ቮƁ೾೥࢓଍ଉƁƄ ƁƋƁ߁ ࢟ ſ ƛ ܸߢࢽ ࡦƁ߁ ዖዄ Ž૊ સၫƋࣝެ࠾ኃ ቮƁƊܸެࢲ ଘଐƁ 114 ። ፎକଐ߁ ެࢲ କଐƁƢ। ्࢟ ࣝࡦƁŻܸࢤ ߾ ଔଐƁ ܸୃกܸ଍ଉŽˍ ໝ ແࡦफउˍ ࠣ‫ݶ‬ Ɖܸࢤ ެߕକଐƁƢࠣ‫ࡦ ࢽޟܸ ࢲޟ‬ƁƢࠣ‫ޟ‬පඓ ൷Ɓ౭౑ࢲ କଐƁƄ ܸࢲ ƁƋኃ ᒔᐺ። ጏ࢓‫ݕ‬ ࡦƁ࣬ ࣊ Ɓ࢓ߢೕಹኃ ቮƁ೾೥࢓଍ଉƁƄ ƁƋƁ߁ ࢟ ſ ƛ ܸߢࢽ ࡦƁ߁ ዖዄ Ž૊ સၫƋࣝެ࠾ኃ ቮƁƊܸެࢲ ଘଐƁ ። ፎକଐ߁ ެࢲ କଐƁƢ। ्࢟ ࣝࡦƁŻܸࢤ ߾ ଔଐƁ žܻࢧ ࠁ ଘଓƄƥ१ ॐࢢ ࣠ࡩƄžܻࢧ ࠁ ଘଓƄܻዙ዗໠ ໄࡺࣀ ߘ፟ ፑƄᄎ ႞ߥ୐෺žܻࢧ ࠁ ଘଓƄࣷࣳၮƣ ܾ‫ހ‬ơ ƥଘଓƄࡣ ࢧ ߦࡩƄ፥ ጿܻ‫ ྲރ‬႖ƀ ࢝ ‫ ߄ࡷࠦࡩޯ࡯ ࠑޢ‬ႀ ޯࣀ ଐଌƄ Ɵ ࢖࣐ ࡩƄ ‫ ࢵܻࡩޢ ݪ‬ଘଓƄƧ ࣠ޯ༨࿻ १ ॐࢢ ࣠ࡩƄžܻࢧ ࠁ ଘଓƄ ƣ ܾ‫ރܻހ‬ഃ೨ƍƥ‫ۑ‬፥ ፑଘଓ߄ ޯࢵ ଘଓƄ ܻ‫ޢ‬ዙኼƎƣ ࣠ࡓ ‫ކ‬ƣ Ƅࡼ ࡓ ೘಼‫ ࢵܻࡩޢ ݪ‬ଘଓƄƤ߃ ࠧƥ‫ܻࠧۑ‬౰ేࣀ ߦ‫ޢ ࡣކ‬ᐳྜྋƓ ࢖‫ݓ‬ ‫ހ‬ᑑ ᐽƍƥࠦ‫ޢ‬ዙኼ༟༎ƄƇ ܻ‫ࡩީޢܻރ‬Ƅߎ ࢢ ƥƥ ơ ᇴᇍ ࢸ ଘଓƄơ ൎ൉ၯ ࡼ ࡷܻࠩࡺࠩߙࡣ ᐳྜཾ‫ޢ ݟ‬༨ဂ ܻ୆คܻ୆กƥƄࠦީࢵ ܻެࡷơ ƥƄ߄ ၴ ፥ ጿܻࠩ‫ࢠݘ‬ƥࡼ ࡓ ೘಼ƥܻ୆กܻࣀ ࠁ ƥ ࡼ ࡓ ๻๬เܻಛಗൎ൉ၯ ࡼ ࡷ཯དྷࡩƄƥ Ɵ ࢖࣐ ࡩƄ ‫ ࢵܻࡩޢ ݪ‬ଘଓƄᑻ ᑧ‫ކ‬Ƨ ࣠ޯ༨࿻ ። ፎକଐ߁ ެࢲ କଐƁŻܸࢤ ߾ କଐƁƥ‫ࠦۑ‬୆คܻ‫ࡩީޢ‬ƄƢ࣠ࢵ ༟༎Ƅƥ ‫ޯࡺ ݊ۑ‬഻വറƄƄ߃ ࠧࡼ ࡷࠦ‫ ࣀ ࠒ࣠ހ‬ଐଌƄࠦެ‫ࡩݘޢ‬Ƅ ݊ ߘƄƥࡼ ࡷᑑ ᐽƍƥ፥ ገܻࢧ ࠁ ଘଓƄ݊ ࡺޯ഻വറƄơ ࣠ࠁ Ƈ ƄƎƄ࢖ߥࡩƄƍܻ༧ဈ Ƅࡼ ࡷ ࠦ୆෷ܻࢧ ࠁ ܺᄣܻᐳႰ ‫ݶ‬ ܻࢧ ࠀ ƥࠦެ‫ࡩݘޢ‬Ƅߎ ࢢ ƥࣩ ࡺఱ ఠƥ‫ ࠁޯࡺܻࡩޢࠦۑ‬ƄƔ ƥྲ ႖‫ ݝ‬༨ဈ ܻࠧƍܻ༧ဈ Ƅযঠ‫ݹ‬Ƈ ƄƎƄ࢖ߥࡩƄƤ߃ ࠧƥ [ Parameters] Ƣࣩ ࡺఱ ఠ݊ ࡺޯ഻വറƄߎ ࢧ ‫཯ކ‬དྷࡩƄ࢝ ‫ݶ‬ ܻࢧ உै ƄܻࡺެყჅ [ sample statistics]ࠦެ‫ࡩݘޢ‬Ƅ ࡼ ࡷ݊ ࡺޯ഻വറƄ ࢝ ‫ݶ‬ ܻࢧ ࠀ Ɖ ܻޯެჴࢵ Ƅ࣠ࠧ ્ ઻‫࡯ޢ‬ႁ ޯࡩƄƥƀ ፥ ፑଘଓ߄ ޯࢵ ଘଓƄžܻࢧ ࠁ ଘଓƄࡼ ࡷ࿃ ႓ܻࢵ ଘଓƄƞ ߄ ಛಗƄƥ ࡼ ‫ݑ‬ơ ଘଓ߄ ޯࢵ ൎᑉᐽ [ statistics] ࠦެ‫ࡩݘޢ‬Ƅࠦ୆෷ܻࢧ ࠁ ƄƢƄ߄ ႀ ޯࢵ Ƅ߄ ެ‫ީ ݝࡷެࠢۑߙݘ‬༨೛ ࠦެ‫ݶ ࢝ݑޢ‬ ܻࢧ ࠀ ࠦ‫࡯ޢ‬ዙኸ ࠦߥ‫ީ ࠒ ݟݑ ࡼ ࣍࢖ݹ‬ᑃᐽܾႅ ࡯ެߙ ،݊ ࡺޯ഻വറƄࡼ ‫ݶ ࢇݑ‬ ܻߥஊै Ƅᇯ ᆝ ܻ୐෺ኆ ቱ࢖ߙ࣠୓ขଘଓࠢ‫ރ‬ଘଓ،፥ ገܻࢧ ࠁ ଘଓƄ݊ ࡺޯ഻വറƄ ࠦ‫ޢ‬ዙኤ ፥ ፕ ࢇ ‫ݶ‬ ܻߥஊै ‫ا‬Ƥ߃ ࠧƥ‫࣠މ࣠ߘޢࠦۑ‬ଐଌƄࡼ ࡷࠦၴ Ǝਔ ৹݊ ࡺޯ഻വറƄࡼ ‫ݶ ࢇݑ‬ ܻߥࠁ ፟ ፑƄơ ࣠ࢢ ࣠ࡩƄܻಛಗൎ൉ၯ ࡼ ࡷ݊ ‫ޯ ࣐ޢ‬ఱ ఝ ‫ܻۑ‬୆෷Ƅ࣠ࣃ ‫ޯࡺ ݊ݑ‬഻വറƄࡼ ࡷࠦ‫ ࣀ ࠒ࣠ހ‬ଐଌƄࠦެ‫ࡩݘޢ‬ƄƇ ܻ‫ܻރ‬୆กơ ൎ൉ၯ ࡼ ࡷ [ parameters ]݊ ࡺޯ഻വറƄᎨ፻ܻ‫ݘ‬ଐଌ ໠ ໄࡺࣀ ߘ፟ ፑƄ፥ ገܻࢧ ࠁ ଘଓƄơ ଘଓ߄ ޯࢵ ଘଓƄᄎ ႞ߥ୐෺ƥ [ Estimation ] मअ߄ ߥޯࡩƄ .1 [ Hypothesis Testing ]Ɠ ƥ࢖ߦࡩƄƇ ƄƎܻީޯࠀ Ƅ .2 ે ઻ၮơ ࣠ࢧ ஊै ƄƌƄमअƥ ،ቿቱ࣠ࠩ༬ீ ‫ݓܻࡩ ݪ‬ ࡩƄ፥ ጿƣ ࣠ࡓ ‫཯ކ‬དྷࡩƄƥ ،፥ ገܻࢧ ࠁ ଘଓƄ݊ ࡺޯ഻വറƄᎨ፻ܻ‫ࡷݘ‬मअ߄ ߥ‫ ࠧ࣠ކ‬फं߁ ߢެࡦƁ ࠦެ‫ݑޢ‬Ƈ ܻ‫ܻރ‬୆กࡼ ࡷ໠ ໄ‫ࡷݘ‬ኆ ቷ‫ހ‬፥ ጿƌ࢖ߦࡩƄࡣ ၯ ƌ࣊ ࢵ ࣠ޯࡷमअ߄ ߥ‫ކ‬ൎ൉޲ዙኽ ࠦެ‫ࡩݘޢ‬ƄƇ ܻ‫ܻރ‬୆กࡼ ࡷܻಛಗƇ Ƅमअ߄ ߥ‫ކ‬ ࠦެ‫୐ࠦࡷݘޢެࠦࡷ ࡼݑޢ‬෺߄ ࡷ፥ ጿƢ࣠ॶॱ५ƄƇ ܻႅ ޯެଐଌƠ ൎ൉ᐱᐢࢵ ଘଓƄ࣊ ೘ೆमअ߄ ߥ‫ކ‬ƥƄ ،݊ ࡺޯ഻വറƄࡼ ࡷƌƄ࢖ߙƄࡣ ‫ ࠀޢ‬Ƅ߄ ଐଌ ࡼ ࡷࠦ‫ܻࡷݘ‬எै Ƅƅ ൎ൉࣐ ࡩ࣯ ൞൙‫ࡩݘ‬Ƅࡣ ‫ࡩޯ ࠒ ࢧޢ‬ƄƦ ࣠ޯࣀ ࡷमअ߄ ߥ‫ކ‬ƥƀࡴ ࠩࡁ ൎ൉ᐱᐢࢵ Ƅ࣊ ೘ೆƥࠦ୐෺߄ ଐଌƄƤ߃ ࠧƣ ᇴᇍ ࢵ ࡼ ࡷ ࣐ࠦ ߥ୐กमअ߄ ߥޯࡩƄܻዙ዗मअ߄ ߥޯᑃᐽƣ ܻޯߥ‫ ߄ ࣍࢖ݹ‬ၴ ࣠‫ކ‬ƥ ‫ ۑ‬ᅵै Ƅ‫ܻࡷݘࠦۑۑ‬எै ƄƤ߃ ࠧ፟ ፑƄƣ ࣠ࡺޯჱႾƅ ൎ൉࣐ ࡩƄࡼ ࡷࠦެ‫ݑޢ‬ [ Interval estimate]Ɔ౰౔ߦ‫ހ‬मअ߄ ߥޯࡩƄƥ [ Point estimate ] ݊ ࡺޯࠖ ൞ᒑᐽቿ ቱ഻࣠ࠩവറƄࠦ൞൙‫ݘ‬ଐଌƄमअ߄ ߥޯࡩƢ߄ ႀ ޯఱ ఓ ƌ߄ ၮ Ɔ߄ ၰ Ƅƥࠦࡺ‫ݑ ࡼߘޢ‬ƆƎܻީ‫୍ߢ ࣍ࠣ ࠧ࣠ݑ‬෾फं߁ ߢެࡦƁ  ‫ࡩެߥ ࣐ ࣯ۑ‬Ƅमअ߄ ߥޯࡩƄࣩ ࡺఱ ఓ ƌ߄ ၮ ࣐ࠦ ߥ‫ݑ ࡼރ‬ƆƎܻީ‫ݑ‬मअ߄ ߥޯࡩƄƄ߃ ࠧƥ‫ۑ‬፥ ገܻࢧ ࠁ ଘଓƄ ௪ ௎߄ ༨ဂ ƌ߄ ඳඣ ൺ Range Ʀ ߄ ࡷે ઻ၮࡣ ࢧ ༨စ Ɔ౰౔ߦࡩƄमअ߄ ߥ‫ކ‬፥ ጿ ƃ౭౑ߣ‫ݽ‬ फं߁ ߢެࡦƁ  115 ܸୃกܸ଍ଉŽ–ໝ ແࡦफउ –ࠣ‫ݶ‬ Ɖܸࢤ ެߕକଐƁƢࠣ‫ࡦ ࢽޟܸ ࢲޟ‬ƁƢࠣ‫ޟ‬පඓ ൷Ɓ౭౑ࢲ କଐƁƄ ܸࢲ ƁƋኃ ᒔᐺ። ጏ࢓‫ݕ‬ ࡦƁ࣬ ࣊ Ɓ࢓ߢೕಹኃ ቮƁ೾೥࢓଍ଉƁƄ ƁƋƁ߁ ࢟ ſ ƛ ܸߢࢽ ࡦƁ߁ ዖዄ Ž૊ સၫƋࣝެ࠾ኃ ቮƁƊܸެࢲ ଘଐƁ ። ፎକଐ߁ ެࢲ କଐƁƢ। ्࢟ ࣝࡦƁŻܸࢤ ߾ ଔଐƁ Ɔ౰౔ߦࡩƄमअ߄ ߥ‫ކ‬ƥ‫ޯࡺ ݊ۑ‬഻വറƄࡼ ࡷࠦ‫ ࣀ ࠒ࣠ހ‬ଐଌƄࠦެ‫ࡩݘޢ‬Ƅࡼ ࡷܻࡺᐱᐕ࡯ၮࡣ ࢧ ༨စ ፟ ጕƌଘଓƄ߄ ஊै Ƅƥે ઻ၮଘଓƄ߄ ஊै Ƅ ‫ޯࡺ ݊ۑ‬഻വറƄᎨ፻ܻ‫ࡺܻࡷݘ‬ᐱᐟყჅ݊ ߥ‫ކ‬ƣ Ƅࡼ ࡓ ೘಼Ǝܻ࣍ ƀƥƀযওߥࡩƄࡼ ࡷƆ౰౔ߙߎ ࢧ ‫ࡷ ࡼߘޢࡺࠦݹ‬౰౗ࡁ Ƅે ઻ၮƦ ࣠༨࿻ ߁ ‫ޟ‬஋ॅ Ɓफं߁ ߢެࡦƁ࢚ ‫ ߽ ࢤܸݳ‬: ‫ۑ‬ [ Unbiased ]౫ౄၾ ެࡦƁƟ߁ ၫ ࠦ୆෷ܻࢧ ࠁ ଗଓƄࠦ୐෺ܻ‫ݘ‬ଐଌƄ݊ ‫ݹ‬Ə࣠‫ࡷޯ࣠ ࢵ ࣊ކ‬ƣ ࣠ࡓ ‫ࠦݑެ ߄ࡷܻݹ‬൞൙‫ݘ‬൞ᒑᐽ౮ేႁ ޯࡷ౰ేၩƨƄƎ߄ ߥࡷƣ ࣠ࡓ ‫ࠦކ‬୆෷ܻࢧ ࠁ ଗଓƄƣ ƀ ࠦެ‫ݘޢ‬ᑃᐽ X̅ ፥ ጒܻࣀ ஊै Ƅ࣊ ࢵ ࣠ࡩƄमअ߄ ߥ‫ކ‬ƣ Ƃ‫ۑ‬Ƅ౮ేႁ ޯࡷ ƨƄƎ߄ ߥࡷࣩ ࡺࣀ ყߙଘଓƀƥ ቿቷ‫ߥܻހ‬ଐଌƄ݊ ࡺޯ഻വറƄࠦ൞൙‫ࡷݘ‬Ƨ ƥܻఱ ఓ ౮ేႁ ޯࡷ౰ేၩमअ߄ ߥޯࡩƄƣ ࣠ࡓ ‫ ࠖޯࡺ ݊ݑެ ߄ܻࠧݹ‬൞ᒑᐽ፥ ጒܻࣀ ஊै Ƅ࣊ ࢵ ࣠ࡩƄ μƨܻ‫ݹ‬ƥܻࣀ ࡷ݊ ࡺޯ༬ீ ࡼ ࡷƨܻ୆෷Ƅ࣠ࣃ ‫ ࣀ ࠒ࣠ހࠦݑ‬ଐଌƄ :‫أي أن‬ X̅= μ E (X̅ )–μ=0 E (X̅) = μ ‫ۑ‬μ݊ ࡺޯ഻വറƄ࣊ ࢵ ࣠ޯଐଌ౮ేႁ ޯࡷ౰ేၩ࣐ࠦ ߥ‫ރ‬Ǝ߄ ߥࡷ X ƣ ƀơ ࣠ߥ‫ރ‬ [ Consistency ]ࢄ ߖƁࣝެࡦƁ  ࡼ ࡷࠢޯჱႾܻ‫ݹ ݊ࡷݘ‬Ə࣠‫ࡷޯ࣠ ࢵ ࣊ކ‬ƅ Ƅ౰౔ߘƄơ ܻၰ ቿቱ഻࣠ࠩവറƄ݊ ࡺޯ഻വറƄ ࠦ൞൙‫ݘ‬ଐଌƨܻߥߙƄ࣠ޯࡷƨƄƎ߄ ߥࡷࠦެ‫ࡩݘޢ‬Ƅ ࡼ ‫ݑ‬ơ ܻߥ‫ݹ‬ ࣊ ࢵ ࣠ޯଐଌƄƣ Ƃ‫ࡩݘޢެࠦۑ‬Ƅࡴ ᄽ஛ ƌƄƌƏƄܻ൞ᇅᆝ ࢖ߦࢧ ࡩƄࡼ ࡷࠢޯჱႾܻ‫ݹ ݊ࡷݘ‬Ə࣠‫ކ‬௪ ௎୍ܻงƅ Ƅ౰౔ߘƄƥቿቱ഻࣠ࠩവറƄࠦ൞൙‫ݘ‬ଐଌƄ ࡴ ᄽ஛ ƌƄ‫ܻز‬൞ᇅᆝ ƥ࢖ߦࢧ ࡩƄࡼ ࡷࠢ୐෺୍ܻงƅ Ƅ౰౔ߘƄƌƄƌबअܻ൞ᇅᆝ μ݊ ࡺޯ഻വറƄ࣊ ࢵ ࣠ޯଐଌƨܻߥࣀ ჴࡷƨƄƎ߄ ߥࡷ X̅ ፥ ጒܻࣀ ஊै Ƅ ‫ܻۑ‬ᐳ႞ߦ‫ࠦ ރ‬൞൙‫ݘ‬ଐଌƄࡼ ࡷƅ Ƅ౰౔ߘଘଓƄƌƄƌबअ፥ ፑܻޯࡩܻ‫ހ‬ƥ n ࠦެ‫ࡩݘޢ‬Ƅ [ Efficiency ]ƃŻܸߣࡐ ᚸࡦƁ  ࡣ ߘଛଓƄ௪ ௎ܻީޯࡩƄƇ Ƅƍࠦ୆෷ܻࢧ ࠁ ଗଓܻߙ፥ ጒܻࣀ ஊै Ƅ࣊ ࢵ ࣠ࡩƄܻ୓෺ƥܻࣀ ჴࡷ໠ ໄޯყႻܻࢧ ࠁ ଗଓࠦ୐෺ܻ‫ݘ‬ଐଌƄ݊ ‫ݹ‬Ə࣠‫ކ‬ƣ ᇴᆝ ƄƍƂ मअ߄ ߥ‫ࡺ ࣩކ‬ఱ ఠࠦ୆෷ܻࢧ ࠁ ്ᑉᐽቿቷ‫ߥܻހ‬ଐଌƄࠦࡺ‫ࡩߥޢ‬Ƅƥž࣠ߦࡁ ౰ేၩƎ߄ ߥࡷࣩ ࡺఱ ఠƦ ༇໩ ଛଓƄƥ ž࣠ߦࡁ Ǝ߄ ߥࡷࣩ ࡺఱ ఠ Ƈ ܻ୆෷ܻࢧ ࠁ ଗଓƄ݊ ‫ޢ‬ସବଣ ƥƣ ƀሯሡሁ ūࠦ൞൙‫ݘ‬൞ᒑᐽ௪ ௎౮ేႁ ޯࡷ౰ేၩ௪ ௎मअ߄ ߥ‫ܻކ‬ዙ዗ൎᇩᆝƥ‫ ݪ‬ყ჋౰౔ࡩƄે ઻ၮž࣠ߦࡁ ౰ేၩƥƀž࣠ߦࡁ ܻ‫ ࠁޢܻރ‬ƀࣩ ࡺఱ ఓࡣ ߘଛଓƄ௪ ௎ܻީޯࡩƄƇ Ƅƍࠦ୆෷ܻࢧ ࠁ ଗଓܻߙ፥ ጒܻࣀ ஊै Ƅ࣊ ࢵ ࣠ࡩƄࢪ ߦ‫ރ‬ቻቱܻಛಗࠦ୐෺ܻ‫ݘ‬ଐଌƄ݊ ‫ݹ‬Ə࣠‫཯ކ‬དྷࡩƄ ‫ۑ‬Ɔžܻߦࡁ ౰౗ࡁ ଛଓƄमअ߄ ߥޯࡩƄ [ sufficiency ] ࠣ‫ݶ‬ ܸߣࡐ ᚸࡦƁ  ᑑ ᐽƍƥ༇໩ ƄƎ߄ ߥࡷܻᐱᏳߎ ࢧ ୓෺ଘଓࠦެ‫ࡩݘޢ‬Ƅࡼ ࡷࠦ‫ࢠܻߙޢ‬ƂƇ ܻࡷॅळ‫ࡷݘ‬Ƣ߄ ႀ ޯࢵ ƄƄƍƄƞ ᇴᆝ ƎƄ߄ ߥ൞ᒑᐽơ ࣠ߥ‫ރ‬ ݊ ޯ๻๬ึ཯དྷࡩƄƆƎ߄ ߥଐଌƄ፟ ፑƄ౰ేఴ ఠࠦ‫ܻ ࢢޢ‬உै ƄƤ߃ ࠧƥ‫ࠢۑ‬ଐଌܻ‫ࡷݘ‬मअ߄ ߥ‫ކ‬ƌƄ࢖ଐଌƄ݊ ࡺޯ഻വറƄࡼ ‫ݑ‬Ƈ ܻࡷॅळ‫ࡷݘ‬ƒ ൎ൉ႀ ޯࢵ ଘଓ ‫ۑ‬Ƈ ܻ‫ࡩީޢܻރ‬Ƅ݊ ࡷࡣ ࡷܻ‫ࡩޯݘ‬Ƅ፥ ጿƣ ࣠୆෷ܻࢧ ࠁ ଘଓƄܻᐱᏳ 116 ܸୃกܸ଍ଉŽ–ໝ ແࡦफउ –ࠣ‫ݶ‬ Ɖܸࢤ ެߕକଐƁƢࠣ‫ࡦ ࢽޟܸ ࢲޟ‬ƁƢࠣ‫ޟ‬පඓ ൷Ɓ౭౑ࢲ କଐƁƄ ܸࢲ ƁƋኃ ᒔᐺ። ጏ࢓‫ݕ‬ ࡦƁ࣬ ࣊ Ɓ࢓ߢೕಹኃ ቮƁ೾೥࢓଍ଉƁƄ ƁƋƁ߁ ࢟ ſ ƛ ܸߢࢽ ࡦƁ߁ ዖዄ Ž૊ સၫƋࣝެ࠾ኃ ቮƁƊܸެࢲ ଘଐƁ ። ፎକଐ߁ ެࢲ କଐƁƢ। ्࢟ ࣝࡦƁŻܸࢤ ߾ ଔଐƁ ݇ ࡷެസലമƁᎥ፸ܸ‫ݕ‬ ଍ଉࠣ࣍ ߢީࡦƁफं߁ ߢ‫ ރ‬ ࠢ୐෺୍ܻงƥ [ μ ]፥ ጒܻࣀ ஊै Ƅ ࣊ ࢵ ࣠ࡩƄࡣ ޲ࡷࠢଐଌܻ‫ࡷݘ‬ƞ ࢖‫ރݘ‬ଘଓ ،፥ ገܻࢧ ࠁ ଘଓƄ݊ ࡺޯ഻വറƄ፥ ጿܻީࡩܻၩ ‫ [ۑ‬x̅]ࠦެ‫ݘޢ‬ᑃᐽ፥ ጒܻࣀ ஊै Ƅ࣊ ࢵ ࣠ࡩƄ߄ ၴ ࣠‫ރ‬ƥ‫ޯࡺ ݊ۑ‬഻വറƄࡼ ࡷࠦެ‫ ߃ݑޢ‬ၯ ƀ፟ ፑƄܾႅ ࡯‫ރ‬ᑑ ᐽƍࡣ ၴ ଘଓ‫ [ۑ‬σ2] ௪ ௎ܻީޯࡩƄमअ߄ ߥ‫ކ‬Ƥ౰౎ޯ‫ރݘ‬ƥ [ S2 ]ࠦެ‫ࡩݘޢ‬Ƅ௪ ௎୍ܻง߄ ၴ ࣠‫ރ‬ƥ‫ ࠖޯࡺ ݊ۑ‬൞ᒑᐽ፥ ጒܻࣀ ஊै Ƅ࣊ ࢵ ॅळࡩƄमअ߄ ߥ‫ކ‬Ƥ౰౎ޯ‫ރݘ‬ƥ ‫ޯࡺ ݊ۑ‬഻വറƄ Ꭸ፻ܻ‫ࠦࡷ ࡼࡷݘ‬൞൙‫ݘ‬ଐଌ࣐ࠦ ߥެࡩƄमअ߄ ߥ‫ ߄ރܻކ‬ၴ ƥƄ߄ ߘƣ ࣠ࡓ ‫ރ‬໠ ໄޯࡩܻஊै Ƅܻޯᆿᆝ ፥ ጿ‫[ۑ‬σ2 ]ࠢ୐෺୍ܻงƥ݊ ࡺޯ഻വറƄ [ Interval Estimate ]ƃ౭౑ߣࡦƁफं߁ ߢ‫ ރ‬ ݊ ࡺޯ഻വറƄƒ ܻࢧ ࠀ ࠦࢵ ƄƎƌ࣠ࠧ [ Sampling]ࠦ୐෺ܻ‫ݘ‬ଐଌƄƅ ॅळࢵ ܾ‫ހ‬Ƈ ܻ‫ࡩީޢܻރ‬Ƅ݊ ዙዋ ࡼ ࡷ࿃ ႓ܻࢵ ଘଓƄƓ ࢖‫ݓ‬ ࡩƄ যওߥࡩƄࡼ ࡷƆ౰౔ߙߎ ࢧ ‫ݹ‬Ɔ౰౔ߦࡩƄमअ߄ ߥ‫ކ‬ƥ‫ۑ‬Ɔ౰౔ߦࡩƄमअ߄ ߥ‫ކ‬ƥƄ࣐ࠦ ߥެࡩƄमअ߄ ߥ‫ކ‬ơ ൎ൉ၯ ࡼ ࡷ࣊ ߥߙࡼ ࡓ ೘಼Ƅ߃ ࠧƥ मअ߄ ߥ‫ެࡺܻކ‬ყჅ ࠦ൞൙‫ݘ‬൞ᒑᐽƆ߄ ၰ Ƅƥࠦࡺ‫ࡩެߥ ࣐ࠦ ݹݘ ࣐ ࣯ߘޢ‬Ƅमअ߄ ߥ‫ކ‬ƥ‫ޯࡺ ݊ۑ‬഻വറƄᎨ፻ܻ‫ެࠢࡷݘ‬ያዧ ݊ ߥ‫ކ‬ƣ Ƅࡼ ࡓ ೘಼཯དྷࡩƄƥ ࠦࡺ‫ࡩߥޢ‬Ƅ ،໠ ໄޯࡺ‫ހߥޢ‬Ɔ౰౔ߦࡩƄमअ߄ ߥ‫ކ‬ƌ߄ ༨࿻ ƥ‫ࠦۑ‬൞൙‫ݘ‬ଐଌƄܻࡺᐱᐟყჅ݊ ߥ‫ކ‬ƣ ƄƎ߄ ߥ‫ݹ‬Ƨ ኅ ቱƄƥ໠ ໄޯࡺ‫ ݹݘ ࣐ ࣯ߘޢ‬Ɔ౰౔ߦࡩƄ ‫ ۑ‬፟ ጕƌଛଓƄ߄ ஊै Ƅ ܻ୆คኆ ቱƄࠦࡺ‫ࡩߥޢ‬Ƅƥ ે ઻ၮଘଓƄ߄ ஊै Ƅ ܻ‫ࡩݘ࡯ޢ‬Ƅ confidence Interval]ࠦߥ޲ࡩƄƆ౰౔ߙࣩ ࡺఱ ఠūࠦ୆෷Ƅ࣠ࣃ ‫ݑ‬Ɔ౰౔ߦࡩࠦ൞൙‫ݘ‬ଐଌƄमअ߄ ߥ‫ކ‬፟ ፑƄ౰ేఴ ఓƆ౰౔ߦࡩƄमअ߄ ߥ‫ކ‬ƣ ƀ [ ࠦ‫ރࠢݹݘ ࣐ ࣯ߘޢࡺࠦ ࠁߥޢߥޢ‬ƀ‫ߙ ࢥ࡯ޢࠦ ࠁޢ ݟ‬ଘଓƄቻቱ࣐ࠦ ߥެࡩƄमअ߄ ߥ‫ކ‬ƣ Ƅ߄ ༬စ मअ߄ߥޯࡩƄ፥ ጿ໠ ໄޯߥ‫ࡩ ࣐࢖ݹ‬Ƅࠦ‫ރ‬Ǝܻߥ೘ೃƥ Ɔƌ࢖ߦެଐଌƄ࣐ࠦ ߥެࡩƄƣ Ƅ‫ࡩ ࣐࢖ݹߥࠦۑ ࠁޢ ݟ‬ƄƤ߃ ࠧƅ ࣠‫ݹ ࢥܻࡷ ࡼݑޢ‬Ƅƣ ࣠ࡓ ‫ކ‬ƣ Ƅࡼ ࡓ ೘಼Ɔ౮ేଐଌƄƤ߃ ࠧƥ‫ࠦۑ‬൞൙‫ݘ‬൞ᒑᐽ ‫ܻࠧۑ‬मअ߄ ߥ‫ކ‬ƌƄ࢖ଐଌƄࠦ൞൙‫ݘ‬ଐଌƄ፟ ፑƄƆƎ߄ ߥଐଌƄࠦࡺ‫ࡩߥޢ‬Ƅƅ ࢖ߘƦ ߄ ࡷܻެࡩၒল ࣠‫ކ‬ଘଓࠦ‫ߥޢߥޢ‬ஊै ƄƌƄ߄ ၮଘଓƄ࣊ ࠀ ે ઻ၮ ƣ Ƅࡣ ࢥ ߦ‫ݹ‬ƥࠦ൞൙‫ݘ‬൞ᒑᐽƆቧሼ࣓ࡷƆ߄ ၰ Ƅƥࠦࡺ‫ߘޢ‬ƣ ࣠ࡓ ‫ކ‬ƣ Ƅ߃ ީ༨࿻ ଘଓࠦ‫ޢ‬൞൙‫ࡩݘ‬Ƅƈ ܻ༨ဂ ଘଓƄƥƇ ܻࢵ ƄƎኆ ቱƄ፥ ጿܻࢥ ‫ݹ‬Ƅƥ ‫ࡩ޲ߥࠦۑ‬ƄƆ౰౔ߙࣩ ࡺఱ ఠƆƌ߄ ༨ீ Ɔ౰౔ߙƥƄƎܻ࣍ Ƅࡼ ያዧ ƆƎ߄ ߥଐଌƄࠦࡺ‫ࡩߥޢ‬Ƅƣ ࣠ࡓ ‫ކ‬ƥࠦߥ‫މ‬Ƈ ܻၴ ƎƌƠ ܻެࠧƣ ࣠ࡓ ‫ކ‬ ௧ ோܸަެࡦƁƟूर‫ݕ‬ ࡴ݇ ࡷެസലമ። ጏܸࢽ இॅ Ɓࣇ ࢲ ࣝࡦƁफं߁ ߢ‫ێ ރ‬ ݊ ࡺޯ༬ீ ܻ୐෺ኆ ቱƣ ᇴᆝ ƄƍƂࠢ‫ހܾރ‬، [ Central Limit Theorem ]ࠦ‫ݹ‬ഁ೨࢖ଐଌƄࠦ‫ܻݹ‬ᐱᐟࡩƄࠦ‫୐ ࢝ރ ࣏࢖ݹ‬ง : ࠣ‫ހ ࣌࢓ݶ‬ ࠦ୐෺ܻ‫ݘ‬ଐଌƄ݊ ‫ݹ‬Ə࣠ޯߙ ،ࠦၰ ܻޯଐଌƄࠦ୆෷Ƅ࣠ࣃ ‫ࡩݘ‬ƄƇ ܻެ‫ࡩݘޢ‬Ƅᇯ ᆝࠢެࡷܻެީ༨೛ ƥ σ2ࠢ୐෺୍ܻงƥ μ፥ ጒܻࣀ ஊै Ƅ࣐ࠢ ࢵ ƥ ƤƎ߄ ߘ௪ ௎୍ܻงƥ μ X X 117 ~ ~ ƤƎ߄ ߘ࣊ ࢵ ࣠‫ހ‬ഭ ୨‫ࡩ ࣐ީޢ‬Ƅ݊ ‫ݹ‬Ə࣠ޯࡩƄࡼ ࡷ୍ܻ෺࢖ߘܻ‫ݹݘ‬Ə࣠‫ކ‬ƣ ࣠ࡓ ‫ ࢵޢ‬x̅፥ ጒܻࣀ ஊै Ƅ࣊ ࢵ ॅळࡩ N (μ ، σ2 ) N (μ ، σ2/n ) ፥ ጘଘଓᇴᆝ ܻ‫ݹܻࢠޢ‬Ǝᑑ ᐽƍࠦၩܻ‫ ࡓ ࡼ ࢢޢ‬೘಼ƥ‫ۑ‬σ2/n Ƅ ܸ୊งକଐƁ  ܸୃกܸ଍ଉŽ–ໝ ແࡦफउ –ࠣ‫ݶ‬ Ɖܸࢤ ެߕକଐƁƢࠣ‫ࡦ ࢽޟܸ ࢲޟ‬ƁƢࠣ‫ޟ‬පඓ ൷Ɓ౭౑ࢲ କଐƁƄ ܸࢲ ƁƋኃ ᒔᐺ። ጏ࢓‫ݕ‬ ࡦƁ࣬ ࣊ Ɓ࢓ߢೕಹኃ ቮƁ೾೥࢓଍ଉƁƄ ƁƋƁ߁ ࢟ ſ ። ፎକଐ߁ ެࢲ କଐƁƢ। ्࢟ ࣝࡦƁŻܸࢤ ߾ ଔଐƁ ƛ ܸߢࢽ ࡦƁ߁ ዖዄ Ž૊ સၫƋࣝެ࠾ኃ ቮƁƊܸެࢲ ଘଐƁ ƣ ܾߙ ، ( n ࡴ ᓉᒴၤ Ƅࢪ ߦ‫ࡷ ࡼރ‬Ƈ ܻެ‫ࡩݘޢ‬Ƅࡼ ࡷƌ߄ ၮࠢެࡷ‫ީ ݝ‬༨೛ ƥ N ݊ ࡺޯ༬ீ ܻ୐෺ኆ ቱƣ ƄƓ ౰౔ߦ‫ރ‬ ፥ ፕ Ƈ ܻެ‫ࡩݘޢ‬ƄƤ߃ ಛಗࠦ୆กܻࣀ ஊै ƄƔ ܻࢵ ƥଘଓƄ ‫ݔ‬ҧ ‫ݔ‬ҧ ‫ݔ‬ҧ … … … … . . ، ‫ݔ‬ҧ ௡ ଵ، ଶ، ଷ، μ ௫ҧ= μ ௫ҧ= ፥ ጘଘଓᇴᆝ ࣩ ࣐ ‫ݹݘ‬Ɣ ܻࢵ ƥଘଓƄ࣊ ࢵ ƥ ‫و‬ ‫ݔ‬ҧ ‫ݔ‬ҧ ‫ݔ‬ҧ … … … … . . ، ଵ، ଶ، ଷ، ݊ =μ ∑ ௫ҧ ௡ ƣ ƀ‫ࠁޢ ݟ‬ ࠦ୆กܻࣀ ஊै ƄƔ ܻࢵ ƥଘଓƄ࣊ ࢵ ƥ = μ ୶ҧ Ƈ ܻެ‫ࡩݘޢ‬Ƅƌ߄ ၮ n ݊ ࡺޯࠖ ൞ᒑᐽ፥ ጒܻࣀ ஊै Ƅ࣊ ࢵ ࣠ࡩƄ μ ࠦެ‫ݘޢ‬ᑃᐽ፥ ጒܻࣀ ஊै Ƅ࣊ ࢵ ࣠ࡩƄ x̅ ࣯ ࡺޯჱႾƇ ܻެ‫ࡩݘޢ‬ƄƤ߃ ಛಗ፥ ጒܻࣀ ஊै Ƅ࣊ ࢵ ࣠ࡩƄƣ ܾߙ N ࠢࡺᄽ஛ ݊ ࡺޯ༬ீ ࡼ ࡷ n ܻࠧƌ߄ ၮƇ ܻެ‫ީެܻݑޢ‬༨೛ ƄƍƄƥ ‫ۑ‬ଘଓƥƄࠢ‫୍ܻކ‬ชƄƅ ॅळ࣐ ଐଌƄ࣠ࠧƄ߃ ࠧƥ μƧ ƥܻఱ ఓƥ݊ ࡺޯ഻വറƄƄ߃ ࠧ፟ ፑƄ ࣯ ࣐ ‫ݹݘ‬፥ ጒܻࣀ ஊै Ƅ࣊ ࢵ ࣠ࡩƄࡼ ‫ࠦݑ‬୆กܻࣀ ஊै ƄƔ ܻࢵ ƥ൑ᑉᐽƇ ܻߙƄ࢖༨စ ଘଓƄ݊ ‫ ୹ހ‬୪ƣ Ƅሯሡሁ ଶ (‫ݔ‬ҧ μ )ଶ ، (‫ݔ‬ҧ ଶ ି ߤ ) (‫ݔ‬ҧ ଶି ଵି ଶ ߤ)ଶ … … … … . . ، (‫ݔ‬ҧ ௡ି ߤ) σଶ௫̅ ୀ (‫ݔ‬ҧ ߤ)ଶ /n ି Ơ Ž‫߾ޟݜ‬ ፥ ጒܻࣀ ஊै Ƅ࣊ ࢵ ࣠ࡩƄ௪ ௎୍ܻง = σଶ௫̅ ፥ ጒܻࣀ ஊै Ƅ࣊ ࢵ ࣠ࡩƄ ‫̅ݔ‬ σଶ௫̅ = = = 118 ∑( ௫̅ ି ஜ)మ ௡ = ഥ ∑ೣ ∑ೣ ቁ ೙ ೙ ∑ቀ ି ௡ ∑{௡( ௫ି௫̅ )మ} ௡మ .௡ ∑( ௫ି௫̅ )మ ௡మ = ∑( ௫ି௫̅ )మ ೙ ೙ = = ஢మ ௡ ∑∑( ௫ି௫̅ )మ ೙మ ೙ ، = ݊ ࡺޯࠖ ൞ᒑᐽ፥ ጒܻࣀ ஊै Ƅ࣊ ࢵ ࣠ࡩƄ μ Ƈ ܻެ‫ࡩݘޢ‬Ƅƌ߄ ၮ n ∑{∑( ௫ି௫̅ )మ} (σଶ = ௡మ .௡ ) ∑( ௫ି௫̅ )మ ௡ ܸୃกܸ଍ଉŽ–ໝ ແࡦफउ –ࠣ‫ݶ‬ Ɖܸࢤ ެߕକଐƁƢࠣ‫ࡦ ࢽޟܸ ࢲޟ‬ƁƢࠣ‫ޟ‬පඓ ൷Ɓ౭౑ࢲ କଐƁƄ ܸࢲ ƁƋኃ ᒔᐺ። ጏ࢓‫ݕ‬ ࡦƁ࣬ ࣊ Ɓ࢓ߢೕಹኃ ቮƁ೾೥࢓଍ଉƁƄ ƁƋƁ߁ ࢟ ſ ። ፎକଐ߁ ެࢲ କଐƁƢ। ्࢟ ࣝࡦƁŻܸࢤ ߾ ଔଐƁ ƛ ܸߢࢽ ࡦƁ߁ ዖዄ Ž૊ સၫƋࣝެ࠾ኃ ቮƁƊܸެࢲ ଘଐƁ ஢మ σଶ௫̅ = ௡ X̅ ~ N (μ ، σ2/n ) [ ፥ ጕܻ޲ࡩƄƅ ॅळ࣐ ଐଌƄ࣠ࠧƥ ] : ‫اي ان‬ ࠢ‫୍ܻކ‬ชƄƅ ॅळ࣐ ଐଌƄ࣠ࠧƥ :. X̅ ~ N (μ ، σ2/n ) ௧ ோܸަެࡦƁƟूर‫ݕ‬ ࡴ݇ ࡷެസലമ። ጏܸࢽ இॅ Ɓࣇ ࢲ ࣝࡦƁफं߁ ߕ :. σଶ௫̅ = σ2/n σ௫̅ = ට ߪ௫̅ ୀ ഑ ఙమ ௡ √೙ ….(standard error ..SE) Ƨ Ǝܻ‫ݘޢ‬ଐଌƄܾ࣐ உै Ƅ X ~ N (μ ، σ2 ) Z = ௫ିஜ X̅ ~ N (μ ، σ2 /n) ఙ Zα/2 = = Zα/2 = = ఙ/√௡ ௫ҧ ିஜ ࠦެ‫ࡩݘޢ‬Ƅ࣊ ࢵ ࣠ޯଐଌࠦީࣀ ჱࡩܻ‫ࡷܻހ‬ƀ ~ N(0،1) ఙ/√௡ Zα/2 . ߪ/√݊ = ‫ݔ‬ҧ −μ μ = x̅ - Zα/2 . ߪ/√݊ μ = x̅ + Zα/2 . ߪ/√݊ x̅ - Zα/2 . ࣌/√࢔ ˂ μ 119 ƣ ܾߙ ~ N( 0، 1) ௫ҧ ିஜ บ คᇴᆝ Ƅƍƀ ໠ ໄ࣐ ࢵ ࣠ࡩƄ፥ ጿ໠ ໄߙ࢖࣐ ࡩƄƅ ྲ Ⴎ ‫ހ‬ …. x̅ ˃ μ …. x̅ ˂ μ Ƥൎ൉ၮƄ໠ ໄޯࡩƌܻ‫ݘ‬ଐଌƄࡼ ࡷƥ ˂ x̅ + Zα/2 . ࣌/√࢔ ܸୃกܸ଍ଉŽ–ໝ ແࡦफउ –ࠣ‫ݶ‬ Ɖܸࢤ ެߕକଐƁƢࠣ‫ࡦ ࢽޟܸ ࢲޟ‬ƁƢࠣ‫ޟ‬පඓ ൷Ɓ౭౑ࢲ କଐƁƄ ܸࢲ ƁƋኃ ᒔᐺ። ጏ࢓‫ݕ‬ ࡦƁ࣬ ࣊ Ɓ࢓ߢೕಹኃ ቮƁ೾೥࢓଍ଉƁƄ ƁƋƁ߁ ࢟ ſ ƛ ܸߢࢽ ࡦƁ߁ ዖዄ Ž૊ સၫƋࣝެ࠾ኃ ቮƁƊܸެࢲ ଘଐƁ ። ፎକଐ߁ ެࢲ କଐƁƢ। ्࢟ ࣝࡦƁŻܸࢤ ߾ ଔଐƁ ௪ ௎ܻީޯࡩƄƢॅळ‫ࡷݘ‬፥ ጒܻࣀ ஊै Ƅ࣊ ࢵ ॅळࡩƆ౰౔ߦࡩƄमअ߄ ߥ‫ކ‬ƣ ࣠‫ࠧ࣠ߘܻރ‬Ƅ߃ ࠧ [ Zα/2 ] ਔ ৲߄ ༨ဈ ܻࠩ‫཯ރ ࢥݘ‬དྷࡩƄࠦߥ޲ࡩƄƦ ࣠ޯࣀ ࡷƎƄ߄ ߥ೘ೃܻᐱᐢࡺ‫ߘޢ‬ƌ߄ ඳඣ ൺƥ‫ݘޢެࠦۑ‬ᑃᐽ፥ ጒܻࣀ ஊै Ƅ࣊ ࢵ ॅळࡩࠦ‫ݹ‬Ǝܻ‫ݘޢ‬ଐଌƄࠦࡺ‫ࡩߥޢ‬Ƅ፥ ፕ Zα/2 मअ߄ ߥޯࡩƄ፥ ጿܻެޯߥ‫މ‬บ คᇴᆝ Ƅƍܺߙ‫ۑ‬ơ ܻࡺޯࠁ ଘଓƄƎƄ߄ ߥࡷ࣠ࠧࠦߥ޲ࡩƄƦ ࣠ޯࣀ ࡷƥ‫ۑ‬፥ ገܻࢧ ࠁ ଘଓƄमअ߄ ߥޯࡩƄ፥ ጿ ơ ܻࡺޯࠁ ଘଓƄƣ ܺߙ P = 1- α = 0.95 α = 1- 0.95 = 0.05 ࠦ‫ݘެ࣠ݹ‬ଐଌƄƦ ࣠ޯࣀ ࡷ፥ ፕ [ α] ƣ Ƃ‫ࠁޢ ݟ‬ ‫ۑ‬ Ƨ ƥܻࣀ ჴࡷཀྵཏࠒ ެଐଌƄષઔ੮ܻၴ ે ઻ၮࠦၰ ܻࣀ ଐଌƄƣ ܺߙࡣ ‫ࡷޯࡺܻމ‬ཀྵཏࠒ ެଐଌƄƣ ƀ‫ࠁޢ ݟ‬ 0.5 – 0.025 = 0.475 [Z 0.025 = Zα/2 ]ਔ ৲߄ ႁ ޯࡩƥ ࠦၰ ܻࣀ ൞ᒑᐽቿቷ‫ߥܻހ‬ଐଌƄ፥ ፕ Z 0.025ƣ ƀ߄ ༬စ ūZơ ƥ߄ ၴ ūƧ Ǝܻ‫ݘޢ‬ଐଌƄഭ ୨‫ࡩ ࣐ީޢ‬Ƅ݊ ‫ݹ‬Ə࣠ޯࡩƄơ ƥ߄ ၴ ࡼ ࡷƥ ƣ ƀƧ Ƅ ‫ ۑ‬ Z 0.025 = 0.475 = 1.96 x̅ - 1.95 . ࣌/√࢔ ˂ μ ơ ܻࡺޯࠁ ܻ‫ހ‬ᑑ ᐽƍƥ ˂ x̅ + 1.96 . ࣌/√࢔ ‫ۑ‬௪ ௎ܻީޯࡩƄƢॅळ‫ ࠖޯࡺ ݊ࡷݘ‬൞ᒑᐽ፥ ጒܻࣀ ஊै Ƅ࣊ ࢵ ॅळࡩ ơ ܻࡺޯࠁ ܻ‫ࡩ޲ߥࠦހ‬ƄƆ౰౔ߙƌܻ༬࿻ ଘଓƢ߄ ႀ ޯఱ ఓƣ ࣠‫ࡩߥܻރ‬ƄƄ߃ ࠧ 120 ܸୃกܸ଍ଉŽˍ ໝ ແࡦफउˍ ࠣ‫ݶ‬ Ɖܸࢤ ެߕକଐƁƢࠣ‫ࡦ ࢽޟܸ ࢲޟ‬ƁƢࠣ‫ޟ‬පඓ ൷Ɓ౭౑ࢲ କଐƁƄ ܸࢲ ƁƋኃ ᒔᐺ። ጏ࢓‫ݕ‬ ࡦƁ࣬ ࣊ Ɓ࢓ߢೕಹኃ ቮƁ೾೥࢓଍ଉƁƄ ƁƋƁ߁ ࢟ ſ ƛ ܸߢࢽ ࡦƁ߁ ዖዄ Ž૊ સၫƋࣝެ࠾ኃ ቮƁƊܸެࢲ ଘଐƁ = x̅ + 1.96 μ . ࣌/√࢔ ። ፎକଐ߁ ެࢲ କଐƁƢ। ्࢟ ࣝࡦƁŻܸࢤ ߾ ଔଐƁ ࡣ ࢥ ߙƄࠦ‫ݹܻࢠޢ‬Ǝࠦ‫ݓ‬ ‫ހ ࢧޢ‬Ƥൎ൉ၮƄƣ ࣠‫ࡩߥܻރ‬Ƅࠦ‫ૻܻހ‬૓ࡼ ࡓ ೘಼ƥ ƣ ƀ‫ࠁޢ ݟ‬ x̅ + 1.96 . σ/√n =ࠦߥ޲ࡩƄƆ౰౔ߦࡩે ઻ၮଘଓƄ߄ ஊै Ƅ x̅ - 1.96 . σ/√n ࠦߥ޲ࡩƄƆ౰౔ߦࡩ፟ ጕƌଘଓƄ߄ ஊै Ƅ ፥ ጒܻࣀ ஊै Ƅ࣊ ࢵ ࣠ࡩƄ x̅ บ คᇴᆝƥ‫ۑ‬ƣ ଘଓ࣠ࠩ༬ீ σ2ࠢ୐෺୍ܻงƥ μ፥ ጒܻࣀ ஊै Ƅ࣐ࠢ ࢵ ƥഭ ୨‫ޯࡺ ݊ ࣍ީޢ‬഻വറƄƣ ᇴᆝ ƄƍƄƥ ƣ ࣠‫ކݘ ࣐ ࣩހܻࡩߥܻރ‬፥ ጒܻࣀ ஊै Ƅ࣊ ࢵ ॅळࡩࠦߥ޲ࡩƄƆ౰౔ߙƥ‫ ۑ‬n˃30 ࠦެ‫ࡩݘޢ‬Ƅࡴ ᄽ஛ nƥܻᐱᐟ᠃୍ܻง S2 ƥࠦެ‫ݘޢ‬ᑃᐽ x̅ - Zα/2 . ࢙/√࢔ ˂ μ = x̅ + 1.96 μ . ˂ x̅ + Zα/2 . ࢙/√࢔ ࢙/√࢔ ፥ ጘଘଓƄ : ‫أي أن‬ ܻ‫ݹ ݊ ࣍ީޢݘޢ‬Ə࣠ޯࡩƄƣ ࣠ࡓ ‫ [ ݑެ ߄ࡷܻݹ‬t ]݇ ‫ݶ‬ƌࣝ‫ރ‬ƟƁ߁ ၽ ެࢲ ܸ‫ݽ‬ ። ጏܸࢽ இॅ Ɓࣇ ࢲ ूरࡦࠣߢޯࡦƁƄ Ɓ౭౑ߖ ‫ێ‬ ơ ࣠ࠁ ࡣ ‫ݹ ݊ࡷޯࡺܻމ‬Ə࣠ޯࡩƄƄ߃ ࠧ‫ ۑ‬t ݊ ‫ݹ‬Ə࣠‫ކ‬Ƣ߄ ႀ ޯఱ ఝܻ୐คܺߙ n˂ 30 ‫ و‬ơ ࣠ࠩ༬ீ σ2௪ ௎ܻީޯࡩƄࡼ ࡓ ᚻࡩƥ ࡼ ࡷ౰ేީࡁ ž༅໮ Ƅ߃ ಛಗƥ‫ۑ‬Ƨ Ǝܻ‫ݘޢ‬ଐଌƄഭ ୨‫ࡩ ࣐ީޢ‬Ƅ݊ ‫ݹ‬Ə࣠ޯࡩƄ݊ ࡷࠦ‫ރ‬Ǝܻߥଐଌܻ‫ܻހ‬ႁ ࣐ ࡯ߦ‫ކ‬౰౗ࡁ Ƅࠢެࡓ ᚻࡩƥ‫ޯ࣠ ࢵ ࣊ۑ‬ଐଌƄ tܻ‫ݹݘ‬Ə࣠‫ކ‬Ơ ܻެࠧƣ ܻߙ ،߄ ၰ ƄƥƧ Ǝܻ‫ࡷݘޢ‬ഭ୨‫ݹ ݊ ࣍ީޢ‬Ə࣠‫ ߄ކ‬ၴ ࣠‫ެࡺܻݹ‬ყჅ‫ۑ‬ƞ Ƅ࢖࣍ ଘଓƄ߄ ެ‫ࡷ ࣀܻ ࠁޯࠢކߥ ݊ݑ‬ t݊ ‫ݹ‬Ə࣠‫ކ‬ƣ ܺߙࠦެ‫ࡩݘޢ‬Ƅࡴ ᄽ஛ ƌƄƏܻ൞ᇅᆝ ࡼ ࡓ ᚻࡩ‫ޯ࡯ߦࠦۑۑ‬༧ீ ࠦ‫ݹ‬༇໪ Ƈ ܻၴ Ǝਔ ৹ƥࠦެ‫ࡩݘޢ‬Ƅࡼ ࡷࡴ ᄽ஛ ᇯ ᇍ ࡩܻߦ࡯ޯ༧ீ ࡼ ࡷƅ ౰౔ߥ‫ ݹ‬t݊ ‫ݹ‬Ə࣠‫ކ‬ƣ ܺߙਓ ৯߄ ެ‫ݑ‬ƥ، n ≥30ƣ ࣠ࡓ ‫ݑެ ߄ࡷܻކ‬Ƨ Ǝܻ‫ݘޢ‬ଐଌƄഭ ୨‫ࡩ ࣐ީޢ‬Ƅ݊ ‫ݹ‬Ə࣠ޯࡩƄࡼ ࡷƅ ౰౔ߥ‫ݹ‬ ‫ݹܻۑ‬ƥܻࣀ ჴႾཀྵདྷࠁ ‫ۑ‬ഭ ୨‫ࡩ ࣐ީޢ‬Ƅ݊ ‫ݹ‬Ə࣠ޯࡩƄ 121 ܸୃกܸ଍ଉŽ–ໝ ແࡦफउ –ࠣ‫ݶ‬ Ɖܸࢤ ެߕକଐƁƢࠣ‫ࡦ ࢽޟܸ ࢲޟ‬ƁƢࠣ‫ޟ‬පඓ ൷Ɓ౭౑ࢲ କଐƁƄ ܸࢲ ƁƋኃ ᒔᐺ። ጏ࢓‫ݕ‬ ࡦƁ࣬ ࣊ Ɓ࢓ߢೕಹኃ ቮƁ೾೥࢓଍ଉƁƄ ƁƋƁ߁ ࢟ ſ ። ፎକଐ߁ ެࢲ କଐƁƢ। ्࢟ ࣝࡦƁŻܸࢤ ߾ ଔଐƁ ƛ ܸߢࢽ ࡦƁ߁ ዖዄ Ž૊ સၫƋࣝެ࠾ኃ ቮƁƊܸެࢲ ଘଐƁ t݇ ‫ݶ‬ƌࣝ‫ ێ ރ‬ᇬ ᇊ ࢵ ࣩ ࣐ ‫ ކݘ‬df ࠦ‫࢖ݹ‬ஊै ƄƇ ܻၴ Ǝƌƥ‫ޯ࡯ߦࠦۑ‬༧ீ ࠦ‫ݹ‬༇໪ Ƈ ܻၴ Ǝƌ‫ ݝ‬༨ဈ t݊ ‫ݹ‬Ə࣠‫ކ‬ơ ƥ߄ ၴ ፥ ጿ t যওߘࣩ ࣐ ‫ކݘ‬ƥ ࠦߘൎ൉‫ހܻࡩݘ‬ df = n-1 ࠦߥ޲ࡩƄƆ౰౔ߙƥ ، μƆ߄ ၰ Ƅƥࠦ൞൙‫ࡷݘ‬मअ߄ ߥ‫ކ‬ਔ ৲मएܻ૸૓ƄƍƄƥ‫ ߄ۑ‬ၰ Ƅƥܻࢧ ߘܻ‫ࡩݘޢެࠦރ‬Ƅࡴ ᄽ஛ ፥ ፕ ࠦ‫࢖ݹ‬ஊै ƄƇ ܻၴ Ǝƌ ࠦߘൎ൉‫ ކݘ ࣐ ࣩހܻࡩݘ‬t݊ ‫ݹ‬Ə࣠‫ކ‬ƢƄ߄ ႀ ޯࢵ ܻ‫ހ‬ơ ഻࣠ࠩവറƄ݊ ࡺޯࠖ ൞ᒑᐽ፥ ጒܻࣀ ஊै Ƅ࣊ ࢵ ॅळࡩ x̅ - μ t α/2 . ࢙/√࢔ ˂ μ = x̅ + t α/2 . ࢙/√࢔ ˂ x̅ + t α/2 . ࢙/√࢔ : ‫أي أن‬ Ƅ ܸ࣍ ࢲ ࣝެ଍ଉƁૄ સၫቼቴޯࡴŽ  ‫ ࡁ ݅ۑ‬7Ƨ ƥܻఱ ఓࠦၮƎ୷ ୪፥ ጿơ ܻߥ‫ކ‬౰౎ࡩƄƎܻ༬ೞ ƄƉ ܻ୓คଗଓƧ Ǝܻ‫ݘޢ‬ଐଌƄƞ Ƅ࢖༨စ ଘଓƄƣ ᇴᆝ ƄƍƂ 9.1 ơ ܻ޲ࡷ ፥ ፕ ܻዙኽ‫ ࡁ ݅ۑ‬20Ɔ࢖ࠖ ࣃ ࡩƄƉ ܻ୓คƀ࣊ ࢵ ࣠ޯࡷƣ ƀ߄ ၴ ƥƥƆ࢖༬ೞ 36ࡼ ࡷƆ౰ేީࡁࠦ୆෷Ƅ࣠ࣃ ‫ݑޢެࠦݑ‬Ƈ ౰ేޯࠀ Ƅ، ‫ۑ‬%90ࠦߥ޲ࡩƄƦ ࣠ޯࣀ ࡷ߄ ެ‫ޯࡺ ݊ݑ‬഻വറƄ࣊ ࢵ ࣠ޯଐଌࠦߥ޲ࡩƄƆ౰౔ߙ 122 ܸୃกܸ଍ଉŽ–ໝ ແࡦफउ –ࠣ‫ݶ‬ Ɖܸࢤ ެߕକଐƁƢࠣ‫ࡦ ࢽޟܸ ࢲޟ‬ƁƢࠣ‫ޟ‬පඓ ൷Ɓ౭౑ࢲ କଐƁƄ ܸࢲ ƁƋኃ ᒔᐺ። ጏ࢓‫ݕ‬ ࡦƁ࣬ ࣊ Ɓ࢓ߢೕಹኃ ቮƁ೾೥࢓଍ଉƁƄ ƁƋƁ߁ ࢟ ſ ። ፎକଐ߁ ެࢲ କଐƁƢ। ्࢟ ࣝࡦƁŻܸࢤ ߾ ଔଐƁ ƛ ܸߢࢽ ࡦƁ߁ ዖዄ Ž૊ સၫƋࣝެ࠾ኃ ቮƁƊܸެࢲ ଘଐƁ μ μ = x̅ + z α/2 ፥ ጘଘଓƄƣ ࣠‫ݹݘ ࣐ ࣩހܻࡩߥܻރ‬௪ ௎ܻީޯࡩƄƢॅळ‫ޯࡺ ݊ࡷݘ‬഻വറƄ࣊ ࢵ ࣠ޯଐଌࠦߥ޲ࡩƄƆ౰౔ߙࡣ ஊै Ƅ . ો/√‫ܖ‬ z α/2 = Z0.05 = 1.64 = x̅ + 1.64 . ો/√‫ܖ‬ = 20 + 1.64 . 7/ √36 = 20 + ( 1.64 x 1.167) = 20 + 1.913 = ‫ ۑ‬ Ƨ ƥܻఱ ఠ %90ࠦߥ‫މ‬Ʀ ࣠ޯࣀ ࡷ߄ ެ‫ݑ‬ơ ܻߥ‫ކ‬౰౎ࡩƄƎܻ༬ೞ ƄƉ ܻ୓คଗଓ݊ ࡺޯ഻വറƄ࣊ ࢵ ࣠ޯଐଌࠦߥ޲ࡩƄƆ౰౔ߙƣ ܺߙᑑ ᐽਓ ৹ƥ ݊ ߥ‫ݹ‬ơ ܻߥ‫ކ‬౰౎ࡩƄƆ࢖ࠖ ࣃ ࡩƉ ܻ୓คଘଓƄ࣊ ࢵ ࣠ޯࡷƣ ܾ‫ ހ‬%90ࠦީࣀ ჱჅƣ ࣠ߥ‫މ‬Ƅƥܻ୐คƀƧ ƀ‫ ۑ – ۑ ۑ‬ ݅ ࡁ )‫] ۑ – ۑ [ۑ‬Ɔ౰౔ߦࡩƄࡼ ያዧ ƣ Əƥ࣊ ࢵ ࣠ޯࡷƣ ƀ໠ ໄ୍งƄƍܺߙƣ ࣠‫ࡩ ࢧܻހ‬ƄƟ ࣠ࠒ ࣀ ଐଌƆ࣠ީ‫ݑ‬64ࡼ ࡷࠦ୆෷Ƅ࣠ࣃ ‫ݑޢެࠦݑ‬Ƈ ౰ేޯࠀ Ƅ ‫ ۑ‬ơ ܻ޲ࡷ ‫ ۑ‬Ƨ ƥܻఱ ఓࠢ‫ރ‬ƀ߄ ၴ ƥƥࠦެ‫ࡩݘޢ‬Ƅࡼ ࡷƧ Ǝܻ‫ݘޢ‬ଐଌƄƞ Ƅ࢖༨စ ଘଓƄ‫ࡩݘޢެࠦࠧ࣠ ۑ ࡁ ݅ۑ ࠁ ࣀ ݪ‬Ƅ፥ ጿƆ࣠ީ‫ࡩݘ‬Ƅ % 95ࠦߥ‫މ‬Ʀ ࣠ޯࣀ ࡷ߄ ެ‫ ࠖޯࡺ ݊ݑ‬൞ᒑᐽ१ ॏ‫ߥޢ‬ஊै Ƅ࣊ ࢵ ࣠ޯ൞ᒑᐽࠦߥ޲ࡩƄƌƥ߄ ၰ ፥ ፕ ܻዙኽ‫ࡁ ݅ۑ‬ ࡠ இॅ Ɓ 30 ≤Ɔ౰ేީࡁࠦެ‫ࡩݘޢ‬Ƅƣ Ƃ‫ࠁޢ ݟ‬ ( 0‫ۑ‬05 = αࠦ‫ݘެ࣠ݹ‬ଐଌƄƦ ࣠ޯࣀ ࡷ %95ࠦߥ‫މ‬Ʀ ࣠ޯࣀ ࡷ߄ ެ‫ݑ‬ = x̅ + 1.96 μ . ߪ௫ҧ ‫ = ۑ‬z ‫= ۑ‬ z α/2 = x̅ + 1.96 . ‫ݏ‬/√݊ μ = 2.95 + 1.96 . 0.3/ √64 = 2.95 + ( 1.96 x 0.0375 ) μ = 2.95 + 0.0735 Ɔ౰౔ߦࡩƄ፥ ጿ݊ ߥ‫ކ‬ƣ ࣠‫ࡩ ࢧܻހ‬ƄƆ࣠ީ‫ݑ‬ƣ Əƥ ،࣊ ࢵ ࣠ޯࡷƣ ܾ‫ ހ‬%95 ࠦީࣀ ჱჅƣ ࣠ߥ‫މ‬Ƅƥܻ୐คƀƧ ƀ ݅ ࡁ ‫) ۑ – ۑ‬ ഭ ୨‫ݹݘࠢ ࣍ީޢ‬Ə࣠‫ޯࡺ ݊ކ‬༬ீ ࡼ ࡷƇ ߃ ၯ ƀ ، 80࣊ ࢵ ࣠ޯ೘ೃƆƌ࢖ߦࡷ 25ࡼ ࡷࠦ୆෷Ƅ࣠ࣃ ‫ ێ ( ݑޢެࠦݑ‬ƞ ܸޯࡴ ‫ۑ‬Ƣॅळ‫ݘ‬ଐଌƄ౰ేၩ݊ ࡺޯ഻വറƄ࣊ ࢵ ࣠ࡩࠦ୆งଘଓƄࠦߥ޲ࡩƄƇ Ƅ౰౔ߙ߄ ၴ ƥƀ‫ۑ‬30Ƨ Ǝܻ‫ࡷݘޢ‬ƞ Ƅ࢖༨စ ܻ‫ ހ‬، %90 .I 123 ܸୃกܸ଍ଉŽ–ໝ ແࡦफउ –ࠣ‫ݶ‬ Ɖܸࢤ ެߕକଐƁƢࠣ‫ࡦ ࢽޟܸ ࢲޟ‬ƁƢࠣ‫ޟ‬පඓ ൷Ɓ౭౑ࢲ କଐƁƄ ܸࢲ ƁƋኃ ᒔᐺ። ጏ࢓‫ݕ‬ ࡦƁ࣬ ࣊ Ɓ࢓ߢೕಹኃ ቮƁ೾೥࢓଍ଉƁƄ ƁƋƁ߁ ࢟ ſ ƛ ܸߢࢽ ࡦƁ߁ ዖዄ Ž૊ સၫƋࣝެ࠾ኃ ቮƁƊܸެࢲ ଘଐƁ ። ፎକଐ߁ ެࢲ କଐƁƢ। ्࢟ ࣝࡦƁŻܸࢤ ߾ ଔଐƁ %95 .II %99 .III ၖ࿏ ܻޯެࡩƄྲ ႞ߙ .IV ‫ۑ‬Ƨ Ǝܻ‫ݘޢ‬ଐଌƄഭ ୨‫ࡩ ࣐ީޢ‬Ƅ݊ ‫ݹ‬Ə࣠ޯࡩƄƢ߄ ႀ ޯఱ ఝܻ୐คܾߙ ،ࠦࡷॅळ‫ ࡷݘ‬σƣ ƀ‫ ࡣ ࠁޢ ݟ‬ஊै Ƅ μ = x̅ + z α/2 . = x̅ + 1.64 . %90ࠣߢ‫ކ‬ƣ ࣝެࢽ ࡴ߁ ީ‫ ݎ‬ ߪ௫ҧ = x̅ + 1.64 . ߪ௫ҧ μ μ = 80 + 1.64 . ߪ/√݊ 30/√25 = 80 + 1.64 x 6 = 80 + 9.84 ࠦߥ‫މ‬Ʀ ࣠ޯࣀ ࡷ߄ ެ‫ݑ‬ᑑ ᐽƍƥ ‫ ۑ و ۑ‬໠ ໄ‫ ݹߥ ݊ހ‬μ݊ ࡺޯࠖ ൞ᒑᐽ፥ ጒܻࣀ ஊै Ƅ࣊ ࢵ ࣠ࡩƄƣ ƀƧ ƀ %90 μ = x̅ + 1. 96 . ߪ/√݊ %95ࠦߥ‫މ‬Ʀ ࣠ޯࣀ ࡷ߄ ެ‫ ݑ‬ = x̅ + 1. 96 . 30/√25 = 80 + 1. 96 x 6 = 80 + 1. 96 x 6 = 80 + 11.76 Ʀ ࣠ޯࣀ ࡷ߄ ެ‫ݑ‬ᑑ ᐽƍƥ ‫ ۑ – ۑ‬Ɔ౰౔ߦࡩƄ፥ ጿ ݊ ߥ‫ ݹ‬μ ݊ ࡺޯࠖ ൞ᒑᐽ፥ ጒܻࣀ ஊै Ƅ࣊ ࢵ ࣠ࡩƄƣ ƀƧ ƀ % 95ࠦߥ‫މ‬ = x̅ + ‫ ۑ‬. σ/√n μ = 80 + ‫ ۑ‬. ( = 80 + ) √ଶହ ‫ ۑ‬x 6 = 80 + 15.48 124 ଷ଴ % 99ࠦߥ‫މ‬Ʀ ࣠ޯࣀ ࡷ߄ ެ‫ ݑ‬ ܸୃกܸ଍ଉŽ–ໝ ແࡦफउ –ࠣ‫ݶ‬ Ɖܸࢤ ެߕକଐƁƢࠣ‫ࡦ ࢽޟܸ ࢲޟ‬ƁƢࠣ‫ޟ‬පඓ ൷Ɓ౭౑ࢲ କଐƁƄ ܸࢲ ƁƋኃ ᒔᐺ። ጏ࢓‫ݕ‬ ࡦƁ࣬ ࣊ Ɓ࢓ߢೕಹኃ ቮƁ೾೥࢓଍ଉƁƄ ƁƋƁ߁ ࢟ ſ ። ፎକଐ߁ ެࢲ କଐƁƢ। ्࢟ ࣝࡦƁŻܸࢤ ߾ ଔଐƁ ƛ ܸߢࢽ ࡦƁ߁ ዖዄ Ž૊ સၫƋࣝެ࠾ኃ ቮƁƊܸެࢲ ଘଐƁ ߄ ެ‫ݑ‬ᑑ ᐽƍƥ ‫ ۑ‬- 64.52 Ɔ౰౔ߦࡩƄ፥ ጿ ݊ ߥ‫ ݹ‬μ ݊ ࡺޯࠖ ൞ᒑᐽ፥ ጒܻࣀ ஊै Ƅ࣊ ࢵ ࣠ࡩƄƣ ƀƧ ƀ %99ࠦߥ‫މ‬Ʀ ࣠ޯࣀ ࡷ Ơ ܻެࠧƣ ƀƧ Ƅ ܻࢥ ‫ݹ‬ƀƌƄƌबअࠦߥ޲ࡩƄƆ౰౔ߙࡴ ᄽ஛ ƣ ܾߙࠦߥ޲ࡩƄࠦၴ ƎƌƆƌܻ‫ݹ‬Ə݊ ࡷࠢ‫ރ‬ƀ፟ ፑƄ౰ేఴ ఠ ،Ƥൎ൉ၮƀၖ࿏ ܻޯެࡩƄ  ࠦߥ޲ࡩƄࠦၴ Ǝƌ ࡹ ࡐ ᚸࡦƢ ‫ߘࠦۑ‬ƌࡣ ߘƄमअ߄ ߥޯࡩƄ߰ ީࢧ ‫ݹ‬ƥ ࠦߥ޲ࡩƄƆ౰౔ߙƥࠦߥ޲ࡩƄࠦၴ Ǝƌ໠ ໄ‫ݹࠦހ‬ƌ࢖࣍ ࠦߘൎ൉ၮ ߄ ‫ࡁޯߥ࡯ޢ‬Ɔƌܻ‫ࡩݘ‬ƄƇ ༇໮ ƥ‫ۑ‬ཀྵཏ‫ݘ‬ଐଌƄܻࠩ‫ࠦކߦߥ ߄ࡷݘ‬ၴ Ǝਔ ৹ࠦࢥ ߦࠑ ެࡷƣ ࣠ࡓ ‫ߘ ߄ކ‬Ƅ߄ ‫މߥࠦࢠޢߥࠦ ࠅޢ‬Ɔ౰౔ߦ‫୍ ࣐ࠦހ‬ง࢖ଐଌƄ %99 ‫ و‬%95 ، %90፥ ፕ ܻࡷƄ߄ ႀ ޯࢵ Ƅ౰౗ࡁ ଘଓƄࠦߥ޲ࡩƄƇ Ƅ౰౔ߙƣ ܾ‫ހ‬ ƣ Əƥ࣊ ࢵ ࣠ޯࡷƣ ƀ߄ ၴ ƥƥߎ ‫୐ ࣏ޢ‬งƟ ࣠ࠒ ࣀ ࡷƆ࣠ީ‫ݑ‬25ࡼ ࡷࠦ୆෷Ƅ࣠ࣃ ‫ݑޢެࠦݑ‬Ƈ ౰ేޯࠀ Ƅ ‫ ۑ‬ơ ܻ޲ࡷ ܻዙኽ‫ރࠢ ۑ ۑ‬ƀ߄ ၴ ƥƥࠦެ‫ࡩݘޢ‬Ƅࡼ ࡷƧ Ǝܻ‫ݘޢ‬ଐଌƄƞ Ƅ࢖༨စ ଘଓƄ‫ ࣀ ݪ‬ჴၰ ƀƥ ݅ ࡁ ‫ࡩݘޢެࠦࠧ࣠ ۑ‬Ƅ፥ ጿƆ࣠ީ‫ࡩݘ‬Ƅ ݊ ‫ݹ‬Ə࣠‫ކ‬ƣ ƀƓ Ƅ౰౔ߙƄે ઻ၮ %95ࠦߥ‫މ‬Ʀ ࣠ޯࣀ ࡷ߄ ެ‫ݑ‬Ɔ࣠ީ‫ࡩݘ‬Ƅƣ Ə࣠ࡩ१ ॏ‫ߥޢ‬ஊै Ƅ࣊ ࢵ ࣠ޯ൞ᒑᐽࠦߥ޲ࡩƄƌƥ߄ ၰ ፥ ፕ ‫ࡩޯެ ࣏ޢ ߎ ࣍ީޢݘޢܻۑ‬ƄƟ ࣠ࠒ ࣀ ࡷƇ Ƅ࣠ީ‫ݑ‬ ࡠ இॅ Ɓ ‫ = ۑ‬αࠦ‫ݘެ࣠ݹ‬ଐଌƄƦ ࣠ޯࣀ ࡷƣ ܾߙ ، %95ࠦߥ‫މ‬Ʀ ࣠ޯࣀ ࡷ߄ ެ‫ݑ‬ƥ 30 >Ɔ౰ే‫ݓ‬ ࢢ ࠦެ‫ࡩݘޢ‬Ƅƣ ƀܻ೘ೃ t 0.025 = t α/2 ࠦ‫ݹ‬༇໪ ࠦၴ Ǝƌ߄ ެ‫ݑ‬df = n-1 = 25-1 = 24 ࠦࡺ‫ߘޢ‬t α/2݊ ‫ݹ‬Ə࣠‫ކ‬ơ ƥ߄ ၴ ࡼ ࡷ ‫( = ۑ‬t μ μ = x̅ + t α/2 . s /√n = x̅ + 2.064 . 0.30 /√25 = 2.95 + 0.124 %95ࠦߥ‫މ‬Ʀ ࣠ޯࣀ ࡷ߄ ެ‫ (ݑ‬3.074 – 2.83 ፥ ፕ ࠦߥ޲ࡩƄƆ౰౔ߙƣ Ƃ 3.074 ˃ μ ˃ ‫ ۑ‬ : ‫أي أن‬ ࠦၮܻࢵ 300࢖ዙኸ ࣊ ࢵ ࣠ޯ೘ೃࠦ୆෷ܻ‫ ߰ ࡁࠩ࢖ހ‬୆กܻࢧ ࡷ 9ࡼ ࡷࠦ‫ࠦࡷ ࡓ࣠ރ‬୆෷Ƅ࣠ࣃ ‫ީ ݝݑޢެࠦݑ‬༨೛ ( ‫ ێ‬ƞ ܸޯࡴ ࢖ዙኸ ƣ Ƅƞ ƥ࢖‫ ߰ࡷݘ‬୆กܻࢧ ଐଌƄࡼ ࡷƆ౰ేީࡁࠦ୆෷ܻ‫ެࠦࡁࠩ࢖ހ‬༨ೞ ࡼ ࡷࠦၮܻࢵ 45Ƨ Ǝܻ‫ࡷݘޢ‬ƞ Ƅ࢖༨စ ƄƥƟ Ƅ౰౔ၰ Ƅ ‫ۑ‬Ƣॅळ‫ݘ‬ଐଌƄ౰ేၩࡣ ‫ݓޢ‬ ࣃ ჴࡩƄ࢖ዙኸ ࣊ ࢵ ࣠ࡩ %90ࠦߥ޲ࡩƄƆ౰౔ߙ߄ ၴ ƥƀ‫ۑ‬ഭ ୨‫ࡩ ࣐ީޢ‬Ƅ݊ ‫ݹ‬Ə࣠ޯࡩƄ݊ ީჴႾܻࠩ࡯‫ݓޢ‬ ఴఠ df = n-1 = 9-1=8 ࠦ‫ݹ‬༇໪ ࠦၴ Ǝƌ߄ ެ‫ ࡣݑ‬ஊै Ƅ μ 125 = x̅ + t α/2 (t ݊ ‫ݹ‬Ə࣠‫ކ‬ơ ƥ߄ ၴ ࡼ ࡷ ‫ = ۑ‬t 0.05 = t α/2ࠦࡺ‫ߘޢ‬ . s /√n ܸୃกܸ଍ଉŽ–ໝ ແࡦफउ –ࠣ‫ݶ‬ Ɖܸࢤ ެߕକଐƁƢࠣ‫ࡦ ࢽޟܸ ࢲޟ‬ƁƢࠣ‫ޟ‬පඓ ൷Ɓ౭౑ࢲ କଐƁƄ ܸࢲ ƁƋኃ ᒔᐺ። ጏ࢓‫ݕ‬ ࡦƁ࣬ ࣊ Ɓ࢓ߢೕಹኃ ቮƁ೾೥࢓଍ଉƁƄ ƁƋƁ߁ ࢟ ſ ƛ ܸߢࢽ ࡦƁ߁ ዖዄ Ž૊ સၫƋࣝެ࠾ኃ ቮƁƊܸެࢲ ଘଐƁ μ ። ፎକଐ߁ ެࢲ କଐƁƢ। ्࢟ ࣝࡦƁŻܸࢤ ߾ ଔଐƁ = 300 + 1.860 x 45 /√9 = 300 + 27.9 %90ࠦߥ‫މ‬Ʀ ࣠ޯࣀ ࡷ߄ ެ‫ (ݑ‬328 - 272 ፥ ፕ ࠦߥ޲ࡩƄƆ౰౔ߙƣ ƀ %90ࠦߥ‫މ‬Ʀ ࣠ޯࣀ ࡷ߄ ެ‫ ݑ‬328 ˃ μ ˃ 272 : ‫أي أن‬ ࠦ‫࡯ޢ‬ዙኸ Əܻ༬စ ଗଓࡣ ࡷܻ‫ࡩݘ‬Ƅܻࠧ߃ ၯ ܾ‫཯ݹ‬དྷࡩƄࢇ ‫ݶ‬ ܻߘኆ ቱƄƌ߄ ၮ࣊ ࢵ ࣠ޯࡷमअ߄ ߥ‫ކ‬፥ ጿ݊ ެࢧ ࡷमअ߄ ࡷ‫ ݌ ݪ‬मअ ( ‫ ێ‬ƞ ܸޯࡴ ࠦ‫ܻࢠޢ‬ଐଌƄࠢ‫ކ‬౰౎ၯ ࡼ ࡷमअ߄ ଐଌƄࣷࣳ‫ݹݘ‬ƥ‫ۑ‬%90ࠦߥ‫ࠦމ‬ၴ Ǝਔ ৹ƥࢇ ‫ݶ‬ ܻߘƌ 3 † ƌƥ߄ ၰ ፥ ጿࠦެ‫ࢢެܻݑޢࠦࡷݘޢ‬ ‫ॅހࠦۑ‬ळ࣐ ଐଌƄࠦެ‫ݘޢ‬ᑃᐽ፟ ጕƌଘଓƄ߄ ஊै Ƅܻ࣠ࠧࡷ‫ߘޢߥࠦۑ‬ƌ 15࣠ࠧƧ Ǝܻ‫ݘޢ‬ଐଌƄƞ Ƅ࢖༨စ ଘଓƄƣ ƀ Zα/2 = ‫ܠ‬ҧ ିૄ 1.64 . ࣌ ࡠ இॅ Ɓ ࣌࢞ҧ Zα/2 . ࣌࢞ҧ= x̅- μ = x̅- μ √࢔ ૚૞ 1.64 . √࢔ =3 3. √࢔ = 1.64 x 15 √࢔ = ૚.૟૝ ࢞ ૚૞ ૜ √࢔ = 8.20 = 8.20 n = 67.24 ~ 68 n = 68‫ۑۑۑۑۑۑۑۑۑ‬ μ = x̅ + ܾ࣐ உै Ƅࡴ ᄽ஛ (ࠦ‫ॅހ‬ळ࣐ ଐଌƄࠦެ‫ࡩݘޢ‬Ƅࡴ ᄽ஛ ) ܾ࣐ உै Ƅࡴ ᄽ஛ ࠦެ‫ࡩݘޢ‬Ƅ࣊ ࢵ ࣠ޯࡷ १ ॏ‫ߥޢ‬ஊै Ƅ࣊ ࢵ ࣠ޯଐଌƄࠦ࣌ ࣠ࠒ ࡯ࡷ ƞ Ƅ࢖༨စ ଘଓƄƥ 1.495 +ƌƥ߄ ၰ ፥ ጿƌ࣠ߘ࣠ࡩƄࡼ ࡷƧ ࢖ᐳႡࡩƄƟ ܻߦ‫ރ‬ଘଓƄ࣊ ࢵ ࣠ޯࡷमअ߄ ߥ‫ކ‬ƌƄमअ ( ‫ ێ‬ƞ ܸޯࡴ ‫ۑ‬%95ࠦߥ‫މ‬Ʀ ࣠ޯࣀ ࡷ߄ ެ‫ݑ‬ƢƏൎᑉᐽƄࠦެ‫ࡩݘޢ‬Ƅࡴ ᄽ஛ ܻ࣠ࠧࡷ‫ۑ‬10Ƨ Ǝܻ‫ݘޢ‬ଐଌƄ x̅- μ = Zα/2 . ࣌࢞ҧ x̅- μ = Zα/2 . 1.495 = 1.96 ࡣ ஊै Ƅ ࣌ √࢔ ૚૙ .. √࢔ √࢔ . 1.495 = 19.6 √࢔ = ૚.૝ૢ૞ ૚ૢ.૟ 126 ܸୃกܸ଍ଉŽ–ໝ ແࡦफउ –ࠣ‫ݶ‬ Ɖܸࢤ ެߕକଐƁƢࠣ‫ࡦ ࢽޟܸ ࢲޟ‬ƁƢࠣ‫ޟ‬පඓ ൷Ɓ౭౑ࢲ କଐƁƄ ܸࢲ ƁƋኃ ᒔᐺ። ጏ࢓‫ݕ‬ ࡦƁ࣬ ࣊ Ɓ࢓ߢೕಹኃ ቮƁ೾೥࢓଍ଉƁƄ ƁƋƁ߁ ࢟ ſ ። ፎକଐ߁ ެࢲ କଐƁƢ। ्࢟ ࣝࡦƁŻܸࢤ ߾ ଔଐƁ ƛ ܸߢࢽ ࡦƁ߁ ዖዄ Ž૊ સၫƋࣝެ࠾ኃ ቮƁƊܸެࢲ ଘଐƁ n = [ 19.6/1.495 ]2 n =171.9 = 172 ݅ ࡁ 7Ƨ ƥܻఱ ఓƆ౰ేީࡁࠦၮƎ୷ ୪፥ ጿơ ܻߥ‫ކ‬౰౎ࡩƄƎܻ༬ೞ ƄƉ ܻ୓คଗଓƧ Ǝܻ‫ݘޢ‬ଐଌƄƞ Ƅ࢖༨စ ଘଓƄƣ ᇴᆝ Ƅƍ ‫ ێ ( إ‬ƞ ܸޯࡴ ơ ܻࡺޯࠁ Ƅܻ࣠ࠧዙኽ‫ۑ‬Ɔ࢖ࠖ ࣃ ᑃᐽࠦ‫୓ܻ ࠅޢ‬คଘଓƄ࣊ ࢵ ࣠ޯࡷે ઻ၮƞ ࢖‫ޯݘ‬ᑃᐽƆ࢖༬ೞ 36ࡼ ࡷࠦ୆෷Ƅ࣠ࣃ ‫ݑޢެࠦݑ‬Ƈ ౰ేޯࠀ Ƅ،‫ۑ‬ ࠦ‫୓ܻ ࠅޢ‬ค്ᑉᐽ१ ॏ‫ߥޢ‬ஊै Ƅ࣊ ࢵ ࣠ޯଐଌƄࡼ ࡷ + 2 ƌƥ߄ ၰ ፥ ጿ࣊ ࢵ ࣠ޯଐଌƄƣ ࣠ࡓ ‫ݹ‬ƣ ƀ ࣌࢞ҧ= ࣌ = ૠ Z== √૜૟ √࢔ ‫ܠ‬ҧ ିૄ Z== ૚.૟ૠ (ૄ ି૛)ି ૄ Z== ࡠ இॅ Ɓ = 1.167 ࣌࢞ҧ (ૄ ି૛)ି ૄ ૚.૟ૠ = ି૛ ૚.૟ૠ …….. = -1.71 ത= ૄ − ૛ ) (‫ܠ‬ z=0.4564 ፥ ፕ Ƨ Ǝܻ‫ݘޢ‬ଐଌƄഭ ୨‫ࡩ ࣐ީޢ‬Ƅ݊ ‫ݹ‬Ə࣠ޯࡩƄơ ƥ߄ ၴ ፥ ጿ ( z= -1.71) ࠦࡺ‫ࡩߥޢ‬ቿቷ‫ߥܻހ‬ଐଌƄࠦၰ ܻࣀ ଐଌƄƥ Z== Z== (ૄା ૛)ି ૄ ૚.૟ૠ (ૄା૛)ି ૄ ૚.૟ૠ = ା૛ ૚.૟ૠ …….. = +1.71 ത= ૄ + ૛ ) (‫ܠ‬ ࠦࡺ‫ࡩߥޢ‬ቿቷ‫ߥܻހ‬ଐଌƄࠦၰ ܻࣀ ଐଌƄƥz= +1.71) (፥ ፕ Ƨ Ǝܻ‫ݘޢ‬ଐଌƄഭ ୨‫ࡩ ࣐ީޢ‬Ƅ݊ ‫ݹ‬Ə࣠ޯࡩƄơ ƥ߄ ၴ ፥ ጿ z=0.4564 ‫ ۑ = ۑ‬+ ‫ ۑ‬໠ ໄޯࠁ ܻࣀ ଐଌƄƖ ࣠ዙಣಝ ࣠ࠧƥ १ ॏ‫ߥޢ‬ஊै Ƅ࣊ ࢵ ࣠ޯଐଌƄࡼ ࡷ + 2 ƌƥ߄ ၰ ፥ ጿ࣊ ࢵ ࣠ޯଐଌƄƣ ࣠ࡓ ‫ݹ‬ƣ ƀơ ܻࡺޯࠁ Ƅࡣ ޲೘಼ ‫) ۑ‬ ‫ۑ‬ơ ܻߥ‫ކ‬౰౎ࡩƄƆ࢖༬ೞ ࠦ‫୓ܻ ࠅޢ‬คƂ࣊ ࢵ ࣠ޯࡷ ݊ ࡺޯࠖ ൞ᒑᐽƢॅळ‫ݘ‬ଐଌƄ౰ేၩ࣊ ࢵ ॅळࡩࠦߥ޲ࡩƄƇ Ƅ౰౔ߙƌܻ༬࿻ ଘଓࠢࡷƄ߄ ႀ ޯࢵ Ƅഭ ୣީჱႾ݊ ‫ݹ‬Ə࣠‫ކ‬Ƨ Ƅഃ೨ƍƀ ( ‫ ێ‬ƞ ܸޯࡴ  n = 64 ، s = 8 ໠ ໄޯ‫ࡩޯܻࡩޢ‬Ƅ໠ ໄޯࡩܻஊै Ƅ፥ ጿ݊ ࡺޯ഻വറƄࡼ ࡷƆƍ࣠ࠀ ܾࡷࠦ୆෷Ƅ࣠ࣃ ‫ࡷ ࡼݑޢެࠦݑ‬  n =20 ، s = 10 (ഭ ୨‫ࡩ ࣐ީޢ‬Ƅ݊ ‫ݹ‬Ə࣠ޯࡩƄ݊ ީჴႾଘଓ݊ ࡺޯ഻വറƄ) {‫ ࣐ܻࡩ ݪ‬ᑃᐽࡣ ஊै ƄƠ ౰౔‫ }ݹ‬ ߭ ‫ࡦެ ࢤ ࠏޟ‬Ɓࡠ ࡴܸ‫ݕ‬ ࡴ  x̅ ƣ ܾߙ ߪƧ Ǝܻ‫ࡷݘޢ‬ƞ Ƅ࢖༨စ Ƅƥ μ࣊ ࢵ ࣠ޯࡷቻቱ݊ ࡺޯ༬ீ ࡼ ࡷƆƍ࣠ࠀ ܾࡷ n ࡴ ᓉᒴၤ ƄƇ Ƅƍࠦ୆෷Ƅ࣠ࣃ ‫ݑ‬Ƈ ܻެ‫ࡩݘޢ‬ : ፥ ጘଘଓƀƣ ࣠‫ݹݘ ࣐ ࣩހܻࡩߥܻރ‬Ƨ Ǝܻ‫ݘޢ‬ଐଌƄܾ࣐ உै Ƅƥ ࣆ݊ ࡺޯ഻വറƄ࣊ ࢵ ࣠ޯଐଌࠦ‫ݹ‬ƥܻࣀ ࡷƣ ࣠ࡓ ‫ކ‬ X   127 ܸୃกܸ଍ଉŽ–ໝ ແࡦफउ –ࠣ‫ݶ‬ Ɖܸࢤ ެߕକଐƁƢࠣ‫ࡦ ࢽޟܸ ࢲޟ‬ƁƢࠣ‫ޟ‬පඓ ൷Ɓ౭౑ࢲ କଐƁƄ ܸࢲ ƁƋኃ ᒔᐺ። ጏ࢓‫ݕ‬ ࡦƁ࣬ ࣊ Ɓ࢓ߢೕಹኃ ቮƁ೾೥࢓଍ଉƁƄ ƁƋƁ߁ ࢟ ſ ። ፎକଐ߁ ެࢲ କଐƁƢ। ्࢟ ࣝࡦƁŻܸࢤ ߾ ଔଐƁ ƛ ܸߢࢽ ࡦƁ߁ ዖዄ Ž૊ સၫƋࣝެ࠾ኃ ቮƁƊܸެࢲ ଘଐƁ X   ....(1) n or X   n N n ....(2) N 1 n  0.05 ƣ ࣠ࡓ ‫ݑެ ߄ࡷܻކ‬Nࡴ ᓉᒴၤ ƄƇ ƄƍƆƌƥ߄ ഷവറƄƇ ܻ‫ ࠖޯࡺݘ‬൞ᒑᐽ 2 ቿቱƌܻ‫ࡷݘ‬Ƣ߄ ႀ ޯఱ ఠ‫ࠁޢ ݟ‬ Ƨ Ǝܻ‫ݘޢ‬ଐଌƄܾ࣐ உै Ƅࣩ ࡺఱ ఓ  X ‫ࡦެ ࢤ ࠏޟ ߭ و‬Ɓࡠ ࡴܸ‫ݕ‬ ࡴ N n ... N 1 ƎƄ߄ ߥଐଌƄࣩ ࡺఱ ఓƥ Ɔ߄ ၰ ƥ 20፥ ጒܻࣀ ࠁ ࣊ ࢵ ࣠‫࢖ހ‬ᚻࢧ ެ‫ݑ‬1000ࡼ ࡷƣ ࣠ࡓ ୓෺݊ ࡺޯ഻വറƄƣ ƀƓ ౰౔ߙƄ ( ‫ ێ‬ƞ ܸޯࡴ ܻࠩࡺᄽ஛ ࠦެ‫ࡩݘޢ‬Ƨ Ǝܻ‫ݘޢ‬ଐଌƄƞ Ƅ࢖༨စ ଘଓƄƥ፥ ጒܻࣀ ஊै Ƅ࣊ ࢵ ࣠ࡩƄ߄ ၴ ƥƀ‫ۑ‬Ɔ߄ ၰ ƥ 14 Ƨ Ǝܻ‫ࡷݘޢ‬ƞ Ƅ࢖༨စ Ƅƥ ‫ ۑ‬36 ࡠ இॅ Ɓ  X    20 X   12  2 n 36 X  ƣ ܺߙū (n > 0.05 N‫ޢ ݟ‬༨ဂ  n 36ࡼ ࡷଘଓਔ ৹ 64 Ʀ ƥܻఱ ఠ nบ คᇴᆝ ࣠ࡩ N n 14 1000  64 14 936    (1.75)(0.9679)  1.69 N 1 8 999 64 1000  1 ‫ێ‬ƃƉƢ߁ ഴലമƁƄ ܸ‫ݕ‬ ࡷެࠓ ൛ᒎᐺ߭ ‫ࡦެ ࢤ ࠏޟ‬Ɓࡠ ࡴܸ‫ݕ‬ ࡴ ƣ ƥਔ ৹ σx̅ = ‫ࡷ ࡼ ۑ‬ଘଓਔ ৹ Ʀ ߄ ၰ Ƅࡼ ࣐ ߥ‫ކ‬Ɔྲ ႓ƀ 700 Ƨ ࣠ެࣀ ࡩƄࡣ ၯ ኆ ቱƄमअ߄ ߥ‫ކ‬ਔ ৲मऒ݊ ࡺޯ഻വറƄࠦࡷ߄ ၯ ࠦ‫ݘޢ‬ዙዋ ( ‫ ێ‬ƞ ܸޯࡴ ࣊ ࢵ ࣠‫ ހ‬50ܻࠩࡺᄽ஛ ࣐ࠦ ‫ޢ‬ఱ చࠦ୆෷Ƅ࣠ࣃ ‫ݑޢެࠦݑ‬Ƈ ߃ ၯ ƀ‫ۑ‬Ƈ ܻၮܻ࣐ ߘࠦ‫ހݘ‬ƎƄ፟ ፑƄࠦࡺࣀ ߥଐଌƄࠦ‫ࡩ ࣃݘީޢ‬Ƅžܻ‫ ࠁޢ‬ଘଓƄ Ɔ౰౔ߦࡩƄमअ߄ ߥ‫ܻކ‬ᐱᏳ ƥ߄ ެࡷࡼ ࡷ‫ݘޢࠦ ࣍࡯ީ ݝ‬ସବଣ Ƅ‫ۑ‬$ 950 = s Ƨ Ǝܻ‫ࡷݘޢ‬ƞ Ƅ࢖༨စ Ƅƥ $ 5000፥ ጒܻࣀ ࠁ ࡣ ၯ ƌ࣊ ࢵ ࣠ޯࡷƣ ܾ‫ ހ‬%90ࠦߥ‫ࠦމ‬ၴ Ǝਔ ৹ࠦߥ‫މ‬Ƅƥܻᐱ᏶Ƅ‫ޢ ݟ‬༨ဂ 700ܻࠧƌ߄ ၮ Ɔྲ ႓൑ᑉᐽƧ ࣠ެࣀ ࡩƄࡣ ၯ ኆ ᒗᐽ ‫ۑ‬Ɔ౰౔ߦࡩƄᑻ ᑧ‫ކ‬፥ ጿ݊ ߥ‫ޯࡺ ݊ݹ‬഻വറƄ ࡠ இॅ Ɓ Ơ Žܸೕೀ n ˃ 0.05 x̅ - μ = Zα/2 ߪ௫ҧ= 128 ఙ √௡ .ට . √௡ ே ି௡ ே ିଵ ఙ .ට : ‫إذا‬ ே ି௡ ே ିଵ ܸୃกܸ଍ଉŽ–ໝ ແࡦफउ –ࠣ‫ݶ‬ Ɖܸࢤ ެߕକଐƁƢࠣ‫ࡦ ࢽޟܸ ࢲޟ‬ƁƢࠣ‫ޟ‬පඓ ൷Ɓ౭౑ࢲ କଐƁƄ ܸࢲ ƁƋኃ ᒔᐺ። ጏ࢓‫ݕ‬ ࡦƁ࣬ ࣊ Ɓ࢓ߢೕಹኃ ቮƁ೾೥࢓଍ଉƁƄ ƁƋƁ߁ ࢟ ſ ። ፎକଐ߁ ެࢲ କଐƁƢ। ्࢟ ࣝࡦƁŻܸࢤ ߾ ଔଐƁ ƛ ܸߢࢽ ࡦƁ߁ ዖዄ Ž૊ સၫƋࣝެ࠾ኃ ቮƁƊܸެࢲ ଘଐƁ ߪ௫ҧ= ߪ௫ҧ= ଽହ଴ √ହ଴ ଽହ଴ ଻.଴଻ .ට ଻଴଴ିହ଴ .ට ଻଴଴ିଵ ଺ହ଴ ଺ଽଽ = (134.37)(0.9643) = 129.57 ߪ௫ҧ= 129.57 … … … … … … … … Ƨ Ǝܻ‫ݘޢ‬ଐଌƄܾ࣐ உै Ƅ μ = x̅ + Zα/2 . ߪ௫ҧ μ = x̅ + (1.64)(129.57) μ = 5000 + (1.64)(129.57) μ = 5000 + 212.50 μ = 5212.5 μ = 5000 - 212.50 …………….‫ېېېېېېېېېېې‬ƅ౯౓ߥᑂᐼ૆ ઺ၭଗ଒ƃ߃ உे ƃ ……………………..ƅ౯౓ߥᑂᐼ፞ ጔƋଗ଒ƃ߃ உे ƃ μ= { 4778.5 – 5012.5 } ‫ و ۑ‬$ ‫ ۑ‬໠ ໄ‫ݹߥ ݊ހ‬Ɔྲ ႓ƀ 700 Ƨ ࣠ެࣀ ࡩƄࡣ ၯ ኆ ቱƄ࣊ ࢵ ࣠ޯࡷƣ ܾߙ %90ࠦߥ‫ࠦމ‬ၴ Ǝਔ ৹ ‫ێ‬$ ໝ ແ࣍ ࢲ ࣝެࡴໝ ແ‫ݽ‬ Ɯ ࢓ߣᑀᐺࠣߢޯࡦƁƃ౭౑ߖफं߁ ߢ‫ێ ރ‬ ، ơ ƥଘଓƄ݊ ࡺޯ഻വറƄ࣊ ࢵ ࣠ޯࡷ ،໠ ໄ‫ࡷݘ‬፥ ፑܻࡺޯࠁ Ƅ݊ ‫ݹ‬Ə࣠୓กܻࡺᐱᐟࡷᇯ ᆝ ໠ ໄ࡯ߥޯࣀ ࡷ໠ ໄ‫ޯࡺݘ‬༬ீ ܻ୐෺ኆ ቱƣ ᇴᆝ ƄƍƂ ે ઻ၮഁ೨౰౔‫ࠧޯࡺܻࡷެܻݹ‬Ƅƣ ᇴᆝ ƄƍƄƥ ، σ22 ࠢ୐෺୍ܻงƥ μ2 ፥ ጕܻ޲ࡩƄ݊ ࡺޯ഻വറƄ࣊ ࢵ ࣠ޯࡷƥ ، σ12 ࠢ୐෺୍ܻงƥ μ1 ݊ ࡺޯ഻വറƄࡼ ࡷ n1໠ ໄޯ࡯ߥޯࣀ ࡷ໠ ໄޯყႻƄ࣠ࣃ ‫ݑ‬໠ ໄޯჱ‫ݑޢ‬Ǝܻޯ༧စ ܻ୐คܺߙ ، ( μ1- μ2 ໠ ໄ‫ࡺݘ‬഻വറƄ࣯ ࣐ ࢵ ࣠ޯࡷ໠ ໄ‫ހ‬Ɵ ࢖ߦࡩƄ ፟ ፑƥଘଓƄࠦެ‫ݘޢ‬ᑃᐽ௪ ௎ܻީޯࡩƄƥ፥ ጒܻࣀ ஊै Ƅ࣊ ࢵ ࣠ޯଐଌƄࡼ ࡷൎᇩᆝ ‫ ࣀ ݪ‬༨စ ƥ፥ ጕܻ޲ࡩƄ݊ ࡺޯ഻വറƄࡼ ࡷ n2Ʀ ༇໩ ƀƥơ ƥଘଓƄ 2 ‫ۑ‬s22 ، x̅ ، x̅ 2፥ ጕܻ޲ࡩƄࠦެ‫ݘޢ‬ᑃᐽ௪ ௎ܻީޯࡩƄƥ፥ ጒܻࣀ ஊै Ƅ࣊ ࢵ ࣠ޯଐଌƄࡼ ࡷൎᇩᆝ ‫ ࣀ ݪ‬༨စ Ꭸ᎜ ، s1 1 ࠦ୆งଛଓƄቿቱƌܻ‫ݘ‬ଐଌƄࢇ ߙƥ໠ ໄ‫ކߥࠢࡷݘ‬Ʀ ࣠ޯࣀ ࡷ߄ ެ‫ݑ‬໠ ໄ࣐ ࢵ ࣠ޯଐଌƄ໠ ໄ‫ހ‬Ɵ ࢖ߦᑃᐽࠦߥ޲ࡩƄƆ౰౔ߙ‫ ࣀ ݪ‬༨စ  12  22  n1 n2 μ1 - μ2 = x̅ + 1 – x̅ 2 যওߘࡼ ࡓ ‫ކ‬Ꭸ፻Ƅƍᓘƥ‫ۑ‬ໝ ແ࣍ ࢲ ࣝެࡴໝ ແ‫ݽ‬ Ɯ ࢓ߣᑀᐺƤ Ƌܸ‫ݕޟ‬ ଍ଉƁܻ࣍ ஆॅ Ɓ ( z σ/ 2  12  22 ƋƁ߁ ߢ଍ଉƁࡠ ޯೕಹ‫߾ޟݜ‬  n1 n2 Ƅमअ߄ ߥ‫ކ‬౰౎ޯ‫ ࠁޢ ݟކݘ‬Ƈ ܻެ‫ࡩݘޢ‬Ƅࡼ ࡷࠦ‫ ࣀ࣠ހ‬ഷവറƄ s12 ،s22௪ ௎ܻީޯࡩƄযওߘƢ߄ ႀ ޯఱ ఝ ،໠ ໄ‫ ࠖޯࡺݘ‬൞ᒑᐽࠦࡷॅळ‫ࡷݘ‬௪ ௎ܻީޯࡩƄ ‫ ۑ‬n ≥ 30 Ɔ౰ేީࡁ Ƈ ܻެ‫ࡩݘޢ‬Ƅࡴ ᄽ஛ ƣ ᇴᆝ ƄƍƄƄ߄ ‫ࠅޢ‬ 129 ܸୃกܸ଍ଉŽ–ໝ ແࡦफउ –ࠣ‫ݶ‬ Ɖܸࢤ ެߕକଐƁƢࠣ‫ࡦ ࢽޟܸ ࢲޟ‬ƁƢࠣ‫ޟ‬පඓ ൷Ɓ౭౑ࢲ କଐƁƄ ܸࢲ ƁƋኃ ᒔᐺ። ጏ࢓‫ݕ‬ ࡦƁ࣬ ࣊ Ɓ࢓ߢೕಹኃ ቮƁ೾೥࢓଍ଉƁƄ ƁƋƁ߁ ࢟ ſ ። ፎକଐ߁ ެࢲ କଐƁƢ। ्࢟ ࣝࡦƁŻܸࢤ ߾ ଔଐƁ ƛ ܸߢࢽ ࡦƁ߁ ዖዄ Ž૊ સၫƋࣝެ࠾ኃ ቮƁƊܸެࢲ ଘଐƁ ƣ ƀ߄ ၴ ƥƥ ،Ɔƌ࢖ߦࡷ 144ࡴ ࠖ ༨ဂ ơ ƥଘଓƄ݊ ࡺޯ഻വറƄࡼ ࡷࠦ୆෷Ƅ࣠ࣃ ‫ݑޢެࠦݑ‬Ƈ ౰ేޯࠀ Ƅ ( ‫ ێ‬ƞ ܸޯࡴ 100ࡴ ࠖ ༨ဂ ơ ƥଛଓƄࡼ ‫ݑ‬ቿቷߥޯࣀ ࡷ፥ ጕܻ޲ࡩƄ݊ ࡺޯ഻വറƄࡼ ࡷƇ ౰ేޯࠀ Ƅƥ‫ۑ‬25௪ ௎ܻީޯࡩƄƥ 65Ƨ ƥܻఱ ఓ࣊ ࢵ ࣠ޯଐଌƄ ໠ ໄ‫ ࠖޯࡺݘ‬൞ᒑᐽࠦߙƥ࢖‫ࡷݘ‬౰ేၩ௪ ௎ܻީޯࡩƄযওߘบ คᇴᆝ Ƅƍܺߙ‫ۑ‬16௪ ௎ܻީޯࡩƄƥ 60Ƨ ƥܻఱ ఓ࣊ ࢵ ࣠ޯଐଌƄƣ Ƅ߄ ၴ ƥƥ Ʀ ࣠ޯࣀ ࡷ߄ ެ‫ݑ‬ᑑ ᐽƍƥ໠ ໄ‫ޯࡺݘ‬഻വറƄ࣯ ࣐ ࢵ ࣠ޯࡷ໠ ໄ‫ހ‬Ɵ ࢖ߦᑃᐽࠦߥ޲ࡩƄƆ౰౔ߙ‫ ࠁ ࣀ ݪ‬ƀ‫ۑ‬ቿቷߥޯࣀ ࡷƇ ܻެ‫ࡩݘޢ‬Ƅบ คᇴᆝƥ ‫ۑ‬%95ࠦߥ‫މ‬ ࡠ இॅ Ɓ μ1 - μ2 = ( x̅1 – x̅2 ) +  12  22  n1 n2 μ1 - μ2 = ( 60- 65 ) + ‫ ۑ‬ = 5 + 1.96 (0.5775) ƣ ƀܻ೘ೃ zσ/2 25 16  144 100 = 5 +1.1319 ፥ ፕ %95ࠦߥ‫މ‬Ʀ ࣠ޯࣀ ࡷ߄ ެ‫ݑ‬໠ ໄ‫ޯࡺݘ‬഻വറƄ࣯ ࣐ ࢵ ࣠ޯࡷ໠ ໄ‫ހ‬Ɵ ࢖ߦᑃᐽࠦߥ޲ࡩƄƆ౰౔ߙƣ ƀƧ ƀ 3‫ێ‬8681≤ μ1 - μ2 ≤ 6.1319 ƃ౭ౄ‫ݐ‬ ࢟ Ƅ ܸީ‫ ݎޟ‬/ໝ ແ࣍ ࢲ ࣝެࡴໝ ແ‫ݽ‬ Ɯ ࢓ߣᑀᐺࠣߢޯࡦƁƉƢ߁ ၭ ࠦ‫ࡩޯܻࡩޢ‬ƄƔ ƥྲ ႡࡩƄ࢖ߙ࣠‫ ކ‬30 ˃ Ɔ౰ే‫ݓ‬ ࢢ Ƈ ܻެ‫ࡩݘޢ‬໠ ໄ࣐ ࢵ ࣠ޯࡷ໠ ໄ‫ހ‬Ɵ ࢖ߦᑃᐽࠦߥ޲ࡩƄƆ౰౔ߙमअ߄ ߥ‫୓ ࣐࡯ ݪކ‬෺ ( n2 ،n1 ഭ ୨‫ ࢵࠦ ࣍ީޢ‬ƄƎኆ ቱƄഭ ୨ࡺޯ༬ீ ݊ ‫ݹ‬Ə࣠‫ކ‬ ቿቷߥޯࣀ ࡷƇ ܻެ‫ࡩݘޢ‬Ƅ ໠ ໄ‫ ࠖޯࡺݘ‬൞ᒑᐽ ƥܻࣀ ჴࡷ Ɠ ౰౔ߦ‫ ࡓެࠢ ݹ‬ᚻࡩƥ ໠ ໄ‫ ࠖޯࡺݘ‬൞ᒑᐽ ƞ ƥ࢖‫ ࡷݘ‬౰ేၩ Ƨ Ǝܻ‫ݘޢ‬ଐଌƄ ƞ Ƅ࢖༨စ ଘଓƄ ƥƀ ௪ ௎ܻީޯࡩƄ ໠ ໄ‫ ࠖޯࡺݘ‬൞ᒑᐽमअ߄ ߥޯࡁ ໠ ໄޯჱ‫ࡩݘޢ‬Ƅࡼ ࡷ௪ ௎ܻީޯࡩƄƢ߄ ႀ ޯఱ ఓᑑ ᐽኅ ቱƥ σ22 =σ12) ࠦ୆งଛଓƄቿቱƌܻ‫ݘ‬ଐଌܻ‫ ހ‬1-α ࠦߥ‫މ‬Ʀ ࣠ޯࣀ ࡷ߄ ެ‫ݑ‬໠ ໄ࣐ ࢵ ࣠ޯଐଌƄ໠ ໄ‫ހ‬Ɵ ࢖ߦᑃᐽࠦߥ޲ࡩƄƆ౰౔ߙ‫ ࣀ ݪ‬༨စ s2 n1  n2 s2 n1 130 μ1 - μ2 = ( x̅1 – x̅2 ) + s2  s2 n2 t σ/ 2 ໠ ໄ࣐ ࢵ ࣠ޯࡷ໠ ໄ‫ހ‬Ɵ ࢖ߦᑃᐽƧ Ǝܻ‫ݘޢ‬ଐଌƄܾ࣐ உै Ƅ df = n1+ n2 ࠦ‫ݹ‬༇໪ ࠦၴ Ǝƌ߄ ެ‫ݑ‬tσ/2 ࠦࡺ‫ߘޢ‬ƌ߄ ༨ဈ ƥ ܸୃกܸ଍ଉŽ–ໝ ແࡦफउ –ࠣ‫ݶ‬ Ɖܸࢤ ެߕକଐƁƢࠣ‫ࡦ ࢽޟܸ ࢲޟ‬ƁƢࠣ‫ޟ‬පඓ ൷Ɓ౭౑ࢲ କଐƁƄ ܸࢲ ƁƋኃ ᒔᐺ። ጏ࢓‫ݕ‬ ࡦƁ࣬ ࣊ Ɓ࢓ߢೕಹኃ ቮƁ೾೥࢓଍ଉƁƄ ƁƋƁ߁ ࢟ ſ  .I .II .III ። ፎକଐ߁ ެࢲ କଐƁƢ। ्࢟ ࣝࡦƁŻܸࢤ ߾ ଔଐƁ ƛ ܸߢࢽ ࡦƁ߁ ዖዄ Ž૊ સၫƋࣝެ࠾ኃ ቮƁƊܸެࢲ ଘଐƁ s2 ࠦ‫ࡩޯܻࡩޢ‬Ƅቿቱƌܻ‫ݘ‬ଐଌܻ‫ ࣀ ݪހ‬ჴ༨ဈ ഭ ୨‫ࡩޯ ࠖࡺޢ‬Ƅ௪ ௎ܻީޯࡩƄ s2ࠦࡺ‫ߘޢ‬ s12 (n1 -1) + s22 ( n2 -1) = __________________ n1 + n2 – 2 ໠ ໄ‫ކ‬ƥ౰౎ࡩƄƦ ࣠ޯ༨ீ ࣊ ࢵ ࣠ޯࡷ໠ ໄ‫ރࠦހ‬Ǝܻߥ൞ᒑᐽࠦࢵ ƄƎਔ ৹ࠦ‫ࡩ ࢧ ࠒޢ‬ƄƇ ܻࣀ ࢵ ࣓ଐଌƄƦ ߄ ၰ Ƅ‫ ߘܻࡷ ݝ‬9.13 ơ ܻ޲ࡷ ࠢެࡓ ᚻࡩƥ ໠ ໄ‫ ࠖޯࡺݘ‬൞ᒑᐽƢॅळ‫ࡷݘ‬౰ేၩ௪ ௎ܻީޯࡩƄযওߘบ คᇴᆝ Ƅƍܺߙ‫ۑ‬ƣ ᇴᇍ ࣀ ࡩƄࡼ ࡷ໠ ໄޯ‫ࡺ࣠ݑ‬഻വറ፥ ጹ࣠‫ࡩޢ‬ƄžƄ߃ ‫ݓ‬ ࡩƄ፥ ጿ ࡼ ࡷƣ ࣠ࡓ ୓ขቿቷߥޯࣀ ࡷࠦ୆෷Ƅ࣠ࣃ ‫ݑ‬Ƈ ܻެ‫ݑޢ‬Ƈ ౰ేޯࠀ Ƅ‫ۑ‬ഭ ୨‫ ࣍ީޢ‬໠ ໄ‫ޯࡺݘ‬഻വറƄ݊ ‫ݹ‬Ə࣠‫ކ‬ƥ‫ݹࠦۑ‬ƥܻࣀ ჴࡷܻࡺᐱ᏶ƀƓ ౰౔ߦ‫ݹ‬ ࡴ ᐱᏩƄ߃ ၩ፥ ጿ໠ ໄ‫ކ‬ƥ౰౎ࡩƄࠦ‫ޢ‬ዙኤ ƅ ܻࣀ ჴၰ ƄᎨ᎙ƥ፥ ጕܻ޲ࡩƄ݊ ࡺޯ഻വറƄࡼ ࡷƄƌ࢖ߙ 15ƥơ ƥଘଓƄ݊ ࡺޯ഻വറƄࡼ ࡷƌ࢖ߙ 10 ƢƄ୹ ‫ ݌‬88Ƨ ƥܻఱ ఓơ ƥଘଓƄ݊ ࡺޯ഻വറƄࡼ ࡷࠦެ‫ݘޢ‬ᑃᐽ཯ཏყ჋ƥ౰౎ࡩƄƦ ࣠ޯഷവറƄ࣊ ࢵ ࣠ޯࡷƣ ƄƇ ܻެ‫ࡩݘޢ‬Ƅࡼ ࡷ߄ ၴ ƥƥ ƢƄ୹ ‫ ݌‬77Ƨ ƥܻఱ ఓ፥ ጕܻ޲ࡩƄ݊ ࡺޯ഻വറƄࡼ ࡷࠦެ‫ݘޢ‬ᑃᐽ཯ཏყ჋ƥ౰౎ࡩƄƦ ࣠ޯഷവറƄ࣊ ࢵ ࣠ޯࡷƥ‫ۑ‬ƢƄ୹ ‫ ݌‬11Ƨ ƥܻఱ ఓ௪ ௎ܻީޯࡩƄƥ ‫ۑ‬%95ࠦߥ‫މ‬Ʀ ࣠ޯࣀ ࡷ߄ ެ‫ݑ‬໠ ໄ‫ޯࡺݘ‬഻വറƄ࣯ ࣐ ࢵ ࣠ޯࡷ໠ ໄ‫ࡩ޲ߥࠦހ‬ƄƆ౰౔ߙ‫ ࠁ ࣀ ݪ‬ƀ‫ۑ‬ƢƄ୹ ‫ ݌‬9Ƨ ƥܻఱ ఓ௪ ௎ܻީޯࡩƄƥ ࡠ இॅ Ɓ s2 11(10-1) + 9(15-1) = _________________ 10+15-2 df = n1+ n2 = 10+15 -2 = 23 μ1 - μ2 ቿቱƌܻ‫ݘ‬ଐଌƄƢƄ߄ ႀ ޯࢵ ܻ‫ހ‬ഭ ୨‫ࡩޯ ࠖࡺޢ‬Ƅ௪ ௎ܻީޯࡩƄଘଓƥƀ‫ ࣀ ݪ‬༨စ = 11(9) + 9(14) = 9.783 : ࠦ‫ݹ‬༇໪ = ( x̅1 – x̅2 ) + tα/2 = ( 88 -77) + 1.714 = 11 + 2.1886 13.1886 ≥ 131 23 ࠦၴ Ǝƌ߄ ެ‫ݑ‬t 0.025 ፥ ፕ ƥ tσ/2ࠦࡺ‫ߘޢ‬ƌ߄ ༨ဈ ƥ ‫ = ۑ‬t 0.05 ࠦࡺ‫ࡩߥޢ‬ƄƤ߃ ࠧƧ ƥܻఱ ఠƥ s2 n1  s2 n2 9.783 9.783  10 15 = (8.8114 – 13.1886 ) μ1 - μ2 ≥ 8.8114 ܸୃกܸ଍ଉŽ–ໝ ແࡦफउ –ࠣ‫ݶ‬ Ɖܸࢤ ެߕକଐƁƢࠣ‫ࡦ ࢽޟܸ ࢲޟ‬ƁƢࠣ‫ޟ‬පඓ ൷Ɓ౭౑ࢲ କଐƁƄ ܸࢲ ƁƋኃ ᒔᐺ። ጏ࢓‫ݕ‬ ࡦƁ࣬ ࣊ Ɓ࢓ߢೕಹኃ ቮƁ೾೥࢓଍ଉƁƄ ƁƋƁ߁ ࢟ ſ ። ፎକଐ߁ ެࢲ କଐƁƢ। ्࢟ ࣝࡦƁŻܸࢤ ߾ ଔଐƁ ƛ ܸߢࢽ ࡦƁ߁ ዖዄ Ž૊ સၫƋࣝެ࠾ኃ ቮƁƊܸެࢲ ଘଐƁ ( n ≥ 30 ƃ౭ౄަ࠾ࠣީ‫ެࡷ ݇ࡴ ࡹ ݎޟ‬സലമࠣަࢽ ხᑀᐺࠣߢޯࡦƁƉƢ߁ ၭ ‫ێ‬ ƃ߁ ၭ ƁƢࠣަࢽ ხࡦࠣߢޯࡦƁƉƢ߁ ၭ ፥ ጿ࣊ ࢵ ࣠ޯଐଌƄơ ࣠ࠁ Ƈ ܻၴ ܻޯެჴࢵ Ƅે ઻ၮơ ࣠ࢧ ஊै Ƅࡼ ࡓ ೘಼ཀྵདྷࠁ Ƈ ܻ࣐ ࢵ ࣠ޯ൞ᒑᐽࠦ୐෺ܻ‫ݘ‬ଐଌƄ݊ ‫ݹ‬Ə࣠‫ݘ࢖ߙࠦކ‬ଐଌƉ ܻޯ༨စ  P̅)፥ ገƄ࣠ࣃ ‫ࡩݘ‬Ƅ౰ే‫ݓ‬ ޯ൞ᒑᐽ፥ ፑܻࡺޯࠁ ଘଓƄ݊ ‫ݹ‬Ə࣠ޯࡩƄƧ ƀ ،‫ ࣀ ݪ‬ჱᑃᐽࠦ୐෺ܻ‫ݘ‬ଐଌƄ݊ ‫ݹ‬Ə࣠‫ކ‬Ɔ࢖‫ݘ‬ଐଌƉ ܻޯ༨စ ࡣ ޲ଐଌܻ‫ހ‬ƥ݊ ࡺޯ഻വറƄ ƢƄ߄ ႀ ޯࢵ ܻ‫ހ‬ᑑ ᐽƍƥ P)݊ ࡺޯ഻വറƄ፥ ጿࠦީࣀ ჱࡩƄơ ࣠ࠁ Ƈ ܻၴ ܻޯެჴࢵ Ƅે ઻ၮơ ࣠ࢧ ஊै Ƅࡼ ࡷࡼ ࡓ ๻๬เཀྵདྷࠁ ࠦީࣀ ჱࡩƄે ઻ၮơ ࣠ࢧ ஊै ƄଘଓƥƄƆƎƥྲ Ⴎ ࡩƄࡼ ࡷᑑ ᐽƍࡣ ၴ ଘଓƥ‫ۑ‬Ƨ Ǝܻ‫ݘޢ‬ଐଌƄഭ ୨‫ࡩ ࣐ީޢ‬Ƅཀྵཏࠒ ެଐଌƄ‫ ݝ‬༨ဈ ࠦၰ ܻࣀ ଐଌƄ मअ߄ ߥ‫ކ‬ቿቱܻၰ ፥ ጿ Z Ƨ Ǝܻ‫ݘޢ‬ଐଌƄഭ ୨‫ࡩ ࣐ީޢ‬Ƅ݊ ‫ݹ‬Ə࣠ޯࡩƄƎܻީޯࠀ ƄƢ߄ ႀ ޯఱ ఝƥ‫ ࣀީࠦۑ‬ჱᑃᐽƧ Ǝܻ‫ݘޢ‬ଐଌƄܾ࣐ உै Ƅƥࠦެ‫ࡩݘޢ‬Ƅࡼ ࡷ tƎܻީޯࠀ ƄƢ߄ ႀ ޯఱ ఝଘଓƥࠦީࣀ ჱᑃᐽƆ౰౔ߦࡩƄ ݊ ࡺޯ഻വറƄࠦީࣀ ჱࡩࠦߥ޲ࡩƄƌƥ߄ ၰ ƌܻ༬࿻ ƀƇ Ƅ࣐࣠ ࠀ Zα/2 ࠦࡺ‫ߘޢ‬ƌܻ༬࿻ ଘଓ Zযওߘơ ƥ߄ ၴ Ƣ߄ ႀ ޯఱ ఝ 1-α ࠦߥ‫މ‬Ʀ ࣠ޯࣀ ଐଌ ƣ ࣠‫ ݹݘ ࣐ ࣩހܻࡩߥܻރ‬P)݊ ࡺޯ഻വറƄ፥ ጿࠦީࣀ ჱᑃᐽࠦߥ޲ࡩƄƌƥ߄ ၰ P = P̅+ Z σp P = P̅+ Z ඨ P = P̅+ Z ඨ ࢖(૚ି࢖) ࢔ .ඨ ࢖(૚ି࢖) ࢔ ࡺ ି࢔ ࡺ −૚ …………… n ˃ 0.05 N ። ፒ (P1 – P2)ໝ ແެძమ చໝ ແ‫ݽ‬ Ɯ ࢓ߣᑀᐺࠣߢޯࡦƁƃ౭౑ߖ ‫ێ‬ (P1 – P2) = (P̅ 1 - P̅ 2 ) + Zα/2ඨ ഥ૚(૚ି ࡼ ഥ૚) ࡼ ࢔૚ + ഥ૛ (૚ି ࡼ ഥ૛) ࡼ ࢔૛ ‫ܻࠩކ ߎۑ‬ᑃᐽƄƏܻࠩࠅ ƣ ࣠ࡓ ᚻ࡯೘಼ࡴ ᐱᐟყჅࡼ ࡷ 34ƣ Ƅ߄ ၴ ƥƥƆྲ ႓ƀ 100ࡼ ࡷࠦެ‫ݑޢ‬Ƈ ߃ ၯ ƀ ‫ ۑ‬ơ ܻ޲ࡷ ‫ۑ‬%95ࠦߥ‫މ‬Ʀ ࣠ޯࣀ ࡷ߄ ެ‫ ࠖޯࡺ ݊ݑ‬൞ᒑᐽࠦީࣀ ჱᑃᐽࠦߥ޲ࡩƄƌƥ߄ ၰ ߄ ၴ ƥƀ ࡣ ஊै Ƅ Ƨ Ǝܻ‫ݘޢ‬ଐଌƄഭ ୨‫ࡩ ࣐ީޢ‬Ƅཀྵཏࠒ ެଐଌƄơ ƥ߄ ၴ Ƣ߄ ႀ ޯఱ ఝƥ‫ۑ‬%5࣠ࠧࠦ‫ݘެ࣠ݹ‬ଐଌƄƦ ࣠ޯࣀ ࡷƣ ܾߙ 1-α) ࠦߥ‫މ‬Ʀ ࣠ޯࣀ ଐଌ Zα/2ࠦ‫ݹ‬Ǝܻ‫ݘޢ‬ଐଌƄࠦࡺ‫ߥޢ‬ᑃᐽཀྵཏࠒ ެଐଌƄ‫ ݝ‬༨ဈ ࠦၰ ܻࣀ ଐଌƄƌܻ༬࿻ ଘଓ،‫ۑ‬Z 132 ܸୃกܸ଍ଉŽ–ໝ ແࡦफउ –ࠣ‫ݶ‬ Ɖܸࢤ ެߕକଐƁƢࠣ‫ࡦ ࢽޟܸ ࢲޟ‬ƁƢࠣ‫ޟ‬පඓ ൷Ɓ౭౑ࢲ କଐƁƄ ܸࢲ ƁƋኃ ᒔᐺ። ጏ࢓‫ݕ‬ ࡦƁ࣬ ࣊ Ɓ࢓ߢೕಹኃ ቮƁ೾೥࢓଍ଉƁƄ ƁƋƁ߁ ࢟ ſ  .I .II ። ፎକଐ߁ ެࢲ କଐƁƢ। ्࢟ ࣝࡦƁŻܸࢤ ߾ ଔଐƁ ƛ ܸߢࢽ ࡦƁ߁ ዖዄ Ž૊ સၫƋࣝެ࠾ኃ ቮƁƊܸެࢲ ଘଐƁ Zα/2 = Z0.025 + ‫ ۑ‬ P̅ = 34/100 = 0.34 ( P)݊ ࡺޯ഻വറƄ፥ ጿࠦީࣀ ჱᑃᐽࠦߥ޲ࡩƄƆ౰౔ߙ P̅ ߎ ‫ܻࠩކ‬ᑃᐽ໠ ໄࡓ ᚻࡩܻଐଌƄࠦީఱ ఝ ƌ࣠ࠅ ࣠ࡷ౰ేၩ݊ ࡺޯࠖ ൞ᒑᐽࠦީࣀ ჱࡩƄƣ ଘଓࠦެ‫ࡩݘޢ‬Ƅࡼ ࡷࠦީࣀ ჱࡩƄƢƄ߄ ႀ ޯࢵ ܻ‫ ࣀީࠦހ‬ჱᑃᐽƧ Ǝܻ‫ݘޢ‬ଐଌƄܾ࣐ உै Ƅ‫ ࣀ ݪ‬༨စ P = P̅ + Zα/2 ඨ (ଵି௣ҧ ) ௣ҧ ݊ P = 0.34+ 1.96 ඨ (଴.ଷସ)(଴.଺଺) 100 P = 0.34 + 1.96 ( 0.047) P = 0.34 + (0.09) P= {0.43 – 0.25 } (0.43 – 0.25 )໠ ໄޯࡺ‫ࡩߥޢ‬Ƅ໠ ໄ‫ߥޢߥޢࠦކߥ ݊ހ‬ஊै Ƅࠦީࣀ ჱࡩƄƣ ƀ ، %95ࠦީࣀ ჱჅࠦߥ‫઻ ેމ‬ၮܻ୐คƀƧ ƀ 60ƣ Ƅ߄ ၴ ƥ ،ࡣ ࡷܻၮ 1200ࠢ‫ܻࡷ ࡣࡷ ࡼࡷ ࢧެ ݊ހ‬ၮ 100ܻࠩࡺᄽ஛ ࠦ୆෷Ƅ࣠ࣃ ‫ݑޢެࠦݑ‬፥ ጿ 9.15 ơ ܻ޲ࡷ ƒ ܻၯ Ƈ ܻࢸ ܻ‫ࡷݘ‬Ɩ ƥྲ Ⴁࡷ፥ ጿƠ Ƅ౰౔ࢸ ଘଓƄࡼ ࡷଘଓਔ ৹ƌƄ࢖ߙᇷᆝ Ƈ ܻࢸ ܻ‫ݘ‬ଐଌƄƢܻ࣏ ‫ރ‬Ƨ Ơ Ƅ౰౔ࢸ ଘଓƄƣ ॅळࢥ ߦ‫ݹ‬ ‫ݹࠦۑ‬ƌ࢖ߙƇ ܻࢸ ܻ‫ࡷݘ‬Ƈ ܻၮƥྲ Ⴁࡷƣ ॅळࢥ ߦ‫ݹ‬௪ ௎ኅ ቱƄ໠ ໄ࡯ࡷܻ‫ࡩݘ‬Ƅࠦީࣀ ჱࡩ %95 ࠦߥ޲ࡩƄƆ౰౔ߙ߄ ၴ ƥƀ‫ۑ‬በሼྲ Ⴁࡩܻ‫ހ‬ ࡣ ஊै Ƅ P̅= 60/100 = 0.60 P = P̅ + Zα/2 σp P = P̅ + Zα/2 ඨ N ≥0.05 ƣ ƀ‫ࠁޢ ݟ‬ . ඨ ே ିଵ (ଵି௣ҧ ) ௣ҧ ݊ P = 0.60 + 1.96 ඨ …………… n ˃ 0.05 N ே ି௡ (଴.଺଴)(଴.ସ଴) .ඨ 100 P = 0.60 + 1.96 (0.048) (0.96) = 0،60 + 0.0903 133 ଵଶ଴଴ିଵ଴଴ ଵଶ଴଴ିଵ ܸୃกܸ଍ଉŽ–ໝ ແࡦफउ –ࠣ‫ݶ‬ Ɖܸࢤ ެߕକଐƁƢࠣ‫ࡦ ࢽޟܸ ࢲޟ‬ƁƢࠣ‫ޟ‬පඓ ൷Ɓ౭౑ࢲ କଐƁƄ ܸࢲ ƁƋኃ ᒔᐺ። ጏ࢓‫ݕ‬ ࡦƁ࣬ ࣊ Ɓ࢓ߢೕಹኃ ቮƁ೾೥࢓଍ଉƁƄ ƁƋƁ߁ ࢟ ſ ። ፎକଐ߁ ެࢲ କଐƁƢ। ्࢟ ࣝࡦƁŻܸࢤ ߾ ଔଐƁ ƛ ܸߢࢽ ࡦƁ߁ ዖዄ Ž૊ સၫƋࣝެ࠾ኃ ቮƁƊܸެࢲ ଘଐƁ ‫ ۑ و ۑ‬Ɔ౰౔ߦࡩƄ፥ ጿ݊ ߥ‫ݹࠦ ކ‬ƌ࢖ߙƇ ܻࢸ ܻ‫ࡷݘ‬ƣ ॅळࢥ ߦ‫ݹ‬௪ ௎ኅ ቱƄ໠ ໄ࡯ࡷܻ‫ࡩݘ‬Ƅࠦީఱ ఝ P ƣ ܾߙࠢ‫࡯ޢ‬ၮƥ ‫ ۑ‬%95ࠦߥ‫މ‬Ʀ ࣠ޯࣀ ࡷ߄ ެ‫ݑ‬ ߄ ၴ ƥƥ࣯ ൞൙‫ࡩݘ‬Ƅᄎ ႞ߥࡩƄࠦࡷܻ‫ࡩݘ‬Ƅࠦ‫ࡩ޲ܻރ࣠ݹ‬Ƅࠦީ࡯࣍ ࡼ ࡷܻީࡩܻ࣍ 80ܻࠩࡺᄽ஛ ࠦ୆෷Ƅ࣠ࣃ ‫ݑޢެࠦݑ‬Ƈ ߃ ၯ ƀ (16 ƞ ܸޯࡴ ࠦࡷܻ‫ࡩݘ‬Ƅࠦ‫ࡩ޲ܻރ࣠ݹ‬Ƅࠦީ࡯࣍ ࡼ ࡷܻީࡩܻ࣍ 60ܻࠩࡺᄽ஛ Ʀ ༇໩ Ƅࠦ୆෷Ƅ࣠ࣃ ‫ݑޢެࠦݑ‬Ƈ ߃ ၯ Ƅƥ‫ۑ‬ƣ ࣠ެࠀ ਔ ৲ࡴ ᐱᐟࡷ 22 ‫أن‬ ‫ۑ‬໠ ໄޯხࣀ ჱࡩƄ໠ ໄ‫ހ‬Ɵ ࢖ߦᑃᐽ %95ࠦߥ޲ࡩƄƆ౰౔ߙ߄ ၴ ƥƀ‫ۑ‬ƣ ࣠ެࠀ ਔ ৲ࡴ ᐱᐟࡷ 25ƣ Ƅ߄ ၴ ƥƥ፥ ጒƌଘଓƄᄎ ႞ߥࡩƄ ࡠ இॅ Ɓ ‫ = ۑ‬60/ 25 =P1 ໠ ໄެࠀ ߄ ଐଌƄ፥ ጒƌଘଓƄᄎ ႞ߥࡩƄƅ ൎ൉࣐ ࡩƄࠦީఱ ఝƣ ƀƓ ࢖ߦ‫ރ‬ 0.275 =80/22 =P2 ໠ ໄެࠀ ߄ ଐଌƄ࣯ ൞൙‫ࡩݘ‬Ƅᄎ ႞ߥࡩƄƅ ൎ൉࣐ ࡩƄࠦީఱ ఝƣ ƀƓ ࢖ߦ‫ރ‬ :፥ ፕ %95ࠦߥ‫މ‬Ʀ ࣠ޯࣀ ࡷ߄ ެ‫ݑ‬໠ ໄޯხఱ ఝ໠ ໄ‫ހ‬Ɵ ࢖ߦᑃᐽࠦߥ޲ࡩƄƆ౰౔ߙ (P1 – P2) = (P̅1 - P̅2 ) + Zα/2ඨ ௉തଵ(ଵି ௉തଵ) ௡ଵ (P1 – P2) = (0.416 – 0.275 ) + 1.96ඨ = 0.141 + 1.96 √ 0.0062659 = 0.141 + ௉തଶ (ଵି ௉തଶ) ௡ଶ ଴.ସଵ଺(଴.ହ଼ସ) ଼଴ + ଴.ଶ଻ହ (଴.଻ଶହ) ଺଴ + 1.96 (0.079) = 0.141 + 0.155 ‫ۑ‬%95ࠦߥ‫މ‬Ʀ ࣠ޯࣀ ࡷ߄ ެ‫ ݑ‬0.296 ،-0.014 ໠ ໄ‫ࡩ޲ߥࠦކߥ ݊ހ‬ƄƆ౰౔ߙƄƍƀ ݇ ࢲ ܸެࡦƁࡠ ࢤ ߣࡦƁ௧ ோƋܸೕೆ ࡼ ࡷƣ ࣠ࡓ ࡷ݊ ࡺޯ༬ீ ࡼ ࡷ 30Ƨ Ǝܻ‫ࡷݘޢ‬ƞ Ƅ࢖༨စ Ƅƥ 80࣊ ࢵ ࣠ޯ೘ೃƆƌ࢖ߦࡷ 25ࡼ ࡷࠦ୆෷Ƅ࣠ࣃ ‫ݑޢެࠦݑ‬Ƈ ߃ ၯ ƀ .1 ƀ ‫ۑ‬Ƣॅळ‫ݘ‬ଐଌƄ౰ేၩ݊ ࡺޯ഻വറƄ࣊ ࢵ ࣠ࡩ ࠦ୆งଘଓƄࠦߥ޲ࡩƄƇ Ƅ౰౔ߙ߄ ၴ ƥƀ‫ۑ‬ഭ ୨‫ࡩ ࣐ީޢ‬Ƅ݊ ‫ݹ‬Ə࣠ޯࡩƄ݊ ީჴႾƥƆƌ࢖ߦࡷ 1000 ‫ۑ‬%99 (‫ )ج‬، %95 (‫ )ب‬، %90 ƣ ƀࡼ ࡷቧሼ ܾޯࡩƄࠢެࡓ ೘಼ཀྵདྷࠁ ࡴ ᓉᒴၤ ƄƄ߃ ܻࠧ࣠ࠧዙኽ n ࠦެ‫ࡩݘޢ‬Ƅࡴ ᄽ஛ मअ߄ ߥ‫ކ‬፥ ጿ݊ ެࢧ ࡷ‫ ݌ ݪ ࢢܻ ࠁ ݪ‬मअ .2 ƞ Ƅ࢖༨စ ଘଓƄƣ ƄࣷࣳၮƄƍƄ‫ࡷݘޢީࠦۑ‬Ƈ Ƅ߄ ၰ ƥ 5ࡼ ࡷ౰౗ࡁ ܾ‫ܻ ࣍ޟܻހ‬ၯ ƣ ࣠ࡓ ‫ ࡩ ࡼݹ‬%95ơ ܻࡺޯࠁ ܻ‫ހ‬Ƥमअ߄ ߥ‫ކ‬ ‫ۑ‬Ɔ߄ ၰ ƥ 20Ƨ Ǝܻ‫ݘޢ‬ଐଌƄ ࡼ ࡷ 10Ƨ Ǝܻ‫ࡷݘޢ‬ƞ Ƅ࢖༨စ Ƅ، 50፥ ጒܻࣀ ࠁ ࣊ ࢵ ࣠ޯ೘ೃƆƌ࢖ߦࡷ 16ࡼ ࡷࠦ‫ࠦࡷ ࡓ࣠ރ‬୆෷Ƅ࣠ࣃ ‫ݑޢެࠦݑ‬Ƈ ߃ ၯ ƀ .3 ‫ ࠖޯࡺ ݊ۑ‬൞ᒑᐽƢॅळ‫ݘ‬ଐଌƄ౰ేၩ࣊ ࢵ ॅळࡩ %95Ɔ౰౔ߙ߄ ၴ ƥƀ‫ۑ‬ഭ ୨‫ࡩ ࣐ީޢ‬Ƅ݊ ‫ݹ‬Ə࣠ޯࡩƄ݊ ީჴႾƄ߄ ၴ ౰ేީࡁ ݊ ࡺޯ༬ீ 134 ܸୃกܸ଍ଉŽ–ໝ ແࡦफउ –ࠣ‫ݶ‬ Ɖܸࢤ ެߕକଐƁƢࠣ‫ࡦ ࢽޟܸ ࢲޟ‬ƁƢࠣ‫ޟ‬පඓ ൷Ɓ౭౑ࢲ କଐƁƄ ܸࢲ ƁƋኃ ᒔᐺ። ጏ࢓‫ݕ‬ ࡦƁ࣬ ࣊ Ɓ࢓ߢೕಹኃ ቮƁ೾೥࢓଍ଉƁƄ ƁƋƁ߁ ࢟ ſ ƛ ܸߢࢽ ࡦƁ߁ ዖዄ Ž૊ સၫƋࣝެ࠾ኃ ቮƁƊܸެࢲ ଘଐƁ ። ፎକଐ߁ ެࢲ କଐƁƢ। ्࢟ ࣝࡦƁŻܸࢤ ߾ ଔଐƁ 60ƤƎƄ߄ ߥࡁ Ƨ Ǝܻ‫ࡷݘޢ‬ƞ Ƅ࢖༨စ Ƅƥ 100ƤƎƄ߄ ߥࡷ፥ ጒܻࣀ ࠁ ࣊ ࢵ ࣠‫ ހ‬144ܻࠩࡺᄽ஛ ࠦ୆෷Ƅ࣠ࣃ ‫ݑޢެࠦݑ‬Ƈ ߃ ၯ ƀ .4 ‫ۑ‬Ƣॅळ‫ݘ‬ଐଌƄ౰ేၩ݊ ࡺޯ഻വറƄ࣊ ࢵ ࣠ࡩ %95ࠦߥ޲ࡩƄƆ౰౔ߙ߄ ၴ ƥƀ‫ۑ‬1000ࠢࡺᄽ஛ ݊ ࡺޯ༬ீ ࡼ ࡷ Ƨ Ǝܻ‫ࡷݘޢ‬ƞ Ƅ࢖༨စ Ƅƥ $ 3000ƤƎ߄ ߘ࣊ ࢵ ࣠ޯࡷ߄ ‫ ࢢޢ‬मऌ࿃ Ⴐ ༧ೞ Ǝܻၯ ƌƄƅ ܻࣀ ࠁ ౰ే‫ݓ‬ ࢢ ્ ઻༨ீ ࠯ ୐กƦ ኆ ቱ .5 Ƥ߃ ಛಗƇ Ƅ༇໩ ߄ ଐଌƄ࣊ ࢵ ࣠ޯࡷƣ ƀơ ܻࡺޯࠁ Ƅܻࡷ ،ƅ ܻࣀ ࠁ 100ࡼ ࡷࠦ୆෷Ƅ࣠ࣃ ‫ࡩީެ ࠯ݑޢެࠦݑ‬Ƅ߃ ၯ ƀƏ $1200 ‫ۑ‬$2800ࡼ ࡷࡣ ߘƄƣ ࣠ࡓ ‫ݶࠦ ࢵޢ‬ ܻଐଌƄƇ ܻ‫ ࣀܻހ‬ஊै Ƅ ࠦၮܻࢵ 300࢖ዙኸ ࣊ ࢵ ࣠ޯ೘ೃࠦ୆෷ܻ‫ ࡓࠩ࢖ހ‬ᚻࡩƄ߰ ୆กܻࢧ ଐଌƄࡼ ࡷƇ Ƅƌ࢖ߦࡷ 9ࡼ ࡷࠦ‫ࠦࡷ ࡓ࣠ރ‬୆෷Ƅ࣠ࣃ ‫ީ ݝݑޢެࠦݑ‬༨೛ .6 ،ഭ ୨‫ࡩ ࣐ީޢ‬Ƅ݊ ‫ݹ‬Ə࣠ޯࡩƄ݊ ީჴႾܻᐱᐕ࡯‫ݓ‬ ఴ ఠ࢖ዙኸ ƣ Ƅƞ ƥ࢖‫ࠦࡷݘ‬୆෷ܻ‫ެࠦࡁࠩ࢖ހ‬༨ೞ ࡼ ࡷࠦၮܻࢵ 45Ƨ Ǝܻ‫ࡷݘޢ‬ƞ Ƅ࢖༨စ Ƅƥ ‫ۑ‬Ƣॅळ‫ݘ‬ଐଌƄ౰ేၩࡣ ‫ݓޢ‬ ࣃ ჴࡩƄ࢖ዙኸ ࣊ ࢵ ࣠ࡩ %90ࠦߥ޲ࡩƄƆ౰౔ߙ߄ ၴ ƥƀ 800ࠢ‫ކ‬Ƅƌ࢖ߦࡷ݊ ࡺޯ༬ீ ࡼ ࡷƇ ߃ ၯ ƀ 20Ƨ Ǝܻ‫ݘޢ‬ଐଌƄܻࠩߙƄ࢖༨စ Ƅƥ 50ܻ࣐ࠩ ࢵ ƥƆƌ࢖ߦࡷ 64ࡼ ࡷࠦ୆෷Ƅ࣠ࣃ ‫ ݑޢެࠦݑ‬.7 ‫ۑ‬%95ࠦߥ‫މ‬Ʀ ࣠ޯࣀ ࡷ߄ ެ‫ݑ‬Ƣॅळ‫ݘ‬ଐଌƄ౰ేၩ݊ ࡺޯ഻വറƄ࣊ ࢵ ࣠ࡩࠦߥ޲ࡩƄƆ౰౔ߙ߄ ၴ ƥƀ ، ࣊ ߥߙ %40ܻࡺެყჅƈ ॅळޯࡩƄࠦஊ஀ ᇴᇍ ࡷ౰ే‫ݘܻݹ‬ଐଌ݊ ࢥ ༧ဈ A যও࡯ߘƄ፥ ጿܻ‫ ࡷ ࢧެݘ‬60ࡼ ࡷ %50ƣ ƀƓ ౰౔ߙƀ .8 ࠦஊ஀ ᇴᇍ ࡷ౰ే‫ݘܻݹ‬ଐଌ݊ ࢥ ༧ဈ ཯དྷࡩƄ݊ ‫ ࢧܻރ‬ଐଌƄࠦީఱ ఝࡣ ࠧ‫ۑ‬౰ే‫ݘܻݹ‬ଐଌƄࢪ ߦެࡩ݊ ࢥ ༧ဈ B যও࡯ߘƄ፥ ጿܻ‫ ࡷ ࢧެݘ‬40ࡼ ࡷ ‫ۑ‬%10 ‫ و‬%5ࠦ‫ࡷݘެ࣠ݹ‬Ʀ ࣠ޯࣀ ࡷ߄ ެ‫ ݑ‬B যও࡯ߘƄ፥ ጿܻᐱᐟ‫ ݑ‬A যও࡯ߘƄ፥ ጿܻ‫ࡷݘެ࣠ݹ‬౰౎ࡁ ƀƈ ॅळޯࡩƄ Ƨ ƥܻఱ ఓ࣊ ࢵ ࣠ޯଐଌƄƣ ƀ߄ ၴ ƥƥ݊ ࡺޯ༬ீ ࡼ ࡷ 100ࡴ ࠖ ༨ဂ ơ ƥଘଓƄ݊ ࡺޯ഻വറƄࡼ ࡷࠦ୆෷Ƅ࣠ࣃ ‫ݑޢެࠦݑ‬Ƈ ౰ేޯࠀ Ƅ .9 ࡴ ࠖ ༨ဂ ơ ƥଘଓƄ݊ ࡺޯ഻വറƄࡼ ‫ݑ‬ቿቷߥޯࣀ ࡷࠦ୆෷Ƅ࣠ࣃ ‫ݑޢެࠦݑ‬፥ ጕܻ޲ࡩƄ݊ ࡺޯ഻വറƄࡼ ࡷƇ ౰ేޯࠀ Ƅƥ‫ۑ‬25௪ ௎ܻީޯࡩ‫ وا‬65 ໠ ໄ‫ ࠖޯࡺݘ‬൞ᒑᐽࠦࡷॅळ‫ࡷݘ‬౰ేၩ௪ ௎ܻީޯࡩƄযওߘบ คᇴᆝ Ƅƍܻߙ‫ۑ‬16௪ ௎ܻީޯࡩƄƥ 60Ƨ ƥܻఱ ఓ࣊ ࢵ ࣠ޯଐଌƄƣ ƀ߄ ၴ ƥƥ 80 ࠦߥ‫މ‬Ʀ ࣠ޯࣀ ࡷ߄ ެ‫ݑ‬໠ ໄ‫ޯࡺݘ‬഻വറƄ࣯ ࣐ ࢵ ࣠ޯࡷ໠ ໄ‫ހ‬Ɵ ࢖ߦᑃᐽࠦߥ޲ࡩƄƆ౰౔ߙ‫ ࠁ ࣀ ݪ‬ƀ ،ቿቷߥޯࣀ ࡷƇ ܻެ‫ࡩݘޢ‬Ƅบ คᇴᆝƥ ‫ۑ‬%90 ௪ ௎Ǝܻޯ༧ீ ࢝ ༧ೞ 100ࡼ ࡷࠦެ‫ݑޢ‬፥ ጿƣ ƀ ،ࠦ‫ ୹ݹ ࡓޢ‬୪ଘଓƄƇ ܻ‫ݹ‬ଘଓ࣠ࡩƄƦ ߄ ၰ ଘଓযও࡯‫ࡩޯݘ‬ƄƆƎƄƌƀƇ ߄ ၴ ƥ Ɔ౰౔ߙ߄ ၴ ƥƀ‫ܻࡷݘޢࠦۑ‬ၴ Ƈ ܻၴ Ǝƌે ઻ၮƄॅळࢧ ࠁ ߄ ߘࡴ ᐱᐟࡷ %40Ƈ ܻ‫ܻࡷݘ‬எै ܻ‫ހ‬໠ ໄߥࠒ ޯ࡯ଐଌƄ໠ ໄ‫ܻࡷ ࡼހ‬୆෷Ƅ࣠ࣃ ‫ݑ‬ ‫ܻࡷݘࠦۑ‬எै ܻ‫ހ‬໠ ໄߥࠒ ޯ࡯ଐଌƄ݊ ‫ޢ‬ዙዋ ໠ ໄ‫ܻࡷݘޢࠦࡷ ࡼހ‬ၴ Ƈ ܻၴ Ǝƌે ઻ၮ໠ ໄ࡯ࢢ ܻஊै Ƅࠦީࣀ ჱࡩࠦߥ‫ މ‬%99 135 ܸୃกܸ଍ଉŽ–ໝ ແࡦफउ –ࠣ‫ݶ‬ Ɖܸࢤ ެߕକଐƁƢࠣ‫ࡦ ࢽޟܸ ࢲޟ‬ƁƢࠣ‫ޟ‬පඓ ൷Ɓ౭౑ࢲ କଐƁƄ ܸࢲ ƁƋኃ ᒔᐺ። ጏ࢓‫ݕ‬ ࡦƁ࣬ ࣊ Ɓ࢓ߢೕಹኃ ቮƁ೾೥࢓଍ଉƁƄ ƁƋƁ߁ ࢟ ſ ƛ ܸߢࢽ ࡦƁ߁ ዖዄ Ž૊ સၫƋࣝެ࠾ኃ ቮƁƊܸެࢲ ଘଐƁ ። ፎକଐ߁ ެࢲ କଐƁƢ। ्࢟ ࣝࡦƁŻܸࢤ ߾ ଔଐƁ ļ Ř ĻǍȋĻ ŉŌļĻ ʫņ ŅŉĿ ő ńŐ Ņŉľ ń [ Hypothesis Testing ] 136 ܸୃกܸ଍ଉŽˍ ໝ ແࡦफउˍ ࠣ‫ݶ‬ Ɖܸࢤ ެߕକଐƁƢࠣ‫ࡦ ࢽޟܸ ࢲޟ‬ƁƢࠣ‫ޟ‬පඓ ൷Ɓ౭౑ࢲ କଐƁƄ ܸࢲ ƁƋኃ ᒔᐺ። ጏ࢓‫ݕ‬ ࡦƁ࣬ ࣊ Ɓ࢓ߢೕಹኃ ቮƁ೾೥࢓଍ଉƁƄ ƁƋƁ߁ ࢟ ſ ƛ ܸߢࢽ ࡦƁ߁ ዖዄ Ž૊ સၫƋࣝެ࠾ኃ ቮƁƊܸެࢲ ଘଐƁ ። ፎକଐ߁ ެࢲ କଐƁƢ। ्࢟ ࣝࡦƁŻܸࢤ ߾ ଔଐƁ Ƅ ܸ‫ࡦߣ࢓ ࢝ޟ‬ƁƋܸަެ߽ ƁƟࣝࠦߣࡴ 1 ‫ۑ‬፥ ገܻࢧ ࠁ ଘଓƄơ ଘଓ߄ ޯࢵ ଘଓƄષઔ੮ܻၴ ࡼ ࡷ፥ ጕܻ޲ࡩƄว คܻஎै Ƅ࣠ࠧ݊ ࡺޯ഻വറƄ࢝ ‫ݶ‬ ܻࢧ ࠀ ࡼ ‫ݑ‬Ƈ ܻ‫ࡩߦ࢖ࢠޢ‬ƄƎܻީޯࠀ Ƅ Ƣ߄ ႀ ޯఱ ఝܻࡷƄ౰ే޲ࡁ ‫ࡩޯܻ ࢵ ݊ۑو‬Ƅࡣ ࢧ ߦࡩƄ፥ ጿࡣ ‫ހܻࡩޯߦ ࢧޢ‬ቻቱƥܻ୐งᎨ᎙Ƨ ኅ ቱƄƥࠢެࡷơ ƥଘଓƄว คܻஎै Ƅमअ߄ ߥޯࡩƄࡣ ޲೘಼ƥ ƥƀ݊ ࡺޯ഻വറƄࠦީఱ ఝƥƀ࣊ ࢵ ࣠ޯࡷơ ࣠ࠁ Ƈ ƄƎƄ࢖ߥࡩƄƍܻ༧ဈ ଘଓ ્ ઻‫࡯ޢ‬ႁ ޯࡩƄžܻࢧ ࠁ ଗଓƄ ፥ ገܻࢧ ࠁ ଘଓƄơ ଘଓ߄ ޯࢵ ଘଓƄ ࡼ ࡷ݊ ࡺޯ഻വറƄ࢝ ‫ݶ‬ ܻࢧ ࠀ ે ઻ၮƞ ࢖‫ޯݘ‬ᑃᐽᑑ ᐽƍƥ‫ۑ‬౰౗ࡁ ƀƥƀ໠ ໄ‫ޯࡺݘ‬഻വറ‫ ࣀ ݪ‬ჱࡩƄƥƀƇ ܻ࣐ ࢵ ࣠ޯଐଌƄ໠ ໄ‫ހ‬Ɵ ƥ࢖ߦࡩƄ ‫ࡩݘޢެࠦۑ‬ƄƇ ܻ୆෷ܻࢧ ࠁ Ƃ݊ ߘƄƥ ࢛ ߦ༧စ ƀ߄ ߘܻࡷ ࠦ୐෺߄ ࡷ፥ ጿƢ࣠ॶॱ५Ƨ ƌ࢖ߦࡩƄƠ ൎ൉ᐱᐢࢵ ଘଓƄ࣊ ࢵ ࣠ޯࡷƣ ᇴᆝ ƄƍƄܻࡺ‫ࡷݘ࢖ߙࠦߙޢ‬፟ ፑƄൎ൉޲ࡷƉ ܻޯ༨စ ߄ ߘ ࡼ ‫ߦ ࢥ ݝݑ‬༧စ Ƅ߄ ߘ໠ ໄ‫ݓ‬ ࡩܻީᑃᐽࠦ‫ࡷޢ‬ଘଓƄࠦީఱ ఝࡣ ࠧƥƀ‫ۑ‬2000Ƣܻ‫ރࠦހݘ‬Ǝܻߥࡷ 2010Ƣܻၮ݅ ࡁ 20ࡼ ‫ݑ‬ ‫ۑ‬Ɔ౰౔ߦࡩƄࢪ ߦެࡩࠦ୐෺߄ ଐଌƄࢪ ߦ‫ރ‬፥ ጿ %40 ࡣ ޲ࡷ ݊ ࡺޯ഻വറƄ Ꭸ፻ܻ‫ ࡷݘ‬যওߘ ơ ࣠ࠁ Ƈ ܻ‫ࡩߦ࢖ࢠޢ‬Ƅ ƥƀ Ƈ ƄžܻၮƌଘଓƄ যও‫ ކߥޢ‬ƥƀ Ǝܻީޯࠀ Ƅ ፟ ፑƄ Ɖ ܻޯ༨စ ᑑ ᐽኅ ቱ ‫ۑ‬௪ ௎ܻީޯࡩƄƥ‫ ࣀ ݪ‬ჱࡩƄƥƇ ܻ࣐ ࢵ ࣠ޯଐଌƄ ݊ ࡺޯ഻വറƄơ ࣠ࠁ Ƈ ƄžܻၮƌଘଓƄƤ߃ ࠧƣ ܾఴ చƎƄ࢖ߘƍܻ༧ဈ ଘଓࠦ‫ޯީݘ‬ଐଌƄƟ ࢖࣐ ࡩƄ߄ ၰ ƀƇ ܻ‫ࡩߦ࢖ࢠޢ‬ƄƎܻީޯࠀ Ƅࡣ ޲೘಼ƥ ࡼ ‫ޯ࡯ ߎࡷݘެ࣠ݹܻݑ‬༧ဈ Ɔ߄ ܻࠧࣃ ଐଌƄၖ࿏ ܻޯެࡩƄบ คᇴᆝ ƄƍƄܻࡷਔ ৲߄ ༨ဈ ƥܻᐳႮ ߙƎƥƀܻಛಗ࣠ީߘ‫ࡷ ࡼ ࠁޢ ݟ‬፥ ገܻࢧ ࠁ ଘଓƄ ‫ۑ‬ƎƄ࢖ߥࡩƄƍܻ༧ဈ ଘଓࠦ‫ ࢵܻ ࢵޢ‬ƄƆ߄ ၮܻߘᇯ ᇍ ఴ ఓƇ ܻ‫ࡩߦ࢖ࢠޢ‬ƄƎܻީޯࠀ Ƅƣ ܾߙ፥ ፑܻޯࡩܻ‫ހ‬ƥ‫ ࠖޯࡺ ݊ۑ‬൞ᒑᐽࠦࡷॅळ‫ݘ‬ଐଌƄᎨ፻ܻ‫ݘ‬ଐଌƄ ࠦެ‫ ߃ݑޢ‬ၯ ܾ‫ރ‬ƥ‫ॅࡷࠦۑ‬ळ‫ݘ‬ଐଌƄ౰ేၩ݊ ࡺޯ഻വറƄࠦ‫ܻ ࢢޢ‬ၯ ࡼ ‫ݑ‬žܻၮƌƄƥƀࠦ‫ހݘࡺ ࡣߙ࢖ࢠޢ‬ƀ߄ ୍คƇ ܻ‫ࡩߦ࢖ࢠޢ‬ƄƎܻީޯࠀ Ƅ፥ ጿ ࠦၴ Ǝਔ ৹ࠦ‫ࡩߦ࢖ࢠޢ‬Ƅ࢛ ߙमएƣ ƀƥƀࡣ ީߥ‫ރ‬ƣ ƀܻࡷƂࠦެ‫ࡩݘޢ‬Ƅ፥ ጿƆ࢖࣌ ܻެଐଌƄࠦ‫ܻ ࢢޢ‬உै ƄƐ ܻࢵ ƀે ઻ၮƥūࠦ୆෷Ƅ࣠ࣃ ‫ݑ‬ Ʀ ߄ ࡷमअ߄ ߥ‫ࡩߦ࢖ࢠޢࠦࠧ࣠ކ‬ƄƎܻީޯࠀ Ƅūܻࡷž࿃ ႖ࠦࡺ‫ߘޢ‬ƥƀࠦ༨༦ ơ ࣠ࠁ žܻၮƌƀ፥ ፕ ࠦ‫ࡩߦ࢖ࢠޢ‬Ƅƥ‫ۑ‬Ɔƌ߄ ༨ீ ࠦߥ‫މ‬ ‫ࡩߦ࢖ࢠޢࠦ ۑ‬Ƅ žܻၮƌଘଓƄƄ߃ ࠧࠦ༨༦ Ɛ Ƣ࢓ߣࡦƁƋܸަެ߽ ƁƄ ܸ‫ێ ߖ࢓ ࢝ޟ‬ ‫ۑ‬ቿቷ‫ࡩީ ߄ݹ‬Ƅࠦ‫ࡩߦ࢖ࢠޢ‬ƄƥƢ߄ ‫ࡩݘ‬Ƅࠦࢠ࢖ߙܻዙ዗໠ ໄޯყࢠ࢖ߙે ઻ၮƓ ƥ࢖ߦࡩƄƎܻީޯࠀ Ƅࡼ ࡺࢥ ୓෺ [ Null Hypothesis] Ɵ߁ ‫ݕ‬ ࡦƁࠣ‫ ߖ࢓ ࢝ޟ‬ Ƈ ࢖ߙ࣠‫ކ‬ƄƍƄଘଓƄܻᐳႮ ߙमएଘଓƥܻᐱᏳ࠯ ࣀ ๻๬เ཯དྷࡩƄࠦ‫ࡩߦ࢖ࢠޢ‬Ƅ፥ ፕ Ho୷ ୪༟༎ܻ‫ܻހ‬ಛಗ୷ ୪मअƥ Ƣ߄ ‫ࡩݘ‬Ƅࠦ‫ߙ࢖ࢠޢ‬ Ɨ Ǝܻߙƥƀࡣ ࣍ ܻ‫ހ‬žܻၮƌଘଓƄƣ ƀ Null)Ƣ߄ ‫ࡩݘ‬Ƅࠦ൞ᇅᆝ ཯ཏ‫ކݘ‬ƥ‫ܻۑ‬ᐳႮ ߙƎ፟ ፑƄƌ࣠ߥ‫ࡩݘޢެࠦކ‬Ƅࡼ ࡷࠦ‫ݶ ࡣߘ࣠ݹ‬ ଘଓƌ ݊ ࢠ࣠ࡷƣ ࣠ࡓ ‫཯ކ‬དྷࡩƄ፥ ፕ Ƣ߄ ‫ࡩݘ‬Ƅࠦ‫ߙ࢖ࢠޢ‬ƥ‫ۑ‬Ɔܻၮ߄ ଐଌƄࠦࡺ‫ࡩߥޢ‬Ƅƥ݊ ࡺޯ഻വറƄࠦ൞൙‫ࡷݘ‬໠ ໄ‫ހ‬Ɵ ࢖ߙƌ࣠ࠅ ƥƢ߄ ၮƥƀ ࡣ ‫ݶ‬ ଘଓኆ ቱƄે ઻ၮžܻ୐ก ࡣ ީߥ‫ܻކ‬ᐱ᏶ƀཀྵཏ‫ݘ‬೘ೃ ࢛ ߙमऒଘଓƢƀቿቷ‫ࡩީ ߄ݹ‬Ƅࠦ‫ࡩߦ࢖ࢠޢ‬Ƅࠦႁ ࡯ࢧ ଐଌ࢛ ߙ౰౔ࢵ ࡣ ࠧƥƎܻީޯࠀ ଘଓƄ Ɔ߄ ၰ Ƅƥࠦࡺ‫ࡺ ࡣߘޢ‬ఴ ఠܻᐱ᏶ܾߙ݊ ࡺޯ഻വറƄᎨ፻ܻ‫ ߄ࡷݘ‬ၰ Ƅơ ࣠ࠁ ࠦ‫ߙ࢖ࢠޢ‬Ǝܻީޯࠀ Ƅቿቱܻၰ ፥ ጿƥ‫ࡩݘޢެࠦۑ‬Ƅܻࠧ࢖ߙ࣠‫཯ކ‬དྷࡩƄ 137 ܸୃกܸ଍ଉŽ–ໝ ແࡦफउ –ࠣ‫ݶ‬ Ɖܸࢤ ެߕକଐƁƢࠣ‫ࡦ ࢽޟܸ ࢲޟ‬ƁƢࠣ‫ޟ‬පඓ ൷Ɓ౭౑ࢲ କଐƁƄ ܸࢲ ƁƋኃ ᒔᐺ። ጏ࢓‫ݕ‬ ࡦƁ࣬ ࣊ Ɓ࢓ߢೕಹኃ ቮƁ೾೥࢓଍ଉƁƄ ƁƋƁ߁ ࢟ ſ ። ፎକଐ߁ ެࢲ କଐƁƢ। ्࢟ ࣝࡦƁŻܸࢤ ߾ ଔଐƁ ƛ ܸߢࢽ ࡦƁ߁ ዖዄ Ž૊ સၫƋࣝެ࠾ኃ ቮƁƊܸެࢲ ଘଐƁ ƣ ߄ ଐଌƄƦ ߄ ၰ ƀ፥ ጿܻ‫ ࢵެ࣠ݹ‬ƌ࢖ߦᑃᐽ౮౎உै ƄƠ ൎ൉ᐱᐢࢵ Ƅ࣊ ࢵ ࣠ޯࡷƣ ƀࠦ‫ߙ࢖ࢠޢ‬Ǝܻީޯࠀ Ƅܻ‫ރ‬ƌƎƄƄƍƄൎ൉޲ዙኽ‫ߙߥ ࣊ۑ‬ ፥ ጘଘଓᇴᆝ Ɨ ܻࢧ ‫ކ‬Ƣ߄ ‫ࡩݘ‬Ƅࠦ‫ߙ࢖ࢠޢ‬ƣ ܾߙ݅ ࡁ 160࣠ࠧ 2002Ƣܻၮ Ho : μ = 160 [ Alternative Hypothesis ] ቼቴ‫ݶ‬ ߁ ަࡦƁࠣ‫ࡦߣ࢓ ࢝ޟ‬Ɓ  ࠦ‫ࡩީܻ ࠁ ݟ ࡁީ ߄ݹ ࡣݑ ࡼߙ࢖ࢠޢ‬Ƅܻࠩ‫཯ݹ ࢥݘ‬དྷࡩƄࠦ‫ࡩߦ࢖ࢠޢ‬Ƅ፥ ፕ ƥ‫ ۑ‬H1୷ ୪༟༎ܻ‫ܻހ‬ಛಗ୷ ୪मअƥ ቿቷ‫ࡩީ ߄ݹ‬Ƅࠦ‫ࡩߦ࢖ࢠޢ‬Ƅ ౮౎உै ƄƠ ൎ൉ᐱᐢࢵ Ƅơ ࣠ࠁ ࢇ ‫ࡩ ࣀܻހ‬Ƅơ ܻ޲ଐଌƄ१ ॐߙ‫ࠦۑ‬ႁ ‫ޢ‬༨༦ ౰ేၩƢ߄ ‫ࡩݘ‬Ƅࠦ‫ߙ࢖ࢠޢ‬ƣ ƀƐ ܻࢵ Ƅે ઻ၮཀྵཏხ჋ƥƢ߄ ‫ࡩݘ‬Ƅ ౰౗ࡁ ƀƥƀ ݅ ࡁ 160ࡼ ࡷࡣ ߘƀ࣠ࠧࠦ୐෺߄ ଐଌƄ፥ ጿ౮౎உै ƄƠ ൎ൉ᐱᐢࢵ Ƅ࣊ ࢵ ࣠ޯࡷƣ ƀ፥ ፕ ቿቷ‫ࡩީ ߄ݹ‬Ƅࠦ‫ࡩߦ࢖ࢠޢ‬Ƅƣ ܾߙ ቿቷ‫ࡩީ ߄ݹ‬Ƅࠦ‫ࡩߦ࢖ࢠޢ‬Ƅƣ ܾߙࠢ‫࡯ޢ‬ၮƥūࠦެ‫ࡩݘޢ‬Ƅࡼ ࡷܻᐱᐕ࡯ၮơ ࣠ࢧ ஊै ƄᎨ᎙Ƈ ܻࡷॅळ‫઻ ેࡷݘ‬ၮžܻ୐ก‫ܻ ۑ‬ᐱᐟ‫ޯ࡯ ߎݑ‬༧ဈ ƥƀ ፥ ጘଘଓᇴᆝ Ɨ ܻࢧ ‫ࡩ ࣀܻހ ࢇ ކ‬Ƅơ ܻ޲ଐଌƄ፥ ጿ H1 : μ ˂ 160 H1 : μ ˃160 ݅ ‫ ߄ ࢢޢ‬ၴ ࣠‫ކ‬ƥ‫މߥࠦࡷݘޢެࠦۑ‬Ʀ ࣠ޯࣀ ࡷ߄ ެ‫ݑ‬ቿቷ‫ࡩީ ߄ݹ‬Ƅࠦ‫ࡩߦ࢖ࢠޢ‬Ƅࡣ ީߥ‫୐ܻރ‬คܾߙƢ߄ ‫ࡩݘ‬Ƅࠦ‫ߙ ࢛ߙ࢖ࢠޢ‬मएܻࡷ߄ ެ‫ݑ‬ƥ ƌƄ࣠ଐଌƄࡼ ࡷ໠ ໄ‫ࡷݘ‬Ɩ ࣠‫ރ‬Ɖ ܻ୓คܺᄣƢ࣠ߥ‫ކ‬ūࠦ୆෷Ƅ߃ ‫ݓ‬ ࡩƄƌƄ࣠൞ᒑᐽበሼྲ ႖Ƈ ƌƄƎƄƄƍƀൎ൉޲ዙኽ‫ۑ‬ቿቷ‫ࡩީ ߄ݹ‬Ƅࠦ‫ߦ࢖ࢠޢ‬ᑃᐽƦ ༇໩ ƀ ƥƄࡣ ߥ‫ݹ‬ଘଓƇ Ƅ࣠ީ‫ࡩݘ‬Ƅƣ Əƥƣ ᇴᆝ ƄƍƄܻࡺ‫ߙޢ‬ቧሼ ܾ୓คƣ Ƅܻ‫ރ‬ƌƎƄƄƍƄƥ‫ۑ‬ƢƄ༇໮ 100Ƈ Ƅ࣠ީ‫ݑ‬፥ ጿࠦ୆෷Ƅ߃ ‫ݓ‬ ࡩƄ H0 : μ =160 ፥ ጘଘଓᇴᆝ Ɨ ܻࢧ ‫ކ‬Ǝܻީޯࠀ ଘଓƄƄ߃ ಛಗࠦ‫ࡩߦ࢖ࢠޢ‬Ƅƣ ܻߙƢƄ୹ ‫ ݌‬100ࡼ ‫ݑ‬ਔ ৲बअ H1 : μ ≠160 ߎ ࡯ޯ༧ဈ ƥƀ ౰౗ࡁ ƀƥƀࡣ ߘƄบ คᇴᆝ ƄžƄ࣠ࢵ ቿቷ‫ࡩީ ߄ݹ‬Ƅࠦ‫ࢠޢ‬༟༎Ƅบ คᇴᆝ ܻࡺᐱᚑƆ߄ ၰ ƄƥƢ߄ ‫ࡩݘ‬Ƅࠦ‫ߙ࢖ࢠޢ‬ƥ Ƅ ܸ‫ࡦߣ࢓ ࢝ޟ‬ƁƋܸަެ߽ Ɓ። ጼŻܸ࣍ ߽ କଐƁߋ ‫ޟ‬ხࢤ ‫ێ ރ‬ žܻ࣐ ࠀ ଘଓƄࡼ ࡷ໠ ໄၮ࣠‫ކ ࡓ ݪރ‬मएƣ ƀࡼ ࡓ ೘಼Ƈ ܻ‫ࡩߦ࢖ࢠޢ‬ƄƎܻީޯࠀ Ƅ፥ ጿ ܾ࣐ உै ƄƄ߃ ࠧƈ ߄ ༨࿻ ƥ α ܾ࣐ உै ܻ‫ࡺ ࣩހ‬ఱ ఓƥ ( Type I error ƞ ƢକଐƁƓ ࣝީࡦƁࡹ ࡴܻ࣍ ஆॅ Ɓ  ‫ࠦۑ‬ႁ ‫ޢ‬༨༦ ݊ ߘƄ࣠ࡩƄ፥ ጿ፥ ፕ ƥƢ߄ ‫ࡩݘ‬Ƅࠦ‫ߙ ࢛ߙ࢖ࢠޢ‬Ǝ߄ ެ‫ݑ‬ ܾ࣐ உै ƄƄ߃ ࠧƈ ߄ ༨࿻ ƥ ß ܾ࣐ உै ܻ‫ࡺ ࣩހ‬ఱ ఓƥ ( Type II error ። ጒܸޯࡦƁƓ ࣝީࡦƁࡹ ࡴܻ࣍ ஆॅ Ɓ  ‫ܻ ࣍ޟࠦۑ‬ၯ ݊ ߘƄ࣠ࡩƄ፥ ጿ፥ ፕ ƥƢ߄ ‫ࡩݘ‬Ƅࠦ‫ߙ࢖ࢠޢ‬ơ ࣠ީߘ߄ ެ‫ݑ‬ ࠣ‫ܸ ࣊ޜ‬ၬ 138 H0Ɵ߁ ‫ݕ‬ ࡦƁƐ ࢓ߖ Ɛ Ƣ࢓ߣࡦƁƋܸަެ߽ ƁŻܸ࣍ ߽ Žߋ ‫ޟ‬ხࢤ ‫ ێ ރ‬ơ ƥ߄ ၴ ࠣၾ ‫ޟ‬༥༣ ƋƁ࢓ߢࡦƁ ( H0 ࢘ ߖƋ ܸୃกܸ଍ଉŽ–ໝ ແࡦफउ –ࠣ‫ݶ‬ Ɖܸࢤ ެߕକଐƁƢࠣ‫ࡦ ࢽޟܸ ࢲޟ‬ƁƢࠣ‫ޟ‬පඓ ൷Ɓ౭౑ࢲ କଐƁƄ ܸࢲ ƁƋኃ ᒔᐺ። ጏ࢓‫ݕ‬ ࡦƁ࣬ ࣊ Ɓ࢓ߢೕಹኃ ቮƁ೾೥࢓଍ଉƁƄ ƁƋƁ߁ ࢟ ſ ƛ ܸߢࢽ ࡦƁ߁ ዖዄ Ž૊ સၫƋࣝެ࠾ኃ ቮƁƊܸެࢲ ଘଐƁ ߭ ‫ޟ‬༥༣ ƋƁ࢓ߕ ßܻ࣍ ஆॅ Ɓ tailed test ። ፎକଐ߁ ެࢲ କଐƁƢ। ्࢟ ࣝࡦƁŻܸࢤ ߾ ଔଐƁ αܻ࣍ ஆॅ Ɓ ߭ ‫ޟ‬༥༣ ƋƁ࢓ߕ ( H0 ƞ ࣝަߕ Ƅ ܸ‫ࡦߣ࢓ ࢝ޟ‬ƁƋܸަެ߽ ƁƓ Ɓࣝ‫ހ‬Žߋ ‫ޟ‬ხࢤ ‫ێ ރ‬ two ໠ ໄߙ࢖࣍ ໠ ໄ୍คܻၴ ፥ ጿƎܻީޯࠀ Ƅ፟ ፑƄƢܻၮᇯ ᇍ ఴ చƇ ܻ‫ࡩߦ࢖ࢠޢ‬ƄƎܻީޯࠀ Ƅߎ ެࢧ ‫ކ‬ Ƅžſ࣠ࢵ Ǝܻީޯࠀ ଘଓƄƖ ࣠‫ރ‬ƣ ᇴᆝ ܻࡺᐱᚑƥ‫ۑ‬one tailed test ߄ ၰ Ƅƥƞ ࢖࣍ ߄ ၰ Ƅƥว คܻၴ ፥ ጿƎܻީޯࠀ Ƅƥ ūቿቷ‫ࡩީ ߄ݹ‬Ƅࠦ‫ࡩߦ࢖ࢠޢ‬Ƅบ คᇴᆝ ܻࡺᐱᚑƆ߄ ၰ Ƅƥƣ ࣠ࡓ ‫ކ‬Ƣ߄ ‫ࡩݘ‬Ƅࠦ‫ߙ࢖ࢠޢ‬ƣ ܾߙ߄ ၰ Ƅƥƞ ࢖࣍ ƥƀ໠ ໄߙ࢖࣍ ƥƍƣ ᇴᆝ ‫ޯ࡯ ߎۑ‬༧ဈ ܻᐱ᏶ƀƌ࢖༬ீ ƥƀ౰౎ࡁ ƀƥƄࡣ ߘƄบ คᇴᆝ Ƅžſ࣠ࢵ ໠ ໄߙ࢖࣍ ࡼ ࡷƎܻީޯࠀ ଘଓƄ፥ ጿ α ܾ࣐ உै Ƅơ ƌܻ‫཯ކݘ‬དྷࡩƄࠦၰ ܻࣀ ଐଌƄ፥ ፕ ƥū ࠦ‫ࡩߦ࢖ࢠޢ‬Ƅ ࢛ ߙƎࠦߥ࣐ ެࡷᄎ ႞ߥ୐งƥ (α ܾ࣐ உै Ƅࠦၰ ܻࣀ ࡷߎ ࢧ ‫ރ‬Ƨ ƥܻఱ ఠܻࡺᐱᐟࡷᇯ ᆝƥƎܻࣀ ყࡩƄ፟ ፑƄƦ ༇໩ ଘଓƄƥ໠ ໄ৤ৈষƄ፟ ፑƄƆ߄ ၰ Ƅƥ ໠ ໄࡺࣀ ߘ፟ ፑƄ ơ ࣠ީߥࡩƄࠦߥ࣐ ެଐଌཀྵཏࠒ ެଐଌƄ‫ ݝ‬༨ဈ ࠦၰ ܻࣀ ଐଌƄƣ ƀ཯ཏ‫ݹݘ‬Ƅ߃ ࠩߙ %90ࠦߥ޲ࡩƄƦ ࣠ޯࣀ ࡷบ คᇴᆝ Ƅƍܻߙ‫ۑ‬α /2 ‫أي‬ ࡼ ࡷᇯ ᆝࠦၰ ܻࣀ ࡷƣ ƀƥ ‫ ۑ‬፥ ፕ αܾ࣐ உै ƄƧ ƥܻఱ ఠ཯དྷࡩƄū࢛ ߙ༟༎Ƅࠦߥ࣐ ެࡷࠦၰ ܻࣀ ࡷƣ Ƅƥ ‫ ۑ‬፥ ፕ Ǝܻީޯࠀ Ƅቿቱܻၰ ፥ ጿࠦ‫ࡩߦ࢖ࢠޢ‬ƄƗ ܻࢧ ‫ކ‬ƥū α /2Ƨ ƥܻఱ ఠƥ ‫ ۑ‬፥ ፕ ໠ ໄߙ࢖࣐ ࡩƄે ઻ၮ࢛ ߙ༟༎Ƅ཯དྷߥ࣐ ެࡷ H0 : μ = μ0 ፥ ጘଘଓᇴᆝ ໠ ໄߙ࢖࣍ ࡼ ࡷƓ ƥ࢖ߦࡩƄ H1 : μ ≠ μ0 ࠦၰ ܻࣀ ࡷƧ ƥܻఱ ఠƥū߄ ၰ Ƅƥว คܻၴ ፥ ጿ݊ ߥ‫ߙ ࢛ކ‬༟༎Ƅࠦߥ࣐ ެࡷƣ ܾߙū߄ ၰ Ƅƥƞ ࢖࣍ ࡼ ࡷƎܻީޯࠀ ଘଓƄ፥ ጿ ቿቱܻၰ ፥ ጿࠦ‫ࡩߦ࢖ࢠޢ‬ƄƗ ܻࢧ ‫ކ‬ƥ‫ཾྜ ྲۑ‬ଘଓƄƥƀࡼ ೘಼ଘଓƄว คܻஎै Ƅ፥ ጿ݊ ߥ‫ކ‬ƥ ൎ൉޲ࡷ ‫) ۑ‬፥ ፕ ƥ αܾ࣐ உै Ƅ ፥ ጘଘଓᇴᆝ ߄ ၰ Ƅƥƞ ࢖࣍ ࡼ ࡷƓ ƥ࢖ߦࡩƄƎܻީޯࠀ Ƅ ࡹ ೕಹଘଐƁƛ ࢓࣍ ࡦƁƋܸަެ߽ Ɓ 139 ܸୃกܸ଍ଉŽ–ໝ ແࡦफउ –ࠣ‫ݶ‬ Ɖܸࢤ ެߕକଐƁƢࠣ‫ࡦ ࢽޟܸ ࢲޟ‬ƁƢࠣ‫ޟ‬පඓ ൷Ɓ౭౑ࢲ କଐƁƄ ܸࢲ ƁƋኃ ᒔᐺ። ጏ࢓‫ݕ‬ ࡦƁ࣬ ࣊ Ɓ࢓ߢೕಹኃ ቮƁ೾೥࢓଍ଉƁƄ ƁƋƁ߁ ࢟ ſ ። ፎକଐ߁ ެࢲ କଐƁƢ। ्࢟ ࣝࡦƁŻܸࢤ ߾ ଔଐƁ ƛ ܸߢࢽ ࡦƁ߁ ዖዄ Ž૊ સၫƋࣝެ࠾ኃ ቮƁƊܸެࢲ ଘଐƁ μ0 = : μ 0H μ0 ˃ : μ 1H H0 ‫رﻓض‬ H0 ‫ﻗﺑول‬ %95   0.05 1.645 ྯ ྙཻକଐƁƛ ࢓࣍ ࡦƁƋܸަެ߽ Ɓ μ0 = : μ 0H μ0 ˃ : μ 1H ‫ ࢽ ݧ‬ხࡦƁƢƄ ܸ࣍ ࢲ ࣝެ൛ᒎᐺƐ Ƣ࢓ߣࡦƁƋܸަެ߽ Ɓ 9.5 [ n ≥ 30 ] ࠣީ‫ݕޟ‬ ࡦƁࣇ ࢲ ࣝެ଍ଉƐ Ƣ࢓ߣࡦƁƋܸަެ߽ Ɓ ‫ێ ێ‬ ࠦ୆งଘଓƄƇ Ƅ࣐࣠ உै Ƅࢇ ߙƥƢܻၮᇯ ᇍ ఴ చƓ ƥ࢖ߦࡩƄƇ ƄƎܻީޯࠀ ƄƦ ࢖༬ဈ ‫ۑ‬ቿቷ‫ࡩީ ߄ݹ‬Ƅࠦ‫ࡩߦ࢖ࢠޢ‬ƄƥƢ߄ ‫ࡩݘ‬Ƅࠦ‫ࢠ ݊ߙ࢖ࢠޢ‬ƥ .1 ūࠦၴ ࢖ஊै ƄযওߥࡩƄਔ ৲߄ ༨ဈ ƞ ߄ ᐱᏳ α ࠦ‫ݘެ࣠ݹ‬ଐଌƄƦ ࣠ޯࣀ ࡷƥƀ 1-α ࠦߥ޲ࡩƄƦ ࣠ޯࣀ ࡷਔ ৲߄ ༨ဈ .2 ‫ۑ‬ᅵै ƀ‫ۑۑۑ‬F .. ، t ، Z Ƈ ܻ‫ެ ࠒެޢ‬ଐଌࠦ‫ࡩޢ‬ƥ߄ எै ƄযওߥࡩƄ፥ ፕ ƥ ፥ ጿযঠ‫ݹ‬Ƨ ኅ ቱƄƎܻ‫ݘޢ‬ଐଌƄƥƀƇ ܻ‫ࡩީޢܻރ‬Ƅࡼ ࡷࠦ‫ ࣀ࣠ހ‬ഷവറƄࠦࡺ‫ࡩߥޢ‬ƄƧ ƀū‫ެܻ ࢵ ݪ‬ଐଌƄ፥ ገܻࢧ ࠁ ଘଓƄƅ ܻࣀ ࠁ .3 ‫ۑ‬Ƣ߄ ‫ࡩݘ‬Ƅࠦ‫ߙ ࢛ߙ࢖ࢠޢ‬Ǝƥƀơ ࣠ީߘࠢ‫ݶ‬ ࣠ࢠ ࠦࡺ‫ރࠦߘޢ‬Ǝܻߥࡷơ ൎ൉ၯ ࡼ ࡷƎƄ࢖ߥࡩƄƆ߄ ၮܻߘƢƄ߄ ႀ ޯࢵ ܻ‫ࡩݘޢެࠦހ‬Ƅ࣊ ࢵ ࣠ޯࡷơ ࣠ࠁ ƎƄ࢖ߥࡩƄƍܻ༧ဈ Ƅ .4 140 ܸୃกܸ଍ଉŽ–ໝ ແࡦफउ –ࠣ‫ݶ‬ Ɖܸࢤ ެߕକଐƁƢࠣ‫ࡦ ࢽޟܸ ࢲޟ‬ƁƢࠣ‫ޟ‬පඓ ൷Ɓ౭౑ࢲ କଐƁƄ ܸࢲ ƁƋኃ ᒔᐺ። ጏ࢓‫ݕ‬ ࡦƁ࣬ ࣊ Ɓ࢓ߢೕಹኃ ቮƁ೾೥࢓଍ଉƁƄ ƁƋƁ߁ ࢟ ſ ። ፎକଐ߁ ެࢲ କଐƁƢ। ्࢟ ࣝࡦƁŻܸࢤ ߾ ଔଐƁ ƛ ܸߢࢽ ࡦƁ߁ ዖዄ Ž૊ સၫƋࣝެ࠾ኃ ቮƁƊܸެࢲ ଘଐƁ ‫ࡩޢࠦۑ‬ƥ߄ எै Ƅࠦၴ ࢖ஊै Ƅযওߥࡩܻ‫ހ‬Ƈ ܻ‫ࡩީޢܻރ‬Ƅࡼ ࡷƅ ࣠ࣀ ഷവറƄ፥ ገܻࢧ ࠁ ଘଓƄ ‫ࡩݘޢެࠦۑ‬Ƅࡴ ᄽ஛ ƥ݊ ‫ݹ‬Ə ࣠ޯࡩƄࠦ‫ ࣍ީޢݘ‬ūࠢࡷ߄ၮࡼ ࡷ݊ ࡺޯ഻വറƄ௪ ௎୍ܻงࠦߙ࢖‫ ࠁ ࣀ ݪࡷݘ‬Ɠ ƥ࢖ߦࡩƄƇ ƄƎܻީޯࠀ Ƅߎ ࡯ޯ༧ဈ Ƨ Ǝܻ‫ݘޢ‬ଐଌƄƞ Ƅ࢖༨စ ଘଓƄ ௪ ௎ܻީޯࡩƄƣ ࣠ࡓ ‫ݑެ ߄ࡷܻݹ‬Ƈ ܻ࣐ ࢵ ࣠ޯ൞ᒑᐽƓ ƥ࢖ߦࡩƄƎܻީޯࠀ Ƅ፥ ጿ Z݊ ‫ݹ‬Ə࣠‫ކ‬Ƣ߄ ႀ ޯఱ ఓƥ Ƣ߄ႀ ޯࣀ ყߙơ ࣠ࠩ༬ீ ௪ ௎ܻީޯࡩƄƣ ᇴᆝ ƄƍƄࡼ ࡓ ᚻࡩ‫ࡩݘޢެࠦۑ‬Ƅࡴ ᄽ஛ ƣ ᇴᆝ ܻࡺᐱᚑūܻ‫ݹ ݊ ࣍ީޢݘޢ‬Ə࣠ޯࡩƄƥܻࡷॅळ‫ ࠖޯࡺ ݊ࡷݘ‬൞ᒑᐽ [ n ≥ 30 ]Ɔ౰ేީࡁࠦެ‫ࡩݘޢ‬Ƅƣ ࣠ࡓ ‫ ߙߥ ࣊ݑެ߄ࡷܻކ‬ZƎܻީޯࠀ Ƅ [ n ≥ 30 ]ࠣީ‫ݕޟ‬ ࡦƁࣇ ࢲ ࣝެ଍ଉƐ Ƣ࢓ߣࡦƁƋܸަެ߽ ƁƄ Ɓࣝ࣍ ߽ ‫ێ ێ‬ ቼቴ‫ݶ‬ ߁ ަࡦƁࠣ‫ࡦߣ࢓ ࢝ޟ‬ƁƢƟ߁ ‫ݕ‬ ࡦƁࠣ‫ߖ࢓ ࢝ޟ‬። ፒ ƢŨࠣ‫ࡦߣ࢓ ࢝ޟ‬Ɓ݇ ࢝ Ƣ  Ƣ߄ ‫ࡩݘ‬Ƅࠦ‫ߙ࢖ࢠޢ‬ ……………………… ቿቷ‫ࡩީ ߄ݹ‬Ƅࠦ‫ࡩߦ࢖ࢠޢ‬Ƅ …………….. μ0 = : μ 0H : μ ≠ μ01H ‫أو‬ μ0 ˃ : μ 1H μ0 ˃ : μ 1H Ɔƌܻၮ፥ ፕ ƥ αܾ࣐ உै Ƅࡣ ޲೘೉཯དྷࡩƄƥܻߦ࡯ࢵ ࠦ‫ݘެ࣠ݹ‬ଐଌƄƦ ࣠ޯࣀ ࡷƌ߄ ༨ဈ (α ࠣ‫ݕީࣝݶ‬ ଍ଉƁƣ ࣝެࢽ ࡴ਑ ৯߁ ༥စ ( %90 ‫ و‬%95 ، %99 ቿቷ‫މߥࠦࡷߥܻހ‬Ƈ ܻ‫ݑެ ߄ࡷ ࣀޯ࣠ݹ‬༇໩ ƀཀྵཏ‫ݘ‬೘ೃƥƀ %10 ‫ أو‬%5 ، %1  ࠦ‫ݹ‬Ǝܻ‫ݘޢ‬ଐଌƄࠦၴ ࢖ஊै Ƅࠦࡺ‫ࡩߥޢ‬Ƅਔ ৲߄ ༨ဈ Ƨ Ƅū Ƅ࢖ߥࡩƄƍܻ༧ဈ ƄƆ߄ ၮܻߘਔ ৲߄ ༨ဈ যঠ‫ݹ‬Ƅ߃ ࠧž࣠ࢠ፥ ጿƥ‫ۑ‬፥ ፑƄ࣠ޯࡩƄે ઻ၮ ࡼ ࡷƎܻީޯࠀ ൎᑉᐽ Z α - ƥ໠ ໄߙ࢖࣍ ࡼ ࡷƎܻީޯࠀ ൎᑉᐽ Z α/2 + ፥ ፕ ࠦࡺ‫ࡩߥޢ‬ƄƤ߃ ࠧƥ ࠦ‫ࡩޢ‬ƥ߄ எै Ƅ ഭ୨‫ࡩ ࣐ީޢ‬Ƅ݊ ‫ݹ‬Ə࣠ޯࡩƄơ ƥ߄ၴ Ƣ߄ ႀ ޯఱ ఝƥ‫ ࡼۑ‬೘಼ଘଓƄƞ ࢖࣐ ᑃᐽƎܻީޯࠀ ൎᑉᐽ Z α ƥྲ ྜཾଘଓƄƞ ࢖࣐ ࡩƄ ‫ࠦۑ‬ၴ ࢖ஊै ƄযওߥࡩƄે ઻ၮơ ࣠ࢧ ॶॱ५ ZƎܻ‫ݘޢ‬ଐଌƄ ࠦ‫ ࣀ࣠ހ‬ഷവറƄࠦ‫ݹ‬Ǝܻ‫ݘޢ‬ଐଌƄࠦࡺ‫ࡩߥޢ‬Ƅƥƀ‫ެܻ ࢵ ݪ‬ଐଌƄ፥ ገܻࢧ ࠁ ଘଓƄƧ ƀࠦެ‫ࡩݘޢ‬Ƅ࣊ ࢵ ࣠ޯଐଌ Zࠦ‫ݹ‬Ǝܻ‫ݘޢ‬ଐଌƄࠦࡺ‫ࡩߥޢ‬Ƅ‫ ࣀ ݪ‬༨စ ࠦ୆งଘଓƄቿቱƌܻ‫ݘ‬ଐଌƄƢƄ߄ ႀ ޯࢵ ܻ‫ހ‬ᑑ ᐽƍƥ‫ۑ‬ Z   X - 0  n ࠦࡺ‫ࡩߥޢ‬Ƅࡼ ࡷࡣ ߘƀࠦ‫ ࣀ࣠ހ‬ഷവറƄࠦ‫ݹ‬Ǝܻ‫ݘޢ‬ଐଌƄࠦࡺ‫ࡩߥޢ‬Ƅบ คᇴᆝ ƄƍƂࠦެ‫ࡩݘޢ‬Ƅ࣊ ࢵ ࣠ޯࡷơ ࣠ࠁ ƎƄ࢖ߥࡩƄƍܻ༧ဈ Ƅ  บ คᇴᆝ Ƅƍᓘƥ‫ ۑ‬H0 Ƣ߄ ‫ࡩݘ‬Ƅࠦ‫ރߥީ ࡣߙ࢖ࢠޢ‬ƥơ ࣠ީߥࡩƄࠦߥ࣐ ެࡷ፥ ጿ݊ ߥ‫ކ‬ᑑ ᐽਓ ৹ܻᐱ᏶ܺߙūࠦ‫ࡩޢ‬ƥ߄ எै Ƅ ū࢛ ߙ༟༎Ƅࠦߥ࣐ ެࡷ፥ ጿ݊ ߥ‫ކ‬ᑑ ᐽਓ ৹ܻᐱ᏶ܺߙūࠦ‫ࡩޢ‬ƥ߄ எै Ƅࠦࡺ‫ࡩߥޢ‬Ƅࡼ ࡷ౰౎ࡁ ƀࠦ‫ ࣀ࣠ހ‬ഷവറƄࠦ‫ݹ‬Ǝܻ‫ݘޢ‬ଐଌƄࠦࡺ‫ࡩߥޢ‬Ƅ ( H1 ቿቷ‫ࡩީ ߄ݹ‬Ƅࠦ‫ࡩߦ࢖ࢠޢ‬Ƅࡣ ީߥ‫ރ‬ƥƢ߄ ‫ࡩݘ‬Ƅࠦ‫ߙ ࢛ߙ࢖ࢠޢ‬मएƥ 141 ܸୃกܸ଍ଉŽ–ໝ ແࡦफउ –ࠣ‫ݶ‬ Ɖܸࢤ ެߕକଐƁƢࠣ‫ࡦ ࢽޟܸ ࢲޟ‬ƁƢࠣ‫ޟ‬පඓ ൷Ɓ౭౑ࢲ କଐƁƄ ܸࢲ ƁƋኃ ᒔᐺ። ጏ࢓‫ݕ‬ ࡦƁ࣬ ࣊ Ɓ࢓ߢೕಹኃ ቮƁ೾೥࢓଍ଉƁƄ ƁƋƁ߁ ࢟ ſ ƛ ܸߢࢽ ࡦƁ߁ ዖዄ Ž૊ સၫƋࣝެ࠾ኃ ቮƁƊܸެࢲ ଘଐƁ ። ፎକଐ߁ ެࢲ କଐƁƢ। ्࢟ ࣝࡦƁŻܸࢤ ߾ ଔଐƁ ( n ˂ 30 ࠣީ‫ݕޟ‬ ࡦƁࣇ ࢲ ࣝެ଍ଉƐ Ƣ࢓ߣࡦƁƋܸަެ߽ Ɓ ‫ێ ێ‬ ݊ ‫ݹ‬Ə࣠‫ކ‬ƥƍࠦެ‫ࡩݘޢ‬ƄࠢެࡷƇ ߃ ၯ Ƅ཯དྷࡩƄ݊ ࡺޯ഻വറƄƥƆƌ࢖ߦࡷ n˂30 ‫ ۑۑ‬30ࡼ ࡷࡣ ߘƄࠦެ‫ࡩݘޢ‬Ƅƣ ࣠ࡓ ‫ݑެ ߄ࡷܻކ‬ ‫ۑ‬t Ǝܻީޯࠀ ƄƢ߄ ႀ ޯఱ ఝơ ࣠ࠩ༬ீ Ƨ Ǝܻ‫ݘޢ‬ଐଌƄƞ Ƅ࢖༨စ ଘଓƄ ௪ ௎ܻީޯࡩƄƥ୍ܻ෺࢖ߥ‫ކ‬ഭ ୨‫࣍ީޢ‬ t Ƌܸަެ߽ ƁƄ Ɓࣝ࣍ ߽  ቿቷ‫ࡩީ ߄ݹ‬Ƅࠦ‫ࡩߦ࢖ࢠޢ‬ƄƥƢ߄ ‫ࡩݘ‬Ƅࠦࢠ࢖ߙ፥ ፕ ƥࠦ‫ࡩߦ࢖ࢠޢ‬Ƅ݊ ࢠƥ  ࠦߦ࡯ޯശവറƄࠦ‫ݘެ࣠ݹ‬ଐଌƄƇ ܻ‫ࡷ ࣀޯ࣠ݹ‬໠ ໄ‫ ࡷ ࣀީߥܻࡷ ࡼހ‬α ܾ࣐ உै Ƅࡣ ޲೘೉Ƨ ኅ ቱƄƥ α ࠦ‫ݘެ࣠ݹ‬ଐଌƄƦ ࣠ޯࣀ ࡷਔ ৲߄ ༨ဈ ، %99 ፥ ፕ ƥ 1-α ) ቿቷ‫ߥܻހ‬ଐଌƄࠦߥ޲ࡩƄƇ ܻ‫ࡷ ࣀޯ࣠ݹ‬ƣ ܾߙ༇໩ Ƅཀྵཏ‫ݘ‬೘ೃƥƀ %10 ‫ او‬%5 ، %1 ) ࠦၴ ࢖ஊै Ƅࠦࡺ‫ࡩߥޢ‬Ƅਔ ৲߄ ༨ဈ Ƨ ƀūƎƄ࢖ߥࡩƄƍܻ༧ဈ ƄƆ߄ ၮܻߘƌ߄ ༨စ ਔ ৲߄ ႁ ޯࡩƄƄ߃ ࠧž࣠ࢠ፥ ጿƥ %90 ‫ و‬%95 Ǝܻީޯࠀ ൎᑉᐽ ( tα + ، tα - ƥ໠ ໄߙ࢖࣍ ࡼ ࡷƎܻީޯࠀ ൎᑉᐽ tα/2 ± ) ፥ ፕ ࠦࡺ‫ࡩߥޢ‬ƄƤ߃ ࠧƥࠦ‫ࡩޢ‬ƥ߄ எै Ƅࠦ‫ݹ‬Ǝܻ‫ݘޢ‬ଐଌƄ ࠦၴ ࢖ஊै ƄযওߥࡩƄે ઻ၮơ ࣠ࢧ ॶॱ५ t ơ ƥ߄ ၴ Ƣ߄ ႀ ޯఱ ఝƥ‫ۑ‬፥ ፑƄ࣠ޯࡩƄે ઻ၮࡼ ೘಼ଘଓƄƞ ࢖࣐ ࡩƄƥྲ ྜཾଘଓƄ ƞ ࢖࣐ ࡩƄࡼ ࡷ ‫ (ۑ‬df = n-1 )ࠦ‫ݹ‬༇໪ Ƈ ܻၴ Ǝƌ߄ ެ‫ݑ‬ ቿቱƌܻ‫ݘ‬ଐଌƄƢƄ߄ ႀ ޯࢵ ܻ‫ހ‬ᑑ ᐽƍƥ ࠦ‫ ࣀ࣠ހ‬ഷവറƄࠦࡺ‫ࡩߥޢ‬Ƅ ࠦެ‫ࡩݘޢ‬Ƅ࣊ ࢵ ࣠ޯଐଌ t ࠦ‫ݹ‬Ǝܻ‫ݘޢ‬ଐଌƄࠦࡺ‫ࡩߥޢ‬Ƅƅ ܻࣀ ࠁ ࠦ୆งଛଓƄ t  X  S n ‫ۑ‬ቿቱ࣠ࠩஎᆋ ƣ ࣠ࡓ ‫ܻކ‬ᐱᐢࡺ‫ߘޢ‬ƣ ଘଓƧ Ǝܻ‫ݘޢ‬ଐଌƄƞ Ƅ࢖༨စ ଘଓƄƅ ܻࣀ ஊै ࠦެ‫ࡩݘޢ‬Ƅࡼ ࡷƆƍ࣠ࠀ ܾଐଌƄƇ ܻ‫ࡩީޢܻރ‬ƄƢ߄ ႀ ޯఱ ఝƥ ፥ ጿࠦ‫ ࣀ࣠ހ‬ഷവറƄࠦࡺ‫ࡩߥޢ‬Ƅ‫ߘݘ ݝ‬ƥƄƍƄūƎƄ࢖ߥࡩƄƆ߄ ၮܻߘƢƄ߄ ႀ ޯࢵ ܻ‫ࡩݘޢެࠦހ‬Ƅ࣊ ࢵ ࣠ޯࡷơ ࣠ࠁ ƎƄ࢖ߥࡩƄƍܻ༧ဈ ƀ  ‫୐ܻރߥީ࡯ܻࠩۑ‬คܾߙᑑ ᐽኅ ቱܻߙൎ൉ၯ ƥƢ߄ ‫ࡩݘ‬Ƅࠦ‫ߙ ࢛ߙ࢖ࢠޢ‬मएū࢛ ߙ༟༎Ƅࠦߥ࣐ ެࡷ [ n ≥30 ƃ౭ౄަ࠾ࠣީ‫ ࠓެࡷ ݇] ݎޟ‬൛ᒎᐺƃ߁ ၭ ƁƢࠣަࢽ ხࡦƐ Ƣ࢓ߣࡦƁƋܸަެ߽ Ɓ ‫ێ ێ‬ Ƈ ܻ࣐ ࢵ ࣠ޯ൞ᒑᐽƓ ƥ ࢖ߦࡩƄƎܻީޯࠀ ଘଓࠦ‫ޯީݘ‬ଐଌƄƇ Ƅ࣐࣠ உै Ƅࡼ ‫ ࣀ ݪݑ‬ჱᑃᐽƓ ƥ࢖ߦࡩƄƎܻީޯࠀ ƄƇ Ƅ࣐࣠ ࠀ ߎ ࡯ޯ༧ဈ ଘଓ Ƨ Ǝܻ‫ݘޢ‬ଐଌƄܾ࣐ உै ƄƥƧ Ǝܻ‫ݘޢ‬ଐଌƄƞ Ƅ࢖༨စ ଘଓƄƅ ܻࣀ ࠁ ࠦߥ‫ ࣍࢖ݹ‬ƞ ൎ൉ޯࠀ Ƅžܻެ޲ჴࢵ ܻ‫ހ‬ūƆ౰ేީࡁࠦެ‫ࡩݘޢ‬Ƅƣ ࣠ࡓ ‫ݑެ ߄ࡷܻކ‬ ƣ ƀƓ Ƅ౰౔ߙܻ‫ހ‬ƥ‫ ۑ‬P̅ ୷ ୪༟༎ܻ‫ࡩݘޢެࠦހ‬Ƅࠦީࣀ ჱࡩ୷ ୪मअܻࡺެყჅ P ୷ ୪༟༎ܻ‫ޯࡺ ݊ހ‬഻വറƄࠦީࣀ ჱࡩ୷ ୪मअƥ‫ ࣀީࠦۑ‬ჱᑃᐽ ‫ۑ‬ഭ ୨‫ݹ ݊ ࣍ީޢ‬Ə࣠‫ࠧ࣠ކ‬Ɔ౰ేީࡓ ᚻࡩƄƇ ܻެ‫ݘޢ‬ᑃᐽࠦީࣀ ჱᑃᐽࠦ୐෺ܻ‫ݘ‬ଐଌƄ݊ ‫ݹ‬Ə࣠‫ކ‬ ࠦ‫ࡩޯܻࡩޢ‬Ƅቿቱƌܻ‫ݘ‬൞ᒑᐽܻߥߙƥࠦީࣀ ჱᑃᐽƧ Ǝܻ‫ݘޢ‬ଐଌƄܾ࣐ உै Ƅ‫ ࣀ ݪ‬༨࿻ ƥ 142 ܸୃกܸ଍ଉŽ–ໝ ແࡦफउ –ࠣ‫ݶ‬ Ɖܸࢤ ެߕକଐƁƢࠣ‫ࡦ ࢽޟܸ ࢲޟ‬ƁƢࠣ‫ޟ‬පඓ ൷Ɓ౭౑ࢲ କଐƁƄ ܸࢲ ƁƋኃ ᒔᐺ። ጏ࢓‫ݕ‬ ࡦƁ࣬ ࣊ Ɓ࢓ߢೕಹኃ ቮƁ೾೥࢓଍ଉƁƄ ƁƋƁ߁ ࢟ ſ ƛ ܸߢࢽ ࡦƁ߁ ዖዄ Ž૊ સၫƋࣝެ࠾ኃ ቮƁƊܸެࢲ ଘଐƁ Sp̅ = ට ። ፎକଐ߁ ެࢲ କଐƁƢ। ्࢟ ࣝࡦƁŻܸࢤ ߾ ଔଐƁ ௣(ଵି௣) ௡ ࣊ ࢵ ࣠ޯଐଌƓ ƥ࢖ߦࡩƄƎܻީޯࠀ ଘଓࠦ‫ޯީݘ‬ଐଌƄƇ Ƅ࣐࣠ உै Ƅࡼ ‫ ࣀ ݪݑ‬ჱᑃᐽƓ ƥ࢖ߦࡩƄƎܻީޯࠀ ƄƇ Ƅ࣐࣠ ࠀ ߎ ࡯ޯ༧ဈ ଘଓ ‫ࡩݘޢެࠦۑ‬Ƅ Zp̅ = p̅ - p/ Sp̅ Zp̅ = ࠦ‫ࡩޯܻࡩޢ‬Ƅቿቱƌܻ‫ݘ‬ଐଌƄ‫ ࠁ ࣀ ݪ‬Z ࠦ‫ݹ‬Ǝܻ‫ݘޢ‬ଐଌƄࠦࡺ‫ࡩߥޢ‬Ƅƅ ܻࣀ ࠁ ௣ҧ ି௣ ට ೛(భష೛) ೙ ࠦީࣀ ჱࡩƄࢪ ყࡩƥࠦީࣀ ჱᑃᐽƧ Ǝܻ‫ݘޢ‬ଐଌƄܾ࣐ உै Ƅƅ ܻࣀ ࠁ ፥ ጿ݊ ࡺޯ഻വറƄࡼ ࡷࠦީࣀ ჱࡩƄƢ߄ ႀ ޯఱ ఝܻ୐คƀࣉ ࠁ ൎ൉‫ݹ‬ƥ  Ƨ Ǝܻ‫ݘޢ‬ଐଌƄܾ࣐ உै Ƅƅ ܻࣀ ࠁ ߄ ެ‫ࡩޯ ࢧ ࠒޢ ߰ݑ‬Ƅࡣ ࡷܻ‫ࡷݘ‬ƢƄ߄ ႀ ޯࢵ Ƅ໠ ໄ‫୓ݘ‬෺ࠢ‫ ࠁ ࣉހܾރ‬ൎ൉‫ݹ‬ሯሡሁ ‫ࡩݘޢެࠦۑ‬Ƅࡼ ࡷ ‫ޯࡺ ݊ۑ‬഻വറƄࡼ ࡷ %5ࠦީఱ ఝࠦެ‫ࡩݘޢ‬Ƅࡴ ᄽ஛ Əƥܻ༬ဈ ƄƍƄ ‫ ࢽ ݧ‬ხࡦƁƢࣇ ࢲ ࣝެ൛ᒎᐺƐ Ƣ࢓ߣࡦƁƋܸަެ߽ Ɓૄ સၫƄ ܸߢ‫ێ ރ ࣍ަޟ‬ ƢƄ୹ ‫ ݌‬100Ƈ Ƅ࣠ީ‫ݑ‬፥ ጿࠦ‫ކ‬ଘଓ࣠ࡁ ࣠ࣃ ࡩƄࡼ ࡷƖ ࣠‫ރ‬Ɖ ܻ୓คܺᄣࠦ୆෷Ƅ߃ ‫ݓ‬ ࡩƄƌƄ࣠൞ᒑᐽበሼྲ ႖Ƣ࣠ߥ‫ ێ ێ (ކ‬ƞ ܸޯࡴ ࢖࣏ ެᑃᐽƣ Ə࣠ࡩƄƐ ܻ‫ࡩߥޢ‬Ɔ࣠ީ‫ݑ‬36ࡼ ࡷࠦ୆෷Ƅ࣠ࣃ ‫ ߃ݑޢެࠦݑ‬ၯ Ƅበሼྲ ႡࡩƄ፥ ጿƇ ܻࢵ ƄƎኆ ቱƄƆ߄ ၰ ƥ‫ߘܻࡷ ݝ‬Ƅƍܻߙ‫ۑ‬ ݊ ࡺޯࠖ ൞ᒑᐽƧ Ǝܻ‫ݘޢ‬ଐଌƄƞ Ƅ࢖༨စ ଘଓƄƣ ƀƓ ࢖ߦ‫ހ‬ƥūࡼ ࡯‫ݘ‬ଐଌƄƣ Ə࣠ࡩƄࡼ ‫ޯ࡯ ߎݑ‬༧࿻ Ƈ Ƅ࣠ީ‫ࡩݘ‬Ƅƣ Əƥƣ ᇴᆝ ƄƍƄܻࡺ‫ߙޢ‬ %95ࠦߥ‫މ‬Ʀ ࣠ޯࣀ ࡷƢƄ߄ ႀ ޯࢵ Ƅ‫ۑ‬ƢƄ୹ ‫ ݌‬110ࠦެ‫ࡩݘޢ‬Ƅࡼ ࡷƣ Ə࣠ࡩƄ࣊ ࢵ ࣠ޯࡷƣ ᇴᆝ Ƅƍܻߙ‫ࠧ࣠ ۑ ݌ ࡴۑ‬ ‫ݘ࡯ ࡼۑ‬ଐଌƄ࣊ ࢵ ࣠ޯଐଌƄࡼ ‫ޯ࡯ ߎݑ‬༧࿻ ࠦެ‫ࡩݘޢ‬Ƅ࣊ ࢵ ࣠ޯࡷƣ ܾ‫ࡩߦ࢖ࢠޢࠦހ‬ƄƎܻީޯࠀ ଘଓ ࡠ இॅ Ɓ H0 : μ = 100 H1 : μ ≠ 100 ࠣ‫ࡦߣ࢓ ࢝ޟ‬Ɓ݇ ࢝ Ƣ  ્ ઻‫ࡩߦݘ‬Ƅ࣊ ࢵ ࣠ޯଐଌƄƣ Ƅે ઻ၮቿቷ‫ࡩީ ߄ݹ‬Ƅࠦ‫ࡩߦ࢖ࢠޢ‬Ƅ߄ ެჴఱ ఠ‫ ࠁޢ ݟ‬ቿቷ‫ࡩީ ߄ݹ‬Ƅࠦ‫ࡩߦ࢖ࢠޢ‬ƄƥƢ߄ ‫ࡩݘ‬Ƅࠦ‫ߙ࢖ࢠޢ‬ਔ ৲߄ ༨ဈ Ꭸ᎙ 143 ࡼ ࡯‫ݘ‬ଐଌƄ࣊ ࢵ ࣠ޯଐଌƄࡼ ‫ݑ‬ਔ ৲बअƥƄࡣ ߥ‫ߘ ߄ݹ‬Ɔ࣠ީ‫ࡩݘ‬Ƅƣ Ə࣠ࡩ ܸୃกܸ଍ଉŽ–ໝ ແࡦफउ –ࠣ‫ݶ‬ Ɖܸࢤ ެߕକଐƁƢࠣ‫ࡦ ࢽޟܸ ࢲޟ‬ƁƢࠣ‫ޟ‬පඓ ൷Ɓ౭౑ࢲ କଐƁƄ ܸࢲ ƁƋኃ ᒔᐺ። ጏ࢓‫ݕ‬ ࡦƁ࣬ ࣊ Ɓ࢓ߢೕಹኃ ቮƁ೾೥࢓଍ଉƁƄ ƁƋƁ߁ ࢟ ſ ። ፎକଐ߁ ެࢲ କଐƁƢ। ्࢟ ࣝࡦƁŻܸࢤ ߾ ଔଐƁ ƛ ܸߢࢽ ࡦƁ߁ ዖዄ Ž૊ સၫƋࣝެ࠾ኃ ቮƁƊܸެࢲ ଘଐƁ ‫ اي‬، %95ࠦߥ‫މ‬Ʀ ࣠ޯࣀ ࡷ߄ ެ‫ݑ‬Zࠦ‫ݹ‬Ǝܻ‫ݘޢ‬ଐଌƄࠦ‫ࡩޢ‬ƥ߄ எै Ƅࠦࡺ‫ࡩߥޢ‬ƄƧ Ƅū ƋƁ࢓ߢࡦƁƊܸ༤စ Žƃ߁ ၫܸߕƉ߁ ༥ဂ  Z = ( Zࠣ‫ݽ‬ ࣝࢽ ഴലമƁࠣࡷ‫ࡦߢޟ‬Ɓ Ƥ Ɓ Zࠣ‫ݶ‬Ƌܸ‫ݕޟ‬ ଍ଉƁࠣࡷ‫ࡦߢޟ‬ƁƂ ܸࢽ ߾  ୶ҧ ିஜ ఙೣҧ Z  ‫ ۑ‬±፥ ፕ Zα/2ࠦࡺ‫ߘޢ‬ X - 0 ‫ߪ( ێێێێ‬௫ҧ= ߪ/√݊ )  Z = (110 – 100 ) n _________ = ߪ/√݊ (53 – 50 ) _________ 10.5/√36 = (3 x 6) _______ = 18 / 10.5 = 1.7 10.5 1. 144 ܸୃกܸ଍ଉŽ–ໝ ແࡦफउ –ࠣ‫ݶ‬ Ɖܸࢤ ެߕକଐƁƢࠣ‫ࡦ ࢽޟܸ ࢲޟ‬ƁƢࠣ‫ޟ‬පඓ ൷Ɓ౭౑ࢲ କଐƁƄ ܸࢲ ƁƋኃ ᒔᐺ። ጏ࢓‫ݕ‬ ࡦƁ࣬ ࣊ Ɓ࢓ߢೕಹኃ ቮƁ೾೥࢓଍ଉƁƄ ƁƋƁ߁ ࢟ ſ ። ፎକଐ߁ ެࢲ କଐƁƢ। ्࢟ ࣝࡦƁŻܸࢤ ߾ ଔଐƁ ƛ ܸߢࢽ ࡦƁ߁ ዖዄ Ž૊ સၫƋࣝެ࠾ኃ ቮƁƊܸެࢲ ଘଐƁ ࠦ‫୐ܻރߥީ ࡣߙ࢖ࢠޢ‬คܾߙ ‫ࡩޢࠦ ۑ‬ƥ߄ எै Ƅࠦࡺ‫ࡩߥޢ‬Ƅࡼ ࡷࡣ ߘƀ ( ‫ = ۑ‬Z )ࠦ‫ ࣀ࣠ހ‬ഷവറƄࠦࡺ‫ࡩߥޢ‬Ƅƣ ƀܻ೘ೃ ܻያዪ‫ޢ ߰ۑ‬༨༦ ౰ేၩࡼ ࡯‫ݘ‬ଐଌƄ࣊ ࢵ ࣠ޯଐଌƄƣ Ƅે ઻ၮൎ൉‫ࡩޢ‬ƌƢ߄ ߥ‫ކ‬ଘଓƆ࢖ߙ࣠ޯଐଌƄƇ ܻ‫ࡩީޢܻރ‬Ƅƣ ƀ‫ (ۑ ࠁޢ ݟ‬H0 Ƣ߄ ‫ࡩݘ‬Ƅ ‫ܻ ۑ‬୆෷ܻࢧ ࠁ Ƃơ Ƅƌ౰ేၩƥ ࠦߙ߄ ࢧ ᑃᐽ݊ ࠅ मअܻ೘ೆƀūࡼ ࡯‫ݘ‬ଐଌƄ࣊ ࢵ ࣠ޯଐଌƄƥࠦެ‫ࡩݘޢ‬Ƅ࣊ ࢵ ࣠ޯࡷ໠ ໄ‫ހ‬Ɵ ࢖ߦࡩƄƣ ƀ౰ేఴ ఓ Ƅƍܻߙū 2000Ƣܻၮ݅ ࡁ 14ƣ Ƅኆ ቷީࡩƄ߄ ၰ Ƅ፥ ጿƢ࣠ॶॱ५ƄƠ ൎ൉ᐱᐢࢵ Ƅ࣊ ࢵ ࣠ޯࡷƣ ᇴᆝ ƄƍƂ ( ‫ ێ ێ‬ƞ ܸޯࡴ Ƣ࣠ॶॱ५ƄƠ ൎ൉ᐱᐢࢵ Ƅ࣊ ࢵ ࣠ޯࡷƣ Ƅࠦެ‫ࡩݘޢ‬Ƅࡼ ࡷ߄ ၴ ƥƥ 2010Ƣܻၮ፥ ጿƄƌ࢖ߙ 49ࡼ ࡷࠦ୆෷Ƅ࣠ࣃ ‫ݑޢެࠦݑ‬Ƈ ߃ ၯ ƀ ࣊ ࢵ ࣠ޯࡷƣ ܾ‫ހ‬ቿቷ‫ݶ‬ ܻߥࡩƄࠦ‫ࡩߦ࢖ࢠޢ‬Ƅ౰౎ޯࠀ ƀ‫ ࡁ ݅ۑ‬6࣠ࠧƧ Ǝܻ‫ݘޢ‬ଐଌƄƞ Ƅ࢖༨စ ଘଓƄƥ݅ ࡁ 16ࠦެ‫ࡩݘޢ‬Ƅࡼ ࡷƌ࢖ߦᑃᐽ ‫ۑ‬%95ࠦߥ‫މ‬Ʀ ࣠ޯࣀ ࡷ߄ ެ‫ݑ‬ᑑ ᐽƍƥ 2000Ƣܻ‫ރࠦހݘ‬Ǝܻߥࡷ 2010Ƣܻၮ݊ ߦ‫ކ‬ƎƄ߄ ߘƢ࣠ॶॱ५Ơ ൎ൉ᐱᐢࢵ ଘଓƄ ࡠ இॅ Ɓ ࠣ‫ࡦߣ࢓ ࢝ޟ‬Ɓ݇ ࢝ Ƣ  H0 : μ = 14 H1 : μ ˃ 14 (1.64= Zα )ࠦࡺ‫ߘޢ‬Ƨ Ƅū %95ࠦߥ‫މ‬Ʀ ࣠ޯࣀ ࡷ߄ ެ‫ݑ‬Zࠦ‫ݹ‬Ǝܻ‫ݘޢ‬ଐଌƄࠦ‫ࡩޢ‬ƥ߄ எै Ƅࠦࡺ‫ࡩߥޢ‬Ƅ Z X - 0 ( Zࠦ‫ ࣀ࣠ހ‬ഷവറƄࠦࡺ‫ࡩߥޢ‬Ƅ Ƨ Ƅ Zࠦ‫ݹ‬Ǝܻ‫ݘޢ‬ଐଌƄࠦࡺ‫ࡩߥޢ‬Ƅƅ ܻࣀ ࠁ  n Z = (16 – 14 ) _________ 6/√49 = 145 2X7 ________ 6 = 14/6 = 2.33 ܸୃกܸ଍ଉŽ–ໝ ແࡦफउ –ࠣ‫ݶ‬ Ɖܸࢤ ެߕକଐƁƢࠣ‫ࡦ ࢽޟܸ ࢲޟ‬ƁƢࠣ‫ޟ‬පඓ ൷Ɓ౭౑ࢲ କଐƁƄ ܸࢲ ƁƋኃ ᒔᐺ። ጏ࢓‫ݕ‬ ࡦƁ࣬ ࣊ Ɓ࢓ߢೕಹኃ ቮƁ೾೥࢓଍ଉƁƄ ƁƋƁ߁ ࢟ ſ ። ፎକଐ߁ ެࢲ କଐƁƢ। ्࢟ ࣝࡦƁŻܸࢤ ߾ ଔଐƁ ƛ ܸߢࢽ ࡦƁ߁ ዖዄ Ž૊ સၫƋࣝެ࠾ኃ ቮƁƊܸެࢲ ଘଐƁ H0 ‫رﻓض‬ H0 ‫ﻗﺑول‬ %95   0.05 1.645 2.33 ࢛ ߙमएū Z = 1.645 ࠦ‫ࡩޢ‬ƥ߄ எै Ƅࠦࡺ‫ࡩߥޢ‬Ƅࡼ ࡷ౰౎ࡁ Ƅ ‫ = ۑ‬Z ࠦ‫ ࣀ࣠ހ‬ഷവറƄࠦࡺ‫ࡩߥޢ‬Ƅƣ ƀܻ೘ೃ ƣ ܾ‫ހ‬౰ేఴ ఠƆ࢖ߙ࣠ޯଐଌƄƇ ܻ‫ࡩީޢܻރ‬Ƅƣ ƀ‫ ۑ ࠁޢ ݟ‬H1 ቿቷ‫ࡩީ ߄ݹ‬Ƅࠦ‫ࡩߦ࢖ࢠޢ‬Ƅࡣ ީߥ‫ރ‬ƥ H0 Ƣ߄ ‫ࡩݘ‬Ƅࠦ‫ߙ࢖ࢠޢ‬ ࣊ ࢵ ࣠ޯࡷƥࠦެ‫ࡩݘޢ‬Ƅ࣊ ࢵ ࣠ޯࡷ໠ ໄ‫ހ‬Ɵ ࢖ߦࡩƄƥ‫ۑ‬2000Ƣܻ‫ރࠦހݘ‬Ǝܻߥࡷ౰ే‫ݓ‬ ‫ ކ‬2010Ƣܻၮƌ࢖ߦᑃᐽƢ࣠ॶॱ५ƄƠ ൎ൉ᐱᐢࢵ Ƅ ‫ۑ‬Ƨ ࢖ࠧ࣠ࠅ ݊ ࡺޯ഻വറƄ ƆƎܻ‫ ࢵޢ‬፥ ፕ ݊ ‫ީޢ‬ᑃᐽܻࠩࠁ ࢖࣐ ‫཯ݹ‬དྷࡩƄƆƎܻ‫ࡩ ࣀޢ‬Ƅƣ ܾ‫ހ‬ƣ ൎ൉ၮƄ፥ ጿƇ ƄƎܻ‫ ࡣ ࢵޢ‬૮૓ƥ፥ ፎ ਔ ৲ ( ‫ ێ ێ‬ƞ ܸޯࡴ Ƈ ൎ൉‫࡯ޢ‬ඳඣ ൴ᑻ ᑧᐱᐢࣀ ଐଌƄࠦ‫ܻݹ‬ዙ዇ ࠦ‫ݘޢ‬ዙዋ ‫ߘ ߄ߘܻࡷ ݝ‬ƥ‫ۑ‬௪ ௎౮౑‫ހ‬౰౔ࡩᇯ ᇍ ࡩ /Ꭸ፫ 12࣊ ࢵ ࣠ޯଐଌܻ‫ހ‬౰ేఱ ఠƥࠦ‫ݹ‬ƌܻࢧ ޯߘƄ Ƈ ߄ ၴ ƥƥƇ ƄƎܻ‫ࡩ ࣀޢ‬Ƅࡣ ૮૓ƥžܻၮƌƄࠦ༨༦ ፥ ጿ࢖࣏ ެᑃᐽƆƎܻ‫ ࢵޢ‬30ࡼ ࡷࠦ୆෷Ƅ࣠ࣃ ‫ݑޢެࠦݑ‬ƢƄ߄ ႀ ޯࢵ ܻ‫ࠦހ‬୆෷ܻࢧ ࠁ Ƅ Ǝܻ‫ݘޢ‬ଐଌƄƞ Ƅ࢖༨စ ଘଓƄƥ௪ ௎౮౑‫ހ‬౰౔ࡩᇯ ᇍ ࡩ /Ꭸ፫ 11፥ ፕ ƆƎܻ‫ࡩ ࣀޢ‬Ƅܻࠩ‫཯ކߥ ࣐ݘ‬དྷࡩƄࠦߙܻࣀ ଐଌƄ࣊ ࢵ ࣠ޯࡷƣ ƀࠦެ‫ࡩݘޢ‬Ƅࡼ ࡷ ‫ۑ‬౰౔ࡩ / 5 ‫ي‬ ࡣ ૮૓࣠ࡩƄࠢ‫ݑޢ‬ਔ ৲ܻያዪࡣ ߘƀ࣠ࠧƆƎܻ‫ࡩ ࣀޢ‬Ƅܻࠩ‫཯ކߥ ࣐ݘ‬དྷࡩƄࠦߙܻࣀ ଐଌƄ࣊ ࢵ ࣠ޯࡷƣ ܾ‫ހ‬ቿቷ‫ݶ‬ ܻߥࡩƄࠦ‫ࡩߦ࢖ࢠޢ‬Ƅ౰౎ޯࠀ ƀ  ‫ۑ‬%90ࠦߥ޲ࡩƄƦ ࣠ޯࣀ ࡷƢƄ߄ ႀ ޯࢵ ܻ‫ހ‬ 146 ܸୃกܸ଍ଉŽ–ໝ ແࡦफउ –ࠣ‫ݶ‬ Ɖܸࢤ ެߕକଐƁƢࠣ‫ࡦ ࢽޟܸ ࢲޟ‬ƁƢࠣ‫ޟ‬පඓ ൷Ɓ౭౑ࢲ କଐƁƄ ܸࢲ ƁƋኃ ᒔᐺ። ጏ࢓‫ݕ‬ ࡦƁ࣬ ࣊ Ɓ࢓ߢೕಹኃ ቮƁ೾೥࢓଍ଉƁƄ ƁƋƁ߁ ࢟ ſ ƛ ܸߢࢽ ࡦƁ߁ ዖዄ Ž૊ સၫƋࣝެ࠾ኃ ቮƁƊܸެࢲ ଘଐƁ ። ፎକଐ߁ ެࢲ କଐƁƢ। ्࢟ ࣝࡦƁŻܸࢤ ߾ ଔଐƁ ࡠ இॅ Ɓ ࠦ‫ࡩߦ࢖ࢠޢ‬Ƅ݊ ࢠƥ  H0 : μ = 12 H1 : μ ˂ 112 ે ઻ၮ݊ ߥ‫཯ކ‬དྷࡩƄ ( Zα )ࠦࡺ‫ߘޢ‬Ƨ Ƅū %90ࠦߥ‫މ‬Ʀ ࣠ޯࣀ ࡷ߄ ެ‫ݑ‬Zࠦ‫ݹ‬Ǝܻ‫ݘޢ‬ଐଌƄࠦ‫ࡩޢ‬ƥ߄ எै Ƅࠦࡺ‫ࡩߥޢ‬Ƅ  ( Z = -1.28 ፥ ፕ ࠦࡺ‫ࡩߥޢ‬ƄƤ߃ ࠧƥࠦၰ ܻࣀ ଐଌƄࡼ ࡷ %10ܻࠧƎܻఱ ఓ ( Zࠦ‫ ࣀ࣠ހ‬ഷവറƄࠦࡺ‫ࡩߥޢ‬Ƅ Ƨ Ƅ Zࠦ‫ݹ‬Ǝܻ‫ݘޢ‬ଐଌƄࠦࡺ‫ࡩߥޢ‬Ƅƅ ܻࣀ ࠁ Z  X - 0  n Z = (12 – 11) 5/√30 = (-1) X (√30 ) 5 = - 5.477 / 5 = -1.10 Ƣ߄ ‫ࡩݘ‬Ƅࠦ‫ރߥީ ࡣߙ࢖ࢠޢ‬ūơ ࣠ީߥࡩƄࠦߥ࣐ ެࡷ፥ ጿ݊ ߥ‫ ۑ ކ‬- = z ࠦ‫ ࣀ࣠ހ‬ഷവറƄࠦࡺ‫ࡩߥޢ‬Ƅƣ ƀܻ೘ೃ  ࡼ ࡯‫ݘ‬ଐଌƄ࣊ ࢵ ࣠ޯଐଌƄƣ ƀે ઻ၮൎ൉‫ࡩޢ‬ƌƢ߄ ߥ‫ކ‬ଘଓƆ࢖ߙ࣠ޯଐଌƄƇ ܻ‫ࡩީޢܻރ‬Ƅƣ ƀ‫ ࠁޢ ݟ‬ū H1 ࢛ ߙमएƥ H0 ‫ࡩ ࢧ ߄ߙࠦۑ‬Ƅ፟ ፑƄ݊ ࠅ मअܻ೘ೆƄࡼ ࡯‫ݘ‬ଐଌƄ࣊ ࢵ ࣠ޯଐଌƄƥࠦެ‫ࡩݘޢ‬Ƅ࣊ ࢵ ࣠ޯࡷ໠ ໄ‫ހ‬Ɵ ࢖ߦࡩƄƣ ƀ౰ేఴ ఓܻያዪ‫ޢ ߰ۑ‬༨༦ ౰ేၩ ‫ ࡣۑ‬૮૓࣠ࡩƄžܻၮƌƄ݊ ࡷࢇ ߦ୓ขƆƎܻ‫ࡩ ࣀޢ‬Ƅܻࠩ‫཯ކߥ ࣐ݘ‬དྷࡩƄࠦߙܻࣀ ଐଌƄƣ Ƅƥ Ƅ߃ ಛಗƢ߄ ߥ‫ކ‬Ƅƍܺߙ‫ۑ‬115࣊ ࢵ ࣠ޯ೘ೃഭ ୨‫ࡩ ࣐ީޢ‬Ƅ݊ ‫ݹ‬Ə࣠ޯࡩƄ݊ ީჴႾ IQ žᇴᆝኅ ቱƄƎܻީޯࠀ Ƅ ( ‫ ێ ێ‬ƞ ܸޯࡴ Ƨ ƥܻఱ ఓƎܻީޯࠀ ଘଓƄƄ߃ ᐱᏳࡴ ᐱᏹܻၴ Ǝƌ࣊ ࢵ ࣠ޯࡷบ คᇴᆝƥࠦެ‫ ࢵࠦࡷݘޢ‬Ǝ߄ ೘ೃܻީࡩܻ࣍ 25ࡼ ࡷƣ ࣠ࡓ ୓ขࠦެ‫ݑޢ‬Ǝܻީޯࠀ ଘଓƄ ࠦࡷܻၮࠦߦࢧ ‫ࡩ ࣐࡯ީࠦހ‬ƄƇ ܻၴ Ǝƌ࣊ ࢵ ࣠ޯࡷƣ ܾ‫ހ‬ơ ࣠ߥࡩƄࡼ ࡓ ೘಼ࡣ ࠧƇ ܻ‫ࡩީޢܻރ‬ƄƤ߃ ࠧࡼ ࡷ‫ۑ‬100௪ ௎୍ܻงƥ 120 ‫ۑ‬%5ࠦ‫ࡷݘެ࣠ݹ‬Ʀ ࣠ޯࣀ ࡷ߄ ެ‫ݑ‬Ƣܻ‫ࡩݘ‬Ƅ࣊ ࢵ ࣠ޯଐଌƄࡼ ‫ޯ࡯ ߎݑ‬༧࿻ ࠦࢵ Ǝ߄ ଐଌƄƤ߃ ࠧ፥ ጿ ࡠ இॅ Ɓ t ݊ ‫ݹ‬Ə࣠‫ ކ‬Ƣ߄ ႀ ޯఱ ఝܻ୐คܾߙ‫ۑۑ‬ơ ࣠ࠩ༬ீ ݊ ࡺޯ഻വറƄ௪ ௎୍ܻงƥ n ˂ 30) 30ࡼ ࡷࡣ ߘƄࠦެ‫ࡩݘޢ‬Ƅࡴ ᄽ஛ ƣ ƀ‫ࠁޢ ݟ‬ 147 ܸୃกܸ଍ଉŽ–ໝ ແࡦफउ –ࠣ‫ݶ‬ Ɖܸࢤ ެߕକଐƁƢࠣ‫ࡦ ࢽޟܸ ࢲޟ‬ƁƢࠣ‫ޟ‬පඓ ൷Ɓ౭౑ࢲ କଐƁƄ ܸࢲ ƁƋኃ ᒔᐺ። ጏ࢓‫ݕ‬ ࡦƁ࣬ ࣊ Ɓ࢓ߢೕಹኃ ቮƁ೾೥࢓଍ଉƁƄ ƁƋƁ߁ ࢟ ſ ƛ ܸߢࢽ ࡦƁ߁ ዖዄ Ž૊ સၫƋࣝެ࠾ኃ ቮƁƊܸެࢲ ଘଐƁ ። ፎକଐ߁ ެࢲ କଐƁƢ। ्࢟ ࣝࡦƁŻܸࢤ ߾ ଔଐƁ H0 : μ = 115 H1 : μ ≠ 115 ߄ ެ‫ݑ‬t ơ ƥ߄ ၴ ࡼ ࡷ tα/2 =0.025 )ƣ ܾߙࠢ‫࡯ޢ‬ၮƥ໠ ໄߙ࢖࣍ ࡼ ࡷƎܻީޯࠀ ଘଓƄƥ ‫ = ۑ‬αƣ ƀ‫ ࠁޢ ݟ‬ƥ ‫ ۑ‬፥ ፕ tα/2 ࠦၴ ࢖ஊै Ƅࠦ‫ࡩޢ‬ƥ߄ எै Ƅࠦࡺ‫ ߙܻࡩߥޢ‬df =n-1 = 25-1 = 24 ࠦ‫ݹ‬༇໪ Ƈ ܻၴ Ǝƌ X  S n – 115 ) t = (120 (t ࠦ‫ ࣀ࣠ހ‬ഷവറƄࠦࡺ‫ࡩߥޢ‬Ƅ Ƨ Ƅ t ࠦ‫ݹ‬Ǝܻ‫ݘޢ‬ଐଌƄࠦࡺ‫ࡩߥޢ‬Ƅƅ ܻࣀ ࠁ t ‫ݏ‬/√݊ (120 -115 ) = =5x5 10 10/√25 = 25/10 = 2.5 Ɠ ࢖ߙ࢛ ߙमएࠢ‫࡯ޢ‬ၮƥ ( ‫ࡩޢࠦ ۑ‬ƥ߄ எै Ƅࠦࡺ‫ࡩߥޢ‬Ƅࡼ ࡷ౰౎ࡁ ƀ ‫ = ۑ‬t ࠦ‫ ࣀ࣠ހ‬ഷവറƄࠦࡺ‫ࡩߥޢ‬Ƅ ፥ ጿࠦީ࡯࣐ ࡩƄƇ ܻၴ Ǝƌ࣊ ࢵ ࣠ޯࡷƣ ƀે઻ၮơ ਔ ৲Ƨ ኅ ቱƄƥ H1 ࡣ ‫ࡩީ ߄ݹ‬ƄƓ ࢖ߦࡩƄࡣ ީߥ‫ރ‬ƥ ( H0 Ƣ߄ ‫ࡩݘ‬Ƅ ‫ۑ‬%5ࠦ‫ࡷݘެ࣠ݹ‬Ʀ ࣠ޯࣀ ࡷ߄ ެ‫ݑ‬Ƣܻ‫ࡩݘ‬Ƅ࣊ ࢵ ࣠ޯଐଌƄࡼ ‫ޯ࡯ ߎݑ‬༧࿻ ࠦࢵ Ǝ߄ ଐଌƄƤ߃ ࠧ ‫ۑ‬Ɔ࣠ީ‫ݑ‬400ࡼ ࡷࠦެ༨ೞ ࡼ ࡷƏƎଘଓƄࡼ ࡷƆ࣠ީ‫ݑ‬50ࡼ ࡷࠦ୆෷Ƅ࣠ࣃ ‫ݑޢެࠦݑ‬Ƈ ߃ ၯ ƀ ( ‫ ێ ێ‬ƞ ܸޯࡴ ࡼ ࡺࢥ ‫ެޯ ߴݹ‬ଐଌƄƣ ᇴᆝ Ƅƍܻߙ‫ۑ‬Ƈ Ƅ࣠ީ‫ݑ‬4࣠ࠧƎ࢖ߥଐଌƄƣ Ə࣠ࡩƄࡼ ‫ܻݑ‬ᐱ᏶Əƥࡣ ߥ‫཯ݹ‬དྷࡩƄƇ Ƅ࣠ީ‫ࡩݘ‬Ƅƌ߄ ၮƣ Ƅ߄ ၴ ƥƥ ‫ أن‬%95ࠦߥ‫މ‬Ʀ ࣠ޯࣀ ೘ೃߴ ޯެჴఱ ఝƣ ƀܻެެࡓ ೘಼ࡣ ࠩߙ‫ۑ‬%3ࡼ ‫ݑ‬ƣ Ə࣠ࡩƄࠦࢧ ߘܻެࡩƄƇ Ƅ࣠ީ‫ࡩݘ‬Ƅࠦީఱ ఝਔ ৲बऒଘଓƣ ƀ ‫ۑ‬ቻቷ༨ீ ౰ేၩ፥ ጿߴ ޯެଐଌƄžܻၮƌƄ ࡠ இॅ Ɓ ࣊ ࢵ ࣠ޯଐଌƓ ƥ࢖ߦࡩƄƎܻީޯࠀ ଘଓࠦ‫ޯީݘ‬ଐଌƄƇ Ƅ࣐࣠ உै Ƅࡼ ‫ ࣀ ݪݑ‬ჱᑃᐽƓ ƥ࢖ߦࡩƄƎܻީޯࠀ ƄƇ Ƅ࣐࣠ ࠀ ߎ ࡯ޯ༧ဈ ଘଓ ‫ࡩݘޢެࠦۑ‬Ƅ H0 : P = 0.03 ࠦ‫ࡩߦ࢖ࢠޢ‬Ƅ݊ ࢠƥ  H1: P ˃ 0.03 148 ܸୃกܸ଍ଉŽ–ໝ ແࡦफउ –ࠣ‫ݶ‬ Ɖܸࢤ ެߕକଐƁƢࠣ‫ࡦ ࢽޟܸ ࢲޟ‬ƁƢࠣ‫ޟ‬පඓ ൷Ɓ౭౑ࢲ କଐƁƄ ܸࢲ ƁƋኃ ᒔᐺ። ጏ࢓‫ݕ‬ ࡦƁ࣬ ࣊ Ɓ࢓ߢೕಹኃ ቮƁ೾೥࢓଍ଉƁƄ ƁƋƁ߁ ࢟ ſ ƛ ܸߢࢽ ࡦƁ߁ ዖዄ Ž૊ સၫƋࣝެ࠾ኃ ቮƁƊܸެࢲ ଘଐƁ ። ፎକଐ߁ ެࢲ କଐƁƢ। ्࢟ ࣝࡦƁŻܸࢤ ߾ ଔଐƁ ࡼ ࡷƎܻީޯࠀ Ƅ࣠ࠧƥ ‫ ۑ‬፥ ፕ ࠦ‫ࡩޢ‬ƥ߄ எै Ƅࠦࡺ‫ࡩߥޢ‬Ƅƣ ܾߙū α =0.5 ࠦ‫ݘެ࣠ݹ‬ଐଌƄƦ ࣠ޯࣀ ࡷ߄ ެ‫ ݑ‬ ‫ެޯ ߴۑ‬ଐଌƄࠢ‫ݑޢ‬ਔ ৲ܻያዪ౰౎ࡁ ƀࠦ‫ࡩߦݘ࡯ޢ‬Ƅࠦީࣀ ჱࡩƄ፥ ፕ ቿቷ‫ࡩީ ߄ݹ‬Ƅࠦ‫ࡩߦ࢖ࢠޢ‬Ƅƥࡼ ೘಼ଘଓƄƞ ࢖࣐ ࡩƄ ࡴ ᄽ஛ ࡼ ࡷ %5ࡼ ࡷ౰౎ࡁ ƀࠦެ‫ࡩݘޢ‬Ƅࡴ ᄽ஛ ƣ ଘଓ߰ ‫ࡩޯ ࢧ ࠒޢ‬Ƅࡣ ࡷܻ‫ࡷݘ‬ƢƄ߄ ႀ ޯࢵ ܻ‫ ࣀީࠦހ‬ჱᑃᐽƧ Ǝܻ‫ݘޢ‬ଐଌƄܾ࣐ உै Ƅ‫ ࣀ ݪ‬༨စ ݊ ࡺޯ഻വറƄ Sp̅ =ට ࢖(૚ି࢖) Sp̅ = ට ࢔ . ට ࡺ ି૚ n/N = 50/400 = 0.125 ˃0.05 ࡺ ି࢔ (૙.૙૜)(૙.ૢૠ) ૞૙ .ට ૝૙૙ି૞૙ ૝૙૙ି૚ Sp̅ = 0.024 x 0.936 =0.0225 Zp̅ = (p̅ - p)/ Sp̅ Zp̅ = (p̅ - p) / Sp̅ ࠦ‫ࡩޯܻࡩޢ‬Ƅቿቱƌܻ‫ݘ‬ଐଌƄ‫ ࠁ ࣀ ݪ‬Z ࠦ‫ݹ‬Ǝܻ‫ݘޢ‬ଐଌƄࠦࡺ‫ࡩߥޢ‬Ƅƅ ܻࣀ ࠁ  ( p̅ = 50/400 = 0.08 0 ) =( 0.08 – 0.03) /0.0225 = 2.2 ࠦ‫ࡩߦ࢖ࢠޢ‬Ƅࡣ ީߥ‫ރ‬ƥ HO Ƣ߄ ‫ࡩݘ‬Ƅࠦ‫ߙ ࢛ߙ࢖ࢠޢ‬मएᑑ ᐽኅ ቱƥ࢛ ߙ༟༎Ƅࠦߥ࣐ ެࡷ፥ ጿ݊ ߥ‫ ࣀ࣠ހࠦކ‬ഷവറƄࠦࡺ‫ࡩߥޢ‬Ƅƣ ƀ߄ ༬စ ߴ ޯެჴఱ ఝƥ ‫ࡩޢࠦ ۑ‬ƥ߄ எै Ƅࠦࡺ‫ࡩߥޢ‬Ƅࡼ ࡷ౰౎ࡁ ƀ ‫ ۑ‬ƅ ࣠ࣀ ഷവറƄ z ࠦࡺ‫ߘޢ‬ƣ ƀ‫ ࠁޢ ݟ‬H1 ቿቷ‫ࡩީ ߄ݹ‬Ƅ ࠦީࣀ ჱࡩƄ໠ ໄ‫ހ‬Ɵ ࢖ߦࡩƄƣ ଘଓ߰ ‫ޢ‬༨༦ ౰ేၩߴ ޯެଐଌƄžܻၮƌƄƣ Ƅે ઻ၮൎ൉‫ࡩޢ‬ƌƢ߄ ߥ‫ކ‬Ɔ࢖ߙ࣠ޯଐଌƄƇ ܻ‫ࡩީޢܻރ‬Ƅƣ ܾ‫ހ‬ᑑ ᐽƍࡼ ࡷ Ƨ ƀ‫ ߄ܻࠧۑ‬ၰ ƥࠦߙ߄ ࢧ ࡩƄࡼ ‫ܻݑ‬༬ဈ ܻ‫ރ‬ƣ ࣠ࡓ ‫ݹ‬ƣ Ƅࡼ ࡷƧ ࢖ࠧ࣠ࠅ ƥ౰ేީࡁ ߴ ޯެଐଌƄܻᐱᐕ‫ݑ‬ਔ ৲཯དྷࡩƄࠦީࣀ ჱࡩƄƥƆ߄ ܻࠧࣃ ଐଌƄ ‫ܻۑ‬୆෷ܻࢧ ࠁ ƂƧ ࢖ࠧ࣠ࠅ Ɵ ࢖ߦࡩƄƣ ƀ ƣ ܾ‫ހ‬žܻၮƌଘଓƄࡼ ࡓ ೘಼ƣ ᇴᆝ ƄƍƄܻࡷ %95ࠦߥ‫ࠦމ‬ၴ Ǝਔ ৹ƞ ࢖‫ކݘ‬ƣ ƀበሼྲ ႖‫ ݌ ݪ‬मऒ ( ‫ ێ ێ‬ƞ ܸޯࡴ ƞ ࢖‫ކݘ‬ƥ‫ۑ‬ƣ ࣠‫ࡩ ࢧܻހ‬Ƅࡼ ࡷƢƄ༇໮ 400ࡼ ࡷ౰౗ࡁ ƀે ઻ၮƧ ࣠ޯ༨࿻ ࠢ‫ޢݘ‬ხ჋Ƨ ኅ ቱƄƟ ࣠ࠒ ࣀ ଐଌƄƣ ࣠‫ࡩ ࢧܻހ‬Ƅࢇ ‫ݹ‬ƌܻެࢢ Ƈ ߃ ၯ Ƅ߄ ߘƥ‫ۑ‬ഭ ୨‫ࡩ ࣐ީޢ‬Ƅ݊ ‫ݹ‬Ə࣠ޯࡩƄ݊ ީჴႾࢇ ‫ݹ‬ƌܻެࢧ ࡩܻ‫ހ‬ƣ ࣠‫ࡩ ࢧܻހ‬Ƅƣ ƄƏƥƀƣ ƀࠦ‫ܻࢠޢ‬ଐଌƄƆ౰౎உै Ƅࡼ ࡷበሼྲ ႡࡩƄ Ƨ Ǝܻ‫ݘޢ‬ଐଌƄƞ Ƅ࢖༨စ ଘଓƄƥƢƄ༇໮ 420፥ ጒܻࣀ ஊै Ƅ࣊ ࢵ ࣠ࡩƄƣ ƀƇ ߄ ၴ ƥƥ 25ܻࠩࡺᄽ஛ ࠦ୆෷Ƅ࣠ࣃ ‫ݑޢެࠦݑ‬በሼྲ ႡࡩƄ ‫ۑ‬75 H0 : μ = 400 ࡠ இॅ Ɓ H1 : μ ˃ 400 149 ܸୃกܸ଍ଉŽ–ໝ ແࡦफउ –ࠣ‫ݶ‬ Ɖܸࢤ ެߕକଐƁƢࠣ‫ࡦ ࢽޟܸ ࢲޟ‬ƁƢࠣ‫ޟ‬පඓ ൷Ɓ౭౑ࢲ କଐƁƄ ܸࢲ ƁƋኃ ᒔᐺ። ጏ࢓‫ݕ‬ ࡦƁ࣬ ࣊ Ɓ࢓ߢೕಹኃ ቮƁ೾೥࢓଍ଉƁƄ ƁƋƁ߁ ࢟ ſ ƛ ܸߢࢽ ࡦƁ߁ ዖዄ Ž૊ સၫƋࣝެ࠾ኃ ቮƁƊܸެࢲ ଘଐƁ ። ፎକଐ߁ ެࢲ କଐƁƢ। ्࢟ ࣝࡦƁŻܸࢤ ߾ ଔଐƁ ݊ ‫ݹ‬Ə࣠‫ ކ‬Ƣ߄ ႀ ޯఱ ఝ ū ࠦࡷॅळ‫ ࡷݘ‬౰ేၩ σ ᑑ ᐽቦሼƥ n ˂ 30 ഭ ୨‫ݹ ݊ ࣍ީޢ‬Ə࣠ޯࡩƄ ƣ Ƅ ‫ࠁޢ ݟ‬ ‫ۑ‬0.05 ࠦ‫ࡷݘެ࣠ݹ‬Ʀ ࣠ޯࣀ ೘ೃƥƎܻީޯࠀ ൎᑉᐽࠦၴ ࢖ஊै Ƅࠦߥ࣐ ެଐଌƄਔ ৲߄ ༨ဈ df =n-1 = 25-1 = 24 ) t (t ࠦ‫ ࣀ࣠ހ‬ഷവറƄࠦࡺ‫ࡩߥޢ‬Ƅ Ƨ Ƅ t ࠦ‫ݹ‬Ǝܻ‫ݘޢ‬ଐଌƄࠦࡺ‫ࡩߥޢ‬Ƅƅ ܻࣀ ࠁ X  S n – 400) t = (420 t = 75/√25 (20) X 5 = 20 75 15 = 1.33 ‫ ࣀ࣠ހࠦ ۑ‬ഷവറƄ tࠦࡺ‫ܻࡷ ݊ߘޢ‬ᐱ᏶Ǝܻߥ‫ރ‬ƥ ‫ = ۑ‬24ࠦ‫࢖ݹ‬ஊै ƄƇ ܻၴ Ǝƌ߄ ެ‫ࡩޢࠦݑ‬ƥ߄ எै Ƅ tࠦࡺ‫ߘޢ‬ ࢛ ߙमएƥ H0ࡣ ީߥ‫ࡩޢࠦ ۑ ރ‬ƥ߄ எै Ƅ t ࠦࡺ‫ߘ ࡣࡷ ࡼߘޢ‬Ƅ ‫ ࣀ࣠ހࠦ ۑ‬ഷവറƄ t ࠦࡺ‫ߘޢ‬ƣ ƀܻ೘ೃ ‫߰ۑ‬ ‫ޢ‬༨༦ ౰ేၩበሼྲ ႡࡩƄžܻၮƌƄƣ ܾ‫཯ހ‬ཏ‫ݹݘ‬Ƅ߃ ࠧƥ H1 ( n 1 , n2 ≥ 30 ໝ ແ࣍ ࢲ ࣝެࡴໝ ແ‫ݽ‬ Ɯ ࢓ߣᑀᐺƐ Ƣ࢓ߣࡦƁƋܸަެ߽ Ž Ǝܻީޯࠀ ଘଓƄƇ Ƅ࣐࣠ ࠀ  ቿቷ‫ࡩީ ߄ݹ‬Ƅࠦ‫ࡩߦ࢖ࢠޢ‬ƄƥƢ߄ ‫ࡩݘ‬Ƅࠦ‫ߙ࢖ࢠޢ‬፥ ፕ ƥƓ ƥ࢖ߦࡩƄ݊ ࢠƥ  ( %10 ، %5 ، %1Ɔƌܻၮ αܾ࣐ உै Ƅࠦࡺ‫ߘޢ‬ਔ ৲߄ ༨ဈ Ƨ ƀ α ࠦ‫ݘެ࣠ݹ‬ଐଌƄƦ ࣠ޯࣀ ࡷਔ ৲߄ ༨ဈ  Ƣ߄ ႀ ޯࣀ ଐଌƄ፥ ገܻࢧ ࠁ ଘଓƄƥ ࠦ‫ ࣀ࣠ހ‬ഷവറƄࠦࡺ‫ࡩߥޢ‬Ƅ ࠦެ‫ࡩݘޢ‬Ƅ࣊ ࢵ ࣠ޯଐଌ Z ࠦ‫ݹ‬Ǝܻ‫ݘޢ‬ଐଌƄࠦࡺ‫ࡩߥޢ‬Ƅƅ ܻࣀ ࠁ  ࠦ‫ࡩޢ‬ƥ߄ எै Ƅࠦၴ ࢖ஊै Ƅࠦࡺ‫ࡩߥޢ‬Ƅਔ ৲߄ ༨ဈ Ꭸ᎜ࡼ ࡷƥ Z Ǝܻީޯࠀ Ƅ࣠ࠧƎܻީޯࠀ ଘଓƄƄ߃ ࠧ፥ ጿ 150 ܸୃกܸ଍ଉŽ–ໝ ແࡦफउ –ࠣ‫ݶ‬ Ɖܸࢤ ެߕକଐƁƢࠣ‫ࡦ ࢽޟܸ ࢲޟ‬ƁƢࠣ‫ޟ‬පඓ ൷Ɓ౭౑ࢲ କଐƁƄ ܸࢲ ƁƋኃ ᒔᐺ። ጏ࢓‫ݕ‬ ࡦƁ࣬ ࣊ Ɓ࢓ߢೕಹኃ ቮƁ೾೥࢓଍ଉƁƄ ƁƋƁ߁ ࢟ ſ ። ፎକଐ߁ ެࢲ କଐƁƢ। ्࢟ ࣝࡦƁŻܸࢤ ߾ ଔଐƁ ƛ ܸߢࢽ ࡦƁ߁ ዖዄ Ž૊ સၫƋࣝެ࠾ኃ ቮƁƊܸެࢲ ଘଐƁ ፥ ጘଘଓᇴᆝ ‫ ࣀ ݪ‬༨࿻ ƥ໠ ໄ࣐ ࢵ ࣠ޯࡷ໠ ໄ‫ހ‬Ɵ ࢖ߦᑃᐽƧ Ǝܻ‫ݘޢ‬ଐଌƄܾ࣐ உै Ƅ  X1  X 2  12  22    n1 n2 Ơ Ž‫߾ޟݜ‬  X1  X 2 s12 s 22  n1 n 2 30ࡼ ‫ݑ‬ਔ ৲बऒƥƀƧ ƥܻఱ ఠ཯དྷࡩƄ ࠦެ‫ࡩݘޢ‬Ƅࡼ ࡷࠦ‫ ࣀ࣠ހ‬ഷവറƄ S22 ، S12Ƨ Ǝܻ‫ݘޢ‬ଐଌƄƞ Ƅ࢖༨စ ଘଓƄযওߘƢ߄ႀ ޯఱ ఠƥ ƣ ࣠ࡓ ‫ކ‬ƆƎƌܻ‫ރ‬Ƈ ଘଓܻၰ ፥ ጿࡼ ࡓ ᚻࡩƥū໠ ໄ‫ ࠖޯࡺݘ‬൞ᒑᐽƧ Ǝܻ‫ݘޢ‬ଐଌƄƞ Ƅ࢖༨စ ൎᑉᐽƆ߄ ‫ ࠅޢ‬Ƈ Ƅमअ߄ ߥ‫ކ‬፥ ፕ ƥūƆƌ࢖ߦࡷ ‫ܻۑ‬ᐱᚑƄ߄ ႀ ޯࢵ Ƅ໠ ໄ‫୓ݘ‬෺ࠢ‫ߙࠦߙܾރ‬ƥ࢖‫ࡷݘ‬໠ ໄ‫ޯࡺݘ‬഻വറƄൎᇩᇍ ࡩࠦ‫ݹ‬Ǝܻ‫ݘޢ‬ଐଌƄƇ ܻߙƄ࢖༨စ ଘଓƄܻᐱᐕߙ ࠦࡺ‫ރࠦߘޢ‬Ǝܻߥࡷơ ൎ൉ၯ ࡼ ࡷଘଓƢƀܻ‫ ࠅ࣠ࠧ࢖ݹ‬໠ ໄ࣐ ࢵ ࣠ޯଐଌƄ໠ ໄ‫ހ‬Ɵ ࢖ߦࡩƄƣ ᇴᆝ ƄƍƄܻࡷơ ࣠ࠁ ƎƄ࢖ߥࡩƄƍܻ༧ဈ ƀ ࠦ‫ࡩޢ‬ƥ߄ எै Ƅࠦࡺ‫ࡩߥޢ‬Ƅ݊ ࡷƇ ܻެ‫ࡩݘޢ‬Ƅࡼ ࡷƅ ࣠ࣀ ഷവറƄ፥ ገܻࢧ ࠁ ଘଓƄ ༇໮ ଘଓƄƣ ᇴᆝ Ƅƍƀܻࡷ %5ࠦ‫ࡷݘެ࣠ݹ‬Ʀ ࣠ޯࣀ ࡷ߄ ެ‫ݑ‬ƌ߄ ༨࿻ ƣ ƀमअ߄ ࡷ‫ ݌ ݪ‬मअ ( ‫ ێ ێ‬ƞ ܸޯࡴ ࡼ ࡷࠦ୆෷Ƅ࣠ࣃ ‫ ߃ݑޢެࠦݑ‬ၯ ܾ‫ݹ‬ᑑ ᐽƍࡣ ࡺ‫ࡩݘ‬ū໠ ໄޯჱႾ߄ ࡷ፥ ጿƣ ܻ‫ݹ‬ƥܻࣀ ჴࡷƆ࢖ࠩଐଌƄߎ ࢧ ‫ރ‬ơ ܻࡺ‫ݘ‬ᑃᐽ ($)ࠦၮܻࣀ ࡩܻ‫ހ‬ X̅1 = 6.00 ، X̅2 = 5.40 ፥ ጘଘଓᇴᆝ ၖ࿏ ܻޯެࡩƄบ คᇴᆝƥ໠ ໄޯჱႾ߄ ଐଌƄࡼ ࡷᇯ ᆝ ࡼ ࡷࠦၮܻࣀ ࡩܻ‫ހ‬༇໮ ଘଓƄ S1 = 2.00 , S2 = 1.80 n1 = 40 , n2 = 54 ࡠ இॅ Ɓ ‫ ۑ‬+ = Z α/2 ࠦ‫ࡩޢ‬ƥ߄ எै Ƅࠦࡺ‫ࡩߥޢ‬Ƅƣ ܾߙ %5ࠦ‫ࡷݘެ࣠ݹ‬Ʀ ࣠ޯࣀ ࡷ߄ ެ‫ݑ‬ ࠦ‫ ࣀ࣠ހ‬ഷവറƄࠦࡺ‫ࡩߥޢ‬Ƅ Zࠦ‫ݹ‬Ǝܻ‫ݘޢ‬ଐଌƄࠦࡺ‫ࡩߥޢ‬Ƅ‫ ࣀ ݪ‬༨စ 151 ܸୃกܸ଍ଉŽ–ໝ ແࡦफउ –ࠣ‫ݶ‬ Ɖܸࢤ ެߕକଐƁƢࠣ‫ࡦ ࢽޟܸ ࢲޟ‬ƁƢࠣ‫ޟ‬පඓ ൷Ɓ౭౑ࢲ କଐƁƄ ܸࢲ ƁƋኃ ᒔᐺ። ጏ࢓‫ݕ‬ ࡦƁ࣬ ࣊ Ɓ࢓ߢೕಹኃ ቮƁ೾೥࢓଍ଉƁƄ ƁƋƁ߁ ࢟ ſ ƛ ܸߢࢽ ࡦƁ߁ ዖዄ Ž૊ સၫƋࣝެ࠾ኃ ቮƁƊܸެࢲ ଘଐƁ  X1X2  ። ፎକଐ߁ ެࢲ କଐƁƢ। ्࢟ ࣝࡦƁŻܸࢤ ߾ ଔଐƁ  X1  X 2 Ƨ Ǝܻ‫ݘޢ‬ଐଌƄܾ࣐ உै Ƅଘଓƥƀ‫ ࣀ ݪ‬༨စ 12  22 s2 s2 2.002 1.802   1  2    0.1 0.06  0.16  0.4 n1 n2 n1 n2 40 54 z  ( X 1  X 2 ) 0. 6   1. 5  X1  X 2 0. 4 ࠦߥ࣐ ެࡷࡣ ၯ Ƅƌ݊ ߥ‫ކ‬ƥ ‫ࡩޢࠦ ۑ‬ƥ߄ எै Ƅࠦࡺ‫ࡩߥޢ‬Ƅࡼ ࡷࡣ ߘƄ ‫ ࣀ࣠ހࠦ ۑ‬ഷവറƄ zࠦࡺ‫ߘޢ‬ƣ ƀ‫ ࠁޢ ݟ‬ƥ ‫ۑ‬5%ࠦ‫ࡷݘެ࣠ݹ‬Ʀ ࣠ޯࣀ ࡷ߄ ެ‫ݑ‬ū 1 = 2 ‫ أي‬،H0ࡣ ީߥ‫୐ܻރ‬คܾߙūơ ࣠ީߥࡩƄ (n1, n2 ˂ 30 ƃ౭ౄ‫ݐ‬ ࢟ Ƅ ܸީ‫ݕޟ‬ ࡦໝ ແ࣍ ࢲ ࣝެࡴໝ ແ‫ݽ‬ Ɯ ࢓ߣᑀᐺƐ Ƣ࢓ߣࡦƁƋܸަެ߽ Ɓ  Ƈ ܻ‫ࡩߦ࢖ࢠޢ‬Ƅ Ɔ౰ే‫ݓ‬ ࢢ Ƈ ܻެ‫ࡩݘޢ‬Ƅ ഭ ୨‫ ࣍ީޢ‬໠ ໄ‫ޯࡺݘ‬഻വറƄ݊ ‫ݹ‬Ə࣠‫ކ‬ ƥܻࣀ ჴࡷࠢެࡓ ᚻࡩƢॅळ‫ࡷݘ‬౰ేၩ໠ ໄ‫ ࠖޯࡺݘ‬൞ᒑᐽƧ Ǝܻ‫ݘޢ‬ଐଌƄƞ Ƅ࢖༨စ ଘଓƄ (X1  X 2 ) t   X1  X 2 S X1  X 2  s2 n1  .I .II .III ( df = n1 + n2 -2 ࠦ‫ݹ‬༇໪ Ƈ ܻၴ Ǝਔ ৹ t ݊ ‫ݹ‬Ə࣠‫ކ‬Ƣ߄ ႀ ޯఱ ఝ ፥ ጘଘଓᇴᆝ ‫ ࣀ ݪ‬༨࿻ ໠ ໄ࣐ ࢵ ࣠ޯࡷ໠ ໄ‫ހ‬Ɵ ࢖ߦᑃᐽƧ Ǝܻ‫ݘޢ‬ଐଌƄܾ࣐ உै Ƅ s2 n2 ໠ ໄޯჱ‫ࡩݘޢ‬Ƅࡼ ࡷᇯ ᆝ ࡼ ࡷ‫ ࣀ ݪ‬ჴ༨࿻ ƥ ࣯ ዙኤ Ƅ౰౔ࡩƄ പ ୥‫ࡦެ ࠓࡷޟ‬Ɓ௧ ோܸަެࡦƁ࣠ࠧቿቱƌܻ‫ݘ‬ଐଌƄ፥ ጿƢ߄ ႀ ޯࣀ ଐଌƄ௪ ௎ܻީޯࡩƄ S2 = S12 ( n1 – 1) + S22 ( n2 – 1) ࠦ୆งଘଓƄቿቱƌܻ‫ݘ‬ଐଌƄ‫ࠁ ࣀ ݪ‬ ________________________ n1 + n 2 – 2 152 ܸୃกܸ଍ଉŽ–ໝ ແࡦफउ –ࠣ‫ݶ‬ Ɖܸࢤ ެߕକଐƁƢࠣ‫ࡦ ࢽޟܸ ࢲޟ‬ƁƢࠣ‫ޟ‬පඓ ൷Ɓ౭౑ࢲ କଐƁƄ ܸࢲ ƁƋኃ ᒔᐺ። ጏ࢓‫ݕ‬ ࡦƁ࣬ ࣊ Ɓ࢓ߢೕಹኃ ቮƁ೾೥࢓଍ଉƁƄ ƁƋƁ߁ ࢟ ſ ። ፎକଐ߁ ެࢲ କଐƁƢ। ्࢟ ࣝࡦƁŻܸࢤ ߾ ଔଐƁ ƛ ܸߢࢽ ࡦƁ߁ ዖዄ Ž૊ સၫƋࣝެ࠾ኃ ቮƁƊܸެࢲ ଘଐƁ मअ߄ ࡷ‫ ݌ ݪ‬मअƥ‫୐ ࣏ޢ ߎۑ‬งƆƌܻࡷࡼ ࡷਔ ৲߄ ၴ ߴ ޯެࡷƉ ܻ୓คܺᄣƇ ᇴᆝྲ ႡࡩƄƦ ߄ ၰ ƀƢ࣠ߥ‫( ކ‬10.6.8)ƞ ܸޯࡴ ƢƄ߄ ႀ ޯࢵ ܻ‫ ހ‬2 ، 1 ໠ ໄޯߥ࣐ ެࡷ፥ ጿࠦ‫࢖ݹ‬ᐳႡࡩƄበሼྲ ႡࡩƄƇ ܻ‫ࡷީޢݘ‬Ƈ ܻ࣐ ࢵ ࣠ޯࡷࠦ‫ރ‬Ǝܻߥࡷ፥ ጿበሼྲ ႡࡩƄ፥ ጿࢇ ‫ ࣀ࣠ݹ‬ჴࡩƄ ໠ ໄ‫ߙ࢖ߘܻހ‬Ơ ܻެࠧƣ ܾ‫ހ‬ቿቷ‫ݶ‬ ܻߥࡩƄࠦ‫ࡩߦ࢖ࢠޢ‬Ƅ౰౎ޯࠀ ƀ ū໠ ໄޯߥ࣐ ެଐଌƄ፥ ጿࠦ‫ݹ‬Ǝܻႅ ޯࡩƄƇ ൎ൉ഷവറƄࡼ ࡷࠦ୆෷Ƅ࣠ࣃ ‫ݑޢެࠦݑ‬ ࠦ୆งଘଓƄࠦެ‫ࡩݘޢ‬Ƅၖ࿏ ܻ୓คࡼ ࡷᑑ ᐽƍƥ %5ࠦ‫ࡷݘެ࣠ݹ‬Ʀ ࣠ޯࣀ ࡷ߄ ެ‫ݑ‬໠ ໄޯߥ࣐ ެଐଌƄ ፥ ጿƇ ܻ‫ީޢݘ‬ଐଌƄƇ ܻ࣐ ࢵ ࣠ޯࡷ X̅1 = 40 , X̅2 = 38 S1 = 15.46 , S2 = 4.34 n1 = 10 , n2 = 15 ( S12 =239.011, S22 =18.84 ) ࡠ இॅ Ɓ H0 : 1 = 2 H1 : 1 ≠2 ࠦ‫ࡩޢࠦࡩީߥޢ‬ƥ߄ எै Ƅࠦࡺ‫ࡩߥޢ‬Ƅƣ ܾߙ αܾ࣐ உै Ƅ࣠ࠧƥ %5ࠦ‫ݘެ࣠ݹ‬ଐଌƄƦ ࣠ޯࣀ ࡷ %95ࠦߥ‫މ‬Ʀ ࣠ޯࣀ ࡷ߄ ެ‫ݑ‬ ‫ ۑ‬፥ ፕ 23ࠦ‫ݹ‬༇໪ Ƈ ܻၴ Ǝƌ߄ ެ‫ݑ‬tࠦၴ ࢖ஊै Ƅ ‫ ۑ‬df = n1 + n2 -2 = 10 +15 -2 =23 ) S2 = S12 ( n1 – 1) + S22 ( n2 – 1) ࣯ ዙኤ Ƅ౰౔ࡩƄ ഭ ୨‫ࡩޯ ࠖࡺޢ‬Ƅ௪ ௎ܻީޯࡩƄ߄ ၴ ࣠‫ރ‬ ________________________ n1 + n 2 – 2 S2 = 239.01(10-1)+18.84( 15 – 1) __________________________ = 103.8 10+15- 2 S2 = 103.8 103.8 103.8  = 10.38  6.92 = 4.16 10 15 (X  X 2 ) (40  38) t 1 = = 0.48  X X 4.16 S X1  X 2  1 153 2 ܸୃกܸ଍ଉŽ–ໝ ແࡦफउ –ࠣ‫ݶ‬ Ɖܸࢤ ެߕକଐƁƢࠣ‫ࡦ ࢽޟܸ ࢲޟ‬ƁƢࠣ‫ޟ‬පඓ ൷Ɓ౭౑ࢲ କଐƁƄ ܸࢲ ƁƋኃ ᒔᐺ። ጏ࢓‫ݕ‬ ࡦƁ࣬ ࣊ Ɓ࢓ߢೕಹኃ ቮƁ೾೥࢓଍ଉƁƄ ƁƋƁ߁ ࢟ ſ ። ፎକଐ߁ ެࢲ କଐƁƢ। ्࢟ ࣝࡦƁŻܸࢤ ߾ ଔଐƁ ƛ ܸߢࢽ ࡦƁ߁ ዖዄ Ž૊ સၫƋࣝެ࠾ኃ ቮƁƊܸެࢲ ଘଐƁ ơ ࣠ީߥࡩƄࠦߥ࣐ ެࡷ፥ ጿ t ፥ ገܻࢧ ࠁ ്ᑉᐽࠦ‫ ࣀ࣠ހ‬ഷവറƄࠦࡺ‫ࡩߥޢ‬Ƅ݊ ߥ‫ކ‬ƥ‫ࡩޢࠦۑ‬ƥ߄ எै Ƅܻᐱᐢࡺ‫ߘ ࡣࡷ ࡼߘޢ‬Ƅࠦ‫ ࣀ࣠ހ‬ഷവറƄ tࠦࡺ‫ߘޢ‬ ܻ‫ߙޢ‬ᇴᆝ ൎ൉‫ࡩޢ‬ƌƢ߄ ߥ‫ކ‬ଘଓƆ࢖ߙ࣠ޯଐଌƄƇ ܻ‫ࡩީޢܻރ‬Ƅƣ ƀᑑ ᐽƍࡼ ࡷߴ ޯެჴఱ ఝƥ‫ۑ‬H0Ƣ߄ ‫ࡩݘ‬Ƅࠦ‫ރߥީ ࡣߙ࢖ࢠޢ‬ᑑ ᐽਓ ৹ƥū ‫ۑ‬໠ ໄޯߥ࣐ ެଐଌƄ ፥ ጿࠦ‫ݹ‬Ǝܻႅ ޯࡩƄƇ ൎ൉ႁ ൞ᒑᐽࠦၮܻީଐଌƄƇ Ƅ࣠ީ‫ࡩݘ‬Ƅƌ߄ ၮ࣯ ࣐ ࢵ ࣠ޯࡷ໠ ໄ‫ހ‬Ƨ ࢖ࠧ࣠ࠅ Ɵ ࢖ߙƠ ܻެࠧƣ Ƅે ઻ၮ ໝ ແެძమ చໝ ແ‫ݽ‬ Ɯ ࢓ߣᑀᐺƐ Ƣ࢓ߣࡦƁƋܸަެ߽ Ɓ  ܻࢥ ‫ݹ‬ƀƠ ܻެࠧūƇ ܻ࣐ ࢵ ࣠ޯଐଌƄ໠ ໄ‫ހ‬Ɵ ƥ࢖ߦࡩƄƥ‫ ࣀ ݪ‬ჱࡩƀƥƀƇ ܻ࣐ ࢵ ࣠ޯଐଌƄ໠ ໄ‫୐ࠦހ‬෺ܻ‫ݘ‬൞ᒑᐽƇ ܻ‫ݹݘ‬Ə࣠‫ކ‬Ơ ܻެࠧƣ ƀሯሡሁ मअ߄ ߥ‫ކ‬Ꭸ᎜ ࡼ ࡷƥ໠ ໄޯხఱ ఝ໠ ໄ‫ހ‬Ɵ ࢖ߦᑃᐽƧ Ǝܻ‫ݘޢ‬ଐଌƄܾ࣐ உै Ƅमअ߄ ߥ୓ก߰ ࡺఱ ఠ‫ ࣀ ݪ‬ჱࡩƄ໠ ໄ‫ހ‬Ɵ ƥ࢖ߦᑃᐽࠦ୐෺ܻ‫ݘ‬൞ᒑᐽƇ ܻ‫ݹݘ‬Ə࣠‫ކ‬ ‫ࡷݘޢެࠦۑ‬Ɵ ƥ࢖ߙơ ࣠ࢧ ஊै Ƈ ଘଓܻࡺޯࠁ ଘଓƄ ፥ ፕ ໠ ໄޯხఱ ఝ໠ ໄ‫ހ‬Ɵ ࢖ߦᑃᐽƧ Ǝܻ‫ݘޢ‬ଐଌƄܾ࣐ உै Ƅቿቱƌܻ‫ࡷݘ‬  p1  p2  p (1  p ) p (1  p )  n1 n2 ࠦ‫ࡩޯܻࡩޢ‬Ƅቿቱƌܻ‫ݘ‬൞ᒑᐽܻߥߙƥ P̅ Ƣܻ‫ࡩݘ‬Ƅࠦީࣀ ჱࡩƄƢƄ߄ ႀ ޯࢵ ܻ‫ހ‬໠ ໄޯხఱ ఝ໠ ໄ‫ހ‬Ɵ ࢖ߦᑃᐽƧ Ǝܻ‫ݘޢ‬ଐଌƄܾ࣐ உै Ƅ‫ ࣀ ݪ‬༨࿻ ƥ ܻ୐คܾߙࠦ‫ ࠅ࣠ࠧ࢖ݹ‬Ɵ ƥ࢖ߙ፥ ፕ ƥƀࠦߙ߄ ࢧ ᑃᐽࠦ‫ ࠅݘ‬ƄƎ‫ ࣀ ݪ‬ჱࡩƄ໠ ໄ‫ހ‬Ɵ ƥ࢖ߦࡩƄบ คᇴᆝ ƄƍƄܻࡷमअ߄ ߥ‫ ࡓ ࡼކ‬೘಼ཀྵདྷࠁ ƥ ࠦࡺ‫ࡩߥޢ‬Ƅƥūࠦ‫ݹ‬Ǝܻ‫ߘޢࡺࠦࡷݘޢ‬፟ ፑƄ ໠ ໄޯხఱ ఝ໠ ໄ‫ހ‬Ɵ ࢖ߦࡩƄ ፥ ገƄ࣠ࣃ ‫ࡩݘ‬Ƅ౰ే‫ݓ‬ ޯଐଌƄࠦࡺ‫ߘޢ‬ơ ࣠༨စ ፥ ፕ Ɠ ƥ࢖ߦࡩƄƎܻީޯࠀ Ƅ፥ ጿƢ߄ ႀ ޯࣀ ଐଌƄ፥ ገܻࢧ ࠁ ଘଓƄࠦࡺ‫޲ ࡣߘޢ‬೘಼Ƨ ኅ ቱƄࠦ‫ݹ‬Ǝܻ‫ݘޢ‬ଐଌƄ ( p  p2) z   p1  p 2 n p n p 1 p 1 1 2 2 n1  n2 ơ ࣠ީߥଐଌƄࠦީఱ ఝบ คᇴᆝ ƄƍƂܻࡷ 1%ࠦ‫ࡷݘެ࣠ݹ‬Ʀ ࣠ޯࣀ ࡷ߄ ެ‫ݑ‬ƌ߄ ༨ဈ ƣ ƀበሼྲ ႖‫ ݌ ݪ‬मऒ : ( ‫ ێ ێ‬ƞ ܸޯࡴ በሼྲ ႡࡩƄƇ ߃ ၯ ƀ߄ ߘƥ‫ۑ‬p2ū્ ઻༨ீ ƌƎ࣠ଐଌܻᐱᐟ‫ݑ‬ਔ ৲बऒ p1ū཯ཌެࠅ ƀƌƎ࣠ଐଌࠦ୆คƥ౰౔ࡓ ᚻࡩଗଓƄƇ ܻ‫ ࡓ࣠ރ‬ᅚଌƄࡼ ࡷ ࡴ ᄽ஛ ࡼ ࡷƇ ܻެ‫ ࡷ ࡼݑޢ‬p2 = 0.7 ‫ و‬p1 = 0.9ƣ ƀƇ ߄ ၴ ƥƥƌƎ࣠ࡷᇯ ᆝࠦެ༨ೞ ࡼ ࡷࠦ୆෷Ƅ࣠ࣃ ‫ݑޢެࠦݑ‬ ‫ۑ‬n2 = 80 ‫ و‬n1 =100 ࡣ ஊै Ƅ 154 ܸୃกܸ଍ଉŽ–ໝ ແࡦफउ –ࠣ‫ݶ‬ Ɖܸࢤ ެߕକଐƁƢࠣ‫ࡦ ࢽޟܸ ࢲޟ‬ƁƢࠣ‫ޟ‬පඓ ൷Ɓ౭౑ࢲ କଐƁƄ ܸࢲ ƁƋኃ ᒔᐺ። ጏ࢓‫ݕ‬ ࡦƁ࣬ ࣊ Ɓ࢓ߢೕಹኃ ቮƁ೾೥࢓଍ଉƁƄ ƁƋƁ߁ ࢟ ſ ። ፎକଐ߁ ެࢲ କଐƁƢ। ्࢟ ࣝࡦƁŻܸࢤ ߾ ଔଐƁ ƛ ܸߢࢽ ࡦƁ߁ ዖዄ Ž૊ સၫƋࣝެ࠾ኃ ቮƁƊܸެࢲ ଘଐƁ Ɠ ƥ࢖ߦࡩƄ H0 : p1 = p2 H1 : p1 > p2 ( ‫ ۑ‬፥ ፕ Ƨ Ǝܻ‫ݘޢ‬ଐଌƄഭ ୨‫ࡩ ࣐ީޢ‬Ƅ݊ ‫ݹ‬Ə࣠ޯࡩƄơ ƥ߄ ၴ ࡼ ࡷ %1ࠦ‫ݘެ࣠ݹ‬ଐଌƄƦ ࣠ޯࣀ ࡷ߄ ެ‫ ݑ‬zα ࠦࡺ‫ߘޢ‬ p  p1 p2  n1 p1  n2 p2 (100)(0.9)  (80)(0.7) 146    0.8 n1  n2 180 180 p(1 p) p(1 p) (0.8)(0.2) (0.8)(0.2)     0.0016 0.002 0.0036 0.06 n1 n2 100 80 z ( p1  p 2 ) 0.9  0.7   0.2 / 0.06  3.33  p1  p2 0.06 ࢛ ߙमए ( ‫ࡩޢࠦ ۑ‬ƥ߄ எै Ƅࠦࡺ‫ࡩߥޢ‬Ƅࡼ ࡷ౰౎ࡁ ƀ ( ‫ ) ۑ‬zα ፥ ገܻࢧ ࠁ ്ᑉᐽࠦ‫ ࣀ࣠ހ‬ഷവറƄࠦࡺ‫ࡩߥޢ‬Ƅƣ ƀܻ೘ೃ ‫ۑ‬1%Ƨ ࣠ެ‫ࡷݘ‬Ʀ ࣠ޯࣀ ࡷ߄ ެ‫ݑ‬p1 > p2 ‫ أي أن‬H1Ɠ ࢖ߦࡩƄࡣ ީߥ‫ރ‬ƥ H0 ƞ ോെߢެࢲ କଐƁƢࢄ ‫ࡦެࣝߖޟ‬ƁƃƉࣝ஋ॅ ݇ ୃ෾फए –ƣ ᇱᆚ Ƌܸަެ߽ Ɓ ‫ێ‬ Ƅ߃ ࠧ Ƈ ܻࡷƄ߄ ႀ ޯࢵ Ƅ ໠ ໄ‫ࡷ ࡼ ހ‬ƥ ‫ۑ‬ቿቷࢧ ޯଐଌƄ ࠦ‫ ࠁޯࡺܻࡩޢ‬ଘଓƄ Ƈ ܻ‫ݹݘ‬Ə࣠ޯࡩƄ ߄ ၰ ƀ ࣠ࠧ ( x2) ݊ ୆กमऒ Ƨ ᇴᆝ ݊ ‫ݹ‬Ə࣠‫ކ‬ ࠦ‫ࡩޢ‬ൎ൉ߥޯࢵ Ƅ ਔ ৲߄ ඳඣ ൴ ள ஭‫ޯݘ‬ଐଌƄ ࠦ‫ࡩޢ‬ൎ൉ߥޯࢵ ଘଓƄƥ ࢇ ‫ࡩޯ࣠ߙޢ‬Ƅ Ɔƌ࣠༬ဂ ࠦߥ࡯‫ޯݘ‬ଐଌƄ Ɠ ƥ࢖ߦࡩƄ Ƈ ƄƎܻީޯࠀ Ƅ ، Ǝܻީޯࠀ ଘଓƄ ፥ ጿ ࠦၮƏ࣠ଐଌƄ Ƈ ܻ‫ީޢܻރ‬ᑃᐽƓ ƥ࢖ߦࡩƄ Ƈ ƄƎܻީޯࠀ Ƅ ፥ ጿ Ǝܻީޯࠀ ଘଓƄ Ƅ߃ ࠧ Ƣ߄ ႀ ޯఱ ఓƥ ‫ޯࡺ ݊ ࡷܻۑ‬഻വറ ౰౗ࡁ ƀ ‫ أو‬໠ ໄޯყࢢ ܻၯ Z ) Ƈ ƄƎܻީޯࠀ Ƅ ƣ ܾࢸ ࠦࢵ ƄƎኆ ቱƄ ݊ ࡺޯ഻വറ ഭ ୨‫ࡩ ࣐ީޢ‬Ƅ ݊ ‫ݹ‬Ə࣠ޯࡩƄ ࠦ‫઻ ે ߙ࢖ࢠޢ‬ၮ Ǝܻީޯࠀ ଘଓƄ Ƅ߃ ࠧ ߄ ެჴఱ ఓ ଘଓƥ ‫ۑ‬Ƈ ܻ‫ߙޟ‬ x2) Ǝܻީޯࠀ Ƅ ƣ ܾߙ ᑑ ᐽኅ ቱƥ ‫(ۑ‬ PARAMETRIC TEST ) ࠦ‫ޢ‬൞൙‫ݘ‬ଐଌƄ Ƈ ƄƎܻީޯࠀ ଘଓܻ‫ࡺ ࣩ ހ‬ఱ ఠ ཯དྷࡩƄƥ ( F ، T ، ፥ ጿ ơ ܻஊै Ƅ ࣠ࠧ ሯሡሁ ƥ ‫ (ۑ‬NON PARAMETRIC TET) ࠦ‫ޢ‬൞൙‫ࡷݘ‬ൎᑉᐽƄ Ƈ ƄƎܻީޯࠀ ଘଓƄ ࡼ ࡷ ࠢ‫ ހܾރ‬ƞ ࢖‫(ݹݘ‬ ‫ ݝ‬༨ဈ Ƈ ܻၰ ܻࣀ ଐଌƄ Ƨ ƥܻఱ ఠ ( x2) ݊ ‫ݹ‬Ə࣠‫ ކ‬ቻቱƧ ኅ ቱƄ ፥ ገƄ࣠ࣃ ‫ࡩݘ‬Ƅ ౰ే‫ݓ‬ ޯ൞ᒑᐽ Ƈ ଘଓܻࡺޯࠁ ଘଓƄ ƣ ܾߙ ഭ ୨‫ࡩ ࣐ީޢ‬Ƅ ݊ ‫ݹ‬Ə࣠ޯࡩƄ ( x2) ơ ƥ߄ ၴ ߰ ࡺఱ ఓƥ ‫ݹ ݊ۑ‬Ə࣠ޯࡩƄ Ƅ߃ ᐱᏳ ƒ ܻၯ ơ ƥ߄ ၴ ƢƄ߄ ႀ ޯࢵ ܻ‫ܻࠧ ހ‬ƌܻ༬࿻ Ƅ ࡼ ࡓ ೘಼ƥ ‫ (ۑ‬x2) Ƈ ܻ‫ࡷެ ࠒެޢ‬ ‫ݹࠦ ࡷݘޢެࠦۑ‬༇໪ Ƈ ܻၴ Ǝƌ ߄ ެ‫ ݊ݑ‬୆กमऒ Ƨ ᇴᆝ যওߘ ƌܻ༬࿻ ܺᄣ 155 ܸୃกܸ଍ଉŽ–ໝ ແࡦफउ –ࠣ‫ݶ‬ Ɖܸࢤ ެߕକଐƁƢࠣ‫ࡦ ࢽޟܸ ࢲޟ‬ƁƢࠣ‫ޟ‬පඓ ൷Ɓ౭౑ࢲ କଐƁƄ ܸࢲ ƁƋኃ ᒔᐺ። ጏ࢓‫ݕ‬ ࡦƁ࣬ ࣊ Ɓ࢓ߢೕಹኃ ቮƁ೾೥࢓଍ଉƁƄ ƁƋƁ߁ ࢟ ſ ። ፎକଐ߁ ެࢲ କଐƁƢ। ्࢟ ࣝࡦƁŻܸࢤ ߾ ଔଐƁ ƛ ܸߢࢽ ࡦƁ߁ ዖዄ Ž૊ સၫƋࣝެ࠾ኃ ቮƁƊܸެࢲ ଘଐƁ ( x2) যওߘ ໠ ໄ‫ݹࠦ ހ‬ƌ࢖࣍ ࠦߘൎ൉ၮ Ơ ܻެࠧ ‫࢖ݹࠦۑ أي أن‬ஊै Ƅ Ƈ ܻၴ Ǝƌ ે ઻ၮ ࠢ‫ݶ‬ Ƅ࣠ޯࡩƄ ߎ ߘ࣠୓෺ ݊ ୆กमऒ Ƨ ᇴᆝ ཀྵཏࠒ ެࡷ ݊ ୆กमऒƧ ᇴᆝ ཀྵཏࠒ ެࡷ : ‫ ێ‬ᇬ ᇊ ࢵ . ࠦ‫࢖ݹ‬ஊै Ƅ Ƈ ܻၴ Ǝƌƥ ݇ ୃ෾फएƤ ᇱᆚ སཌࠏ ީࡴ࢚ ‫߽ ࢤܸݳ‬ 1 ཀྵཏࠒ ެଐଌƄ‫ ݝ‬༨ဈ ࠦ‫ޢ‬ᆿᇍ ࡩƄࠦၰ ܻࣀ ଐଌƄ .1 ࡼ ࡷ१ ॏߙଘଓƄƎ࣠ഷവറƄࡼ ࡷܻ‫ހ‬౰౔ߥࡷ໠ ໄ৤ৈষƄ፟ ፑƄࠦ‫ܻݹ‬ᐱ᏶ଘଓܻࡷ፟ ፑƄ߄ ޯ೘಼ƥ࢖ߦࢧ ࡩƄ࣐ࠦ ߥ‫ࡷ ࡼރ‬ཀྵཏࠒ ެଐଌƄƀ߄ ୍෺ .2 ‫ࡷ ࣀࠢۑ‬ൎ൉‫ݹ‬ƣ Ƅƣ ƥƌ ໠ ໄ৤ৈষƄƤܻ༬ဈ Ƅ࣭ ࣐ ୍กƎ߄ ඳච ൭Ꭸ᎜ࠦࡺߥࡩƄ፟ ፑƄࠦၮྲ ྜ྅݊ ߦ‫ކ‬౰౔ߙ໠ ໄ৤ৈষƄ፟ ፑƄƧ ࣠ޯ࡯‫ݹ‬ƥࡣ ‫ࡷޯࡺܻމ‬౰ేၩཀྵཏࠒ ެଐଌƄ .3 ‫ۑ‬ਔ ৲Ƅ౮౔ࡷᇯ ᇍ ఴ చഭ ୨‫ࡩ ࣐ީޢ‬Ƅཀྵཏࠒ ެଐଌƄᇯ ᇍ ࢸ ߃ ၯ ܾ‫ކ‬Ƈ ܻ‫ެ ࠒެޢ‬ଐଌƄƣ ܻߙࠦ‫࢖ݹ‬ஊै ƄƇ ܻၴ Ǝƌƌ߄ ၮƌƄƏܻ൞ᇅᆝƥ (GOODNESS OF FIT ) ࢄ ‫ࡦެࣝߖޟ‬Ɓ ƃƉࣝ஋ॅ ( x2)Ƃ ܸࢽ ߾  Ǝܻީޯࠀ ଘଓ ( x2 ݊ ୆กमऒ –Ƨ ᇴᆝ ݊ ‫ݹ‬Ə࣠‫ކ‬Ƣ߄ ႀ ޯఱ ఓ ƌ߄ ၮƣ ࣠ࡓ ‫ޯ࣠ߘݘࠦݑެ ߄ࡷܻݹ‬ଐଌƄƇ ƄƎƄಒ౷ޯࡩƄࡼ ‫ ݑ‬ƨܻ‫ޯ࡯ ߎ ࡷݘެ࣠ݹ‬༧ဈ Ɔ߄ ܻࠧࣃ ଐଌƄƇ ƄƎƄಒ౷ޯࡩƄบ คᇴᆝ ƄƍƂ 1) ū໠ ໄ୐ชƀࡼ ࡷ౰౗ࡁ ƀࠦެࡓ ࡺଐଌƄၖ࿟ Ƅ࣠ެࡩƄ 156 ܸୃกܸ଍ଉŽ–ໝ ແࡦफउ –ࠣ‫ݶ‬ Ɖܸࢤ ެߕକଐƁƢࠣ‫ࡦ ࢽޟܸ ࢲޟ‬ƁƢࠣ‫ޟ‬පඓ ൷Ɓ౭౑ࢲ କଐƁƄ ܸࢲ ƁƋኃ ᒔᐺ። ጏ࢓‫ݕ‬ ࡦƁ࣬ ࣊ Ɓ࢓ߢೕಹኃ ቮƁ೾೥࢓଍ଉƁƄ ƁƋƁ߁ ࢟ ſ ። ፎକଐ߁ ެࢲ କଐƁƢ। ्࢟ ࣝࡦƁŻܸࢤ ߾ ଔଐƁ ƛ ܸߢࢽ ࡦƁ߁ ዖዄ Ž૊ સၫƋࣝެ࠾ኃ ቮƁƊܸެࢲ ଘଐƁ (3 ū༇໩ ƀ݊ ‫ݹ‬Ə࣠‫ކ‬Ƨ ƀƥƀūഭ ୨‫ࡩ ࣐ީޢ‬Ƅƥƀū௪ ௎߄ ஊै ƄƄƍࠦެ‫ࡩݘޢ‬ƄࠢެࡷƇ ߃ ၯ ſƧ ኅ ቱƄ݊ ‫ݹ‬Ə࣠ޯࡩƄƣ ᇴᆝ ƄƍƂ 2) ‫ۑ‬ଘଓƢƀ໠ ໄ࡯ߥޯࣀ ࡷƣ Ƅ౰ే‫ݓ‬ ޯࡷƣ ᇴᆝ ƄƍƂ : ࠦ‫ݓ‬ ‫ ހܻࡩ ࢧޢ‬Ɔܻ࣐ ‫ࡩݘޢެࠦ ࡷݘ‬Ƅ Ƈ ܻ‫ܻރ‬୆ก ࡼ ࡷ ࠦ‫ ࣀ࣠ހ‬ഷവറƄ x2ࠦ୆෷ܻࢧ ࠁ ᓘƥ ( f0  fe )2 x  fe 2 Ɔ߄ ܻࠧࣃ ଐଌƄƇ ƄƎƄಒ౷ޯࡩƄ = f0 ࠦ‫ޯ࣠ߘݘ‬ଐଌƄƇ ƄƎƄಒ౷ޯࡩƄ = fe ‫߾ޟݜ‬ ࠦ‫࢖ݹ‬ஊै ƄƇ ܻၴ Ǝƌƥࠦ‫ݘެ࣠ݹ‬ଐଌƄƦ ࣠ޯࣀ ࡷ߄ ެ‫ݑ‬x2ࠦ‫ࡩޢ‬ƥ߄ எै Ƅࠦࡺ‫ࡩߥޢ‬Ƅࡼ ࡷ౰౎ࡁ ƀࠦ‫ ࣀ࣠ހ‬ഷവറƄ x2บ คᇴᆝ Ƅƍܺߙ ‫ۑ‬H1ࡣ ‫ࡩީ ߄ݹ‬ƄƓ ࢖ߦࡩƄᅶै ܻࢧ ࡩ H0፥ ጹ߄ ‫ࡩݘ‬ƄƓ ࢖ߦࡩƄ࢛ ߙमअūƆƌƥ߄ ഷവറƄ ࠦ‫ݓ‬ ‫ހܻࡩ ࢧޢ‬Ɔܻ࣐ ‫ࡩޯ࣠ߙޢ ࢇࡷݘ‬ƄƆƌ࣠ࠅ Ǝܻީޯࠀ ଘଓࠦ‫࢖ݹ‬ஊै Ƅ Ƈ ܻၴ Ǝƌ df = c - m – 1 Ƈ ܻ‫ࡩߦޟ‬Ƅƌ߄ ၮ = c ‫ࡩݘޢެࠦۑ‬ƄƇ ܻ୆෷ܻࢧ ࠁ Ƃࡼ ࡷܻࠧमअ߄ ߥ‫ކ‬Ƨ ࢖༬࿻ ཯དྷࡩƄ݊ ࡺޯ഻വറƄᎨ፻ܻ‫ࡷݘ‬ƌ߄ ၮ = m Ơ Ž‫߾ޟݜ‬ ࠦ‫ݓ‬ ‫ހܻࡩ ࢧޢ‬Ɔܻ࣐ ‫ࡷݘ‬ƣ Ƅ౰౔ߘଘଓƄơ ƥƄ߄ ၴ Ƈ ƄƎܻީޯࠀ Ƅƥƀơ ൎ൉ߥޯࢵ ଘଓƄƇ ƄƎܻީޯࠀ ଘଓࠦ‫࢖ݹ‬ஊै ƄƇ ܻၴ Ǝƌ df = (r - 1)(c - 1) ƣ Ƅ౰౔ߘଘଓƄơ ƥ߄ ၴ ፥ ጿƞ ࣠ߦࢧ ࡩƄƌ߄ ၮ = r ‫ۑ‬Ɔ߄ ዙኸ ଛଓƄƌ߄ ၮ = n fe  157  r c n ‫߾ޟݜ‬ ƣ Ƅ౰౔ߘଘଓƄơ ƥ߄ ၴ ࡼ ࡷࠦ‫࡯ޢ‬ၯ ᇯ ᆝ ፟ ጿ݊ ߘ࣠ޯଐଌƄƎƄಒ౷ޯࡩƄƣ ࣠ࡓ ‫ݹ‬ƥ ‫ۑ‬፥ ፑܻዙዋ ଗଓƄࠦެ‫ࡩݘޢ‬Ƅࡴ ᄽ஛ = n ‫ࠁޢ ݟ‬ ܸୃกܸ଍ଉŽ–ໝ ແࡦफउ –ࠣ‫ݶ‬ Ɖܸࢤ ެߕକଐƁƢࠣ‫ࡦ ࢽޟܸ ࢲޟ‬ƁƢࠣ‫ޟ‬පඓ ൷Ɓ౭౑ࢲ କଐƁƄ ܸࢲ ƁƋኃ ᒔᐺ። ጏ࢓‫ݕ‬ ࡦƁ࣬ ࣊ Ɓ࢓ߢೕಹኃ ቮƁ೾೥࢓଍ଉƁƄ ƁƋƁ߁ ࢟ ſ ። ፎକଐ߁ ެࢲ କଐƁƢ। ्࢟ ࣝࡦƁŻܸࢤ ߾ ଔଐƁ ƛ ܸߢࢽ ࡦƁ߁ ዖዄ Ž૊ સၫƋࣝެ࠾ኃ ቮƁƊܸެࢲ ଘଐƁ ࡼ ࡷࠦၮܻީଐଌƄƇ ܻ‫ࡩޯ࡯ޢߦ࢔ݹ࣠ރ‬Ƅࡼ ࡷ 30%ƣ ƀࠦ‫ܻࢠޢ‬ଐଌƄࠢ‫ކ‬౰౎ၯ ࡼ ࡷƧ Ǝܻ༬ဈ ࡣ ༨ீ ߄ ၴ ƥ ( ‫ ێ ێ‬ƞ ܸޯࡴ ‫ ࠅ ݪ‬Ƅ࣠ࡩƄƣ ƥ࢔ശവറƄࡴ ࠅ ਔ ৲߄ ႁ ޯࡩ‫ۑ‬౰ేީࡓ ᚻࡩƄࡴ ᓉᒴၤ Ƅࡼ ࡷ 30%ź࣊ ࢵ ࣠ޯଐଌƄࡴ ᓉᒴၤ Ƅࡼ ࡷ 40%ź౰ే‫ݓ‬ ࢧ ࡩƄࡴ ᓉᒴၤ Ƅ ƣ ࣠‫ޯ࡯ޢߦ࢔ݹ‬ᑃᐽࠦୖ෺߄ ஊै ƄƇ ܻ‫ީޢݘ‬ଐଌƄࡼ ࡷ 100ࡼ ࡷࠦ୆෷Ƅ࣠ࣃ ‫ݑޢެࠦݑ‬मअ߄ ଐଌƄ߃ ၯ ƀūƖ ࣠‫ރ‬ᇯ ᆝ ࡼ ࡷࠢ‫ހ‬ƕ ܻߦޯࠁ ଘଓƄ ƢƄ߄ ႀ ޯࢵ ܻ‫ۑހ‬౰ేީࡓ ᚻࡩƄƖ ࣠ެࡩƄࡼ ࡷ 40ź࣊ ࢵ ࣠ޯଐଌƄƖ ࣠ެࡩƄࡼ ࡷ 40ź౰ే‫ݓ‬ ࢧ ࡩƄƖ ࣠ެࡩƄࡼ ࡷ 20ܻᐱᐟࡷƣ ƀ߄ ၴ ࣠ߙ ‫ۑ‬ƨƄਔ ৯ܻࢵ ơ ƄƏଘଓ࿃ Ⴉ ܻଐଌƄ፥ ጿƇ ܻ‫ީޢݘ‬ଐଌƄ࣊ ೘ೆƣ ƀቿቷ‫ݶ‬ ܻߥࡩƄࠦ‫ࡩߦ࢖ࢠޢ‬Ƅ౰౎ޯࠀ ƀū5%ࠦ‫ࡷݘެ࣠ݹ‬Ʀ ࣠ޯࣀ ࡷ ࡣ ஊै Ƅ Ƈ ܻ‫ࡩߦ࢖ࢠޢ‬Ƅ H0 Ƅਔ ৯ܻࢵ ơ ƄƏଘଓ࿃ Ⴉ ܻଐଌƄ፥ ጿƇ ܻ‫ީޢݘ‬ଐଌƄ࣊ ೘ೆ H1 ࿃ Ⴉ ܻଐଌܻ‫ރࠦހ‬ƎܻߥࡷƇ ܻ‫ީޢݘ‬ଐଌƄ࣊ ೘ೆ፥ ጿ࣊ ೘ೆ፥ ጿƞ ൎ൉ޯࠀ Ƅ߄ ၴ ࣠‫ݹ‬ ࠣࢵ ܸࣀ ࡦƁࡱ ᄺ஘ ‫ ߾ ࢽ ݧ‬Ơ ࣝ‫ࡦެ࡬ޟߣ࢑ݶ‬Ɓƃ࢑ࠦࠂ ଘଐࠣ‫ݕ‬ ߕࣝެ଍ଉƁƢƃ߁ ࠤܸࣀ ଍ଉƁƄ ܸ‫ݶ‬౭౑ࣀ ଍ଉƁ ‫ ێ‬ƞ Ƣ߁ ၱ f0߄ ܻࠧࣃ ଐଌƄ࣊ ৤৒ষƄ ࿃ Ⴉ ܻଐଌƄ፥ ጿ࣊ ৤৒ষƄfE ౰ేީࡁ 20 30 Ƈ ܻࢸ ܻࣃ ࡩƄࡴ ᄽ஛ ࣊ ࢵ ࣠ޯࡷ ౰ే‫ݓ‬ ࢢ 40 30 40 40 ፥ ፑܻዙዋ ଗଓƄ 100 100 df = c – m – 1 = 3 – 0 – 1 = 2 ( ‫ ۑ‬፥ ፕ 2ࠦ‫ݹ‬༇໪ ࠦၴ Ǝਔ ৹ %5ࠦ‫ࡷݘެ࣠ݹ‬Ʀ ࣠ޯࣀ ࡷ߄ ެ‫ࡩޢࠦݑ‬ƥ߄ எै Ƅ x2ࠦࡺ‫ߘޢ‬ ( f  f )2 (2030)2 (40 40)2 (4030)2 102 02 102 100 100 x2   0 e          6.66 fe 30 40 30 30 40 30 30 30 ࠦ‫ࡷݘެ࣠ݹ‬Ʀ ࣠ޯࣀ ೘ೃ x2 = 5.99ࠦ‫ࡩޢ‬ƥ߄ எै Ƅࠦࡺ‫ࡩߥޢ‬Ƅࡼ ࡷ࢖‫ݓ‬ ࢢ ƀ x2 = 5.83ࠦ‫ ࣀ࣠ހ‬ഷവറƄࠦࡺ‫ࡩߥޢ‬Ƅƣ ƀ‫ ࠁޢ ݟ‬ƥ ፥ ጿƞ ൎ൉ޯࠀ Ƅ߄ ၴ ࣠‫ݹ‬Ƨ ƀ‫ۑ‬ቿቷ‫ࡩީ ߄ݹ‬Ƅ H1ࡣ ީߥ‫ރ‬ƥū Ho࢛ ߙमएܻ୐คܺߙ 2ࠦ‫ݹ‬༇໪ Ƈ ܻၴ Ǝƌƥ  = 0.05 ‫ۑ‬࿃ Ⴉ ܻଐଌƄ፥ ጿƄਔ ৯ܻࢵ ƣ ᇴᆝ ܻ೘ೃࠦ‫ރ‬Ǝܻߥࡷܻ‫ܻࡩޢ‬ၰ Ƈ ܻ‫ࡩޯ࡯ߦ࢔ݹ࣠ރ‬Ƅ፥ ጿƇ ܻ‫ީޢݘ‬ଐଌƄ࣊ ೘ೆ Ƈ ƄƎܻ‫ࡩ ࣀޢ‬Ƅƌ߄ ၮࡼ ‫ ۑ ݑ‬ơ ƥ߄ ၴ ፥ ጿࠦ༨༤ ࣠ଐଌƄƇ ܻ‫ࡩީޢܻރ‬ƄƇ ƄƎܻ‫ ࢵޢ‬༇໮ ܻ‫ ݊ކ‬ዙዋ :( ‫ ێ ێ‬ƞ ܸޯࡴ ࡴ ࠧƎܻዙኸ ƀžൎ൉ዙኸ ܻᐱᏬ ౰౔ఴ ఓ཯དྷࡩƄƥūࠦެࢵ 30ࡼ ࢵ ‫ ݝ‬༨ဈ ࡴ ࠧƎܻዙኸ ƀžൎ൉ዙኸ ܻࡺᐱᏬ ౰౔ఴ ఓ཯དྷࡩƄࠦ‫࡯ޢ‬ഷവറƄƥࠦ‫ޢ‬ხެࠅ ଛଓƄ 158 ܸୃกܸ଍ଉŽ–ໝ ແࡦफउ –ࠣ‫ݶ‬ Ɖܸࢤ ެߕକଐƁƢࠣ‫ࡦ ࢽޟܸ ࢲޟ‬ƁƢࠣ‫ޟ‬පඓ ൷Ɓ౭౑ࢲ କଐƁƄ ܸࢲ ƁƋኃ ᒔᐺ። ጏ࢓‫ݕ‬ ࡦƁ࣬ ࣊ Ɓ࢓ߢೕಹኃ ቮƁ೾೥࢓଍ଉƁƄ ƁƋƁ߁ ࢟ ſ ። ፎକଐ߁ ެࢲ କଐƁƢ। ्࢟ ࣝࡦƁŻܸࢤ ߾ ଔଐƁ ƛ ܸߢࢽ ࡦƁ߁ ዖዄ Ž૊ સၫƋࣝެ࠾ኃ ቮƁƊܸެࢲ ଘଐƁ ࡼ ࢵ ࡼ ‫ݑ‬ƨൎ൉ީߥޯࣀ ࡷ ࠦ‫࡯ޢ‬༨ீ ƥƀࠦ‫ޢ‬ხެࠅ ƀ ƤƄ౰౔ࣃ ଐଌƄƆƎܻ‫ࡩ ࣀޢ‬ƄƖ ࣠‫ރ‬ƣ ᇴᆝ ƄƍƂܻࡷ౰౎ޯࠀ ƀ‫ۑ‬౰౗ࡁ ܾߙࠦެࢵ 30ࡼ ࢵ ‫ۑ‬%ࠦ‫ݑެ ߄ࡷݘެ࣠ݹ‬1Ƨ ౰౔ࣃ ଐଌƄ Ƅ ƁƋܸ‫ࡦ ࢽޟ‬ƁƤ ౭౑ࣀ ଍ଉƠ Ɓ౭౑ߕକଐƁƞ Ƣ߁ ၱ ‫ ێ‬ƞ Ƣ߁ ၱ ࡼ ࣀ ࡩƄ ࠦ‫࡯ޢ‬༨ீ ، 30‫ ݝ‬༨ဈ ƆƎܻ‫ࡩ ࣀޢ‬ƄƖ ࣠‫ރ‬ 21 ౰౗ࡁ ܾߙū 30 29 ፥ ፑܻዙዋ Ƃ 50 ࠦ‫ޢ‬ხެࠅ ƀ 40 80 120 ፥ ፑܻዙዋ ଗଓƄ 70 100 170 ࡣ ஊै Ƅ Ƈ ܻ‫ࡩߦ࢖ࢠޢ‬Ƅ HoƧ ౰౔ࣃ ଐଌƄࡼ ࢵ ࡼ ‫࡯ޢࠦ ࡷ ࣀޯߥ ࡣݑ‬༨ீ ƥƀࠦ‫ޢ‬ხެࠅ ƀ ƤƄ౰౔ࣃ ଐଌƄƆƎܻ‫ࡩ ࣀޢ‬ƄƖ ࣠‫ރ‬ H1Ƨ ౰౔ࣃ ଐଌƄࡼ ࢵ ࡼ ‫ݑ‬ൎ൉ߥޯࣀ ࡷࢪ ყࡩ ࠦ‫࡯ޢ‬༨ீ ƥƀࠦ‫ޢ‬ხެࠅ ƀ ƤƄ౰౔ࣃ ଐଌƄƆƎܻ‫ࡩ ࣀޢ‬ƄƖ ࣠‫ރ‬ 1ƌ࣠ዙኸ ƥ 1ߎ ࢢ ፟ ፑƥଛଓƄࠦ‫࡯ޢ‬உै Ƅ፥ ጿࠦࡺ‫ࡩߥޢ‬Ƅ ‫ ۑ‬ơ ƥ߄ ၴ ࠦ‫ޯ࣠ߘݘ‬ଐଌƄƇ ƄƎƄಒ౷ޯࡩƄơ ƥ߄ ၴ ࿁ Ⴁჱ჈ fe   r  c  (70)(50)  21 n 170 ࠦߥ‫ࡩ ࣐࢖ݹ‬Ƅࢪ ߦ୐กܻ‫ݹ‬ൎ൉உै Ƅࠦ‫ހߥޢ‬যওߘે ઻ၮơ ࣠ࢧ ஊै Ƅࡼ ࡓ ೘಼ƥ ࡼ ࣀ ࡩƄ ، 30‫ ݝ‬༨ဈ ౰౗ࡁ ܾߙū 30 159 Ƅ ƁƋƁಏ౴ެᑀᐺƃ࢓ࣉ ܸީ଍ଉƁࠣ‫ݕ‬ ߕࣝެ଍ଉƁƄ ƁƋƁಏ౴ެࡦƁƞ Ƣ߁ ၱ ‫ ێ‬ƞ Ƣ߁ ၱ ƆƎܻ‫ࡩ ࣀޢ‬ƄƖ ࣠‫ރ‬ ࠦ‫࡯ޢ‬༨ீ ࠦ‫ޢ‬ხެࠅ ƀ 29 71 21 49 ፥ ፑܻዙዋ ଗଓƄ 70 100 ܸୃกܸ଍ଉŽ–ໝ ແࡦफउ –ࠣ‫ݶ‬ Ɖܸࢤ ެߕକଐƁƢࠣ‫ࡦ ࢽޟܸ ࢲޟ‬ƁƢࠣ‫ޟ‬පඓ ൷Ɓ౭౑ࢲ କଐƁƄ ܸࢲ ƁƋኃ ᒔᐺ። ጏ࢓‫ݕ‬ ࡦƁ࣬ ࣊ Ɓ࢓ߢೕಹኃ ቮƁ೾೥࢓଍ଉƁƄ ƁƋƁ߁ ࢟ ſ ƛ ܸߢࢽ ࡦƁ߁ ዖዄ Ž૊ સၫƋࣝެ࠾ኃ ቮƁƊܸެࢲ ଘଐƁ ፥ ፑܻዙዋ Ƃ 50 ። ፎକଐ߁ ެࢲ କଐƁƢ। ्࢟ ࣝࡦƁŻܸࢤ ߾ ଔଐƁ 120 170 df = (r – 1)(c – 1) = (2 – 1) (2 – 1) = 1 x2   ( f 0  f e )2 (30  21) 2 (40  49)2 (20  29) 2 (80  71)2      9.44 fe 21 49 29 71  ߄ ެ‫ݑ‬ᑑ ᐽƍƥ ‫ࡩޢࠦ ۑ‬ƥ߄ எै Ƅ x2ࠦࡺ‫ࡩߥޢ‬Ƅࡼ ࡷ౰౎ࡁ ƀ ‫ ࣀ࣠ހࠦ ۑ‬ഷവറƄ x2ࠦࡺ‫ߘޢ‬ƣ ƀ‫ ࠁޢ ݟ‬ƥ ƆƎܻ‫ࡩ ࣀޢ‬ƄƖ ࣠‫ރ‬ਔ ৲߄ ༨ဈ ፥ ጿƨൎ൉ࡷܻၮࢪ ყࡩࡼ ࣀ ࡩƄƣ ܾ‫ݶ ࡣހ‬ ܻߥࡩƄ H0࢛ ߙमएƄ߃ ಛಗƥū df = 1 ‫ = و‬0.01 ൎ൉ߥޯࣀ ࡷࢪ ყࡩ ࠦ‫࡯ޢ‬༨ீ ƥƀࠦ‫ޢ‬ხެࠅ ƀ ƤƄ౰౔ࣃ ଐଌƄƆƎܻ‫ࡩ ࣀޢ‬ƄƖ ࣠‫ރ‬ƣ ƀ཯ཏ‫ݹݘ‬Ƨ ኅ ቱƄƥ H1 ࡣ ީߥ‫ރ‬ƥƤƄ౰౔ࣃ ଐଌƄ ‫ޢࠦۑ‬ხެࠅ ଛଓƄƇ ƄƎܻ‫ࡩ ࣀޢ‬ƄžƄྲ ႖፟ ፑƂƥ߄ ୍෺ܻࡺ‫ߙޢ‬ƣ ॅळ‫ޢ‬೘಼ƨܻެჱࢵ ࢖‫ݓ‬ ࢢ ଛଓƄƣ ƀ፟ ፑƂ཯᝵ᐢჱ჋ƥ Ƨ ౰౔ࣃ ଐଌƄࡼ ࢵ ࡼ ‫ݑ‬ ௧ ோܸަެࡦƁࡠ ‫࡬ޟ‬༥စ ໝ ແ࣍ ࢲ ࣝެࡴࡹ ࡴ౭౔࠾ ଘଐƐ Ƣ࢓ߣࡦƁƋܸަެ߽ Ɓ ‫ێ‬ Analysis of variance (ANOVA) ܻࡷ߄ ެ‫ޯ࡯ߦࠦݑ‬༧ீ ƥƀࠦ‫ݹ‬ƥܻࣀ ჴࡷ໠ ໄ‫ޯࡺݘ‬༬ீ ࡼ ࡷ౰౗ࡁ ƀƇ ܻ࣐ ࢵ ࣠ޯࡷƣ ƀƓ ࢖ߙƎܻީޯࠀ ଘଓ௪ ௎ܻީޯࡩƄࡣ ‫࡯ޢ‬༨ဈ Ƣ߄ ႀ ޯఱ ఓ ƣ ƀ፟ ፑƄ ௪ ௎ܻީޯࡩƄࡣ ‫࡯ޢ‬༨ဈ ࠦ‫ࡺޢ‬ఱ ఠƌ࣠‫ކݘ‬ƥ‫ۑ‬௪ ௎ܻީޯࡩƄƧ ƥܻఱ ఠ݊ ࡷƨܻ‫ ࣍ީޢݘޢ‬ƨܻ‫ݹݘ‬Ə࣠‫ࠦ ކ‬ၮƏ࣠ࡷƇ ܻ‫ޯࡺݘ‬഻വറƄƣ ࣠ࡓ ‫ކ‬ ࡣ ‫࡯ޢ‬ႁ ޯࡩƄƞ ߄ ᐱᐢఱ ఓƥ‫ࡩݘޢެࠦۑ‬ƄƇ ܻ‫ܻރ‬୆ก໠ ໄ‫ހ‬௪ ௎ܻީޯᑃᐽൎ൉‫࡯ޢ‬༨ဈ ࡼ ࡺࢥ ୓ขƇ ܻ࣐ ࢵ ࣠ޯଐଌƄ໠ ໄ‫ރࠦހ‬ƎܻߥଐଌƄƇ ƄžƄ༇໮ Ƃ ƌ࣠‫ܻކݘ‬೘ೆƀܻࡺᐱᐟყჅƇ ܻߙൎ൉ޯࠀ ƄƧ ƀƣ ƀƥ୍ܻ෺࢖ߥ‫ݹࠦކ‬ƥܻࣀ ჴࡷƇ ܻ‫ޯࡺݘ‬഻വറƄƇ ܻ࣐ ࢵ ࣠ޯࡷบ คᇴᆝ ƄƍƄܻࡺ‫ߙޢ‬ƣ ܻ‫ޢ‬ხ჋ ࡼ ࡷၖ࿟ ܻ‫ރ‬௪ ௎ܻީޯࡩƄƣ ƀƣ ƄƢƀūഭ ୨‫ݹݘࠢ ࣍ީޢ‬Ə࣠‫ޯࡺ ݊ކ‬༬ீ ࡼ ࡷࠦ୆෷Ƅ࣠ࣃ ‫ݑ‬Ƈ ܻެ‫ ࡓ ࡼކ࣠ߘݘܻࠩࡷ ࡼݑޢ‬೘಼ƥࠦߙ߄ ࢧ ᑃᐽ ‫ࡺܻ ࠅ࣠ࠧ࢖ݹࠦۑ‬ᐱᐟყჅƟ ƥ࢖ߦࡩƄƥ ࠦ‫ݹ‬ƥܻࣀ ჴࡷ౰ేၩ Ƈ ܻ࣐ ࢵ ࣠ޯଐଌƄƞ ൎ൉ޯࠀ Ƅ ௧ ோܸަެࡦƁࡠ ‫࡬ޟ‬༥စ ŻƁ༄໫ ŽƄ Ɓࣝ࣍ ߽  ( MSA Ƈ ܻެ‫ࡩݘޢ‬ƄƇ ܻ࣐ ࢵ ࣠ޯࡷ໠ ໄ‫ހ‬௪ ௎ܻީޯࡩƄࡼ ࡷ݊ ࡺޯ഻വറƄ௪ ௎୍ܻงƎ߄ ߥ‫ ރ‬.1 (MSE Ƈ ܻެ‫ࡩݘޢ‬Ƅࡣ ၯ Ƅƌ௪ ௎ܻީޯࡩƄࡼ ࡷ݊ ࡺޯ഻വറƄ௪ ௎୍ܻงƎ߄ ߥ‫ ރ‬.2 ( MSA / MSE ) Fࠦީࣀ ჱࡩƄ‫ ࣀ ݪ‬༨စ .3 F= Ƈ ܻެ‫ࡩݘޢ‬Ƅ࣊ ࢵ ࣠ޯࡷ໠ ໄ‫ހ‬௪ ௎ܻީޯࡩƄ Ƈ ܻެ‫ࡩݘޢ‬Ƅࡣ ၯ Ƅƌ௪ ௎ܻީޯࡩƄ ࠦެ‫ݘޢ‬ଐଌƄࠦ‫࢖ݹ‬ஊै ƄƇ ܻၴ Ǝƌƥࠦ‫ݘެ࣠ݹ‬ଐଌƄƦ ࣠ޯࣀ ࡷ߄ ެ‫ࡩޢࠦݑ‬ƥ߄ எै Ƅ Fࠦࡺ‫ࡷ ࡼߘޢ‬౰౎ࡁ ƀࠦ‫ ࣀ࣠ހ‬ഷവറƄ F บ คᇴᆝ ƄƍƂ ‫ۑ‬H1ūࡣ ‫ࡩީ ߄ݹ‬ƄƓ ࢖ߦࡩƄᅶै ܻࢧ ࡩ࢛ ߙमअūƇ ܻ‫ޯࡺݘ‬഻വറƄƇ ܻ࣐ ࢵ ࣠ޯࡷƧ ƥܻఱ ఠࡼ ‫ݑ‬H0፥ ጹ߄ ‫ࡩݘ‬ƄƓ ࢖ߦࡩƄƣ ܺߙ 160 ܸୃกܸ଍ଉŽ–ໝ ແࡦफउ –ࠣ‫ݶ‬ Ɖܸࢤ ެߕକଐƁƢࠣ‫ࡦ ࢽޟܸ ࢲޟ‬ƁƢࠣ‫ޟ‬පඓ ൷Ɓ౭౑ࢲ କଐƁƄ ܸࢲ ƁƋኃ ᒔᐺ። ጏ࢓‫ݕ‬ ࡦƁ࣬ ࣊ Ɓ࢓ߢೕಹኃ ቮƁ೾೥࢓଍ଉƁƄ ƁƋƁ߁ ࢟ ſ ። ፎକଐ߁ ެࢲ କଐƁƢ। ्࢟ ࣝࡦƁŻܸࢤ ߾ ଔଐƁ ƛ ܸߢࢽ ࡦƁ߁ ዖዄ Ž૊ સၫƋࣝެ࠾ኃ ቮƁƊܸެࢲ ଘଐƁ ௧ ோܸަެࡦƁࡠ ‫࡬ޟ‬༥စ ƞ Ƣ߁ ၱ ‫ ێ‬ƞ Ƣ߁ ၱ Ƈ ܻၴ Ǝƌ ౰ే‫ݓ‬ ޯࡩƄƎ߄ ࢧ ࡷ Ƈ ܻ‫࢖ހݘ‬ଐଌƄƖ ࣠ዙಣಝ (Aࡣ ࡷܻ‫ࡩݘ‬ƄƤྲ ႞ߦ‫ݹ‬ SSA  r  ( X J  X ) 2 c–1 SSE   ( X iJ  X j ) 2 (r – 1)C Ɣ ܻࢵ ƥଛଓƄ໠ ໄ‫ހ‬ Ƈ ܻެ‫ࡩݘޢ‬Ƅࡣ ၯ Ƅƌ ౰ేၩƥƀܾ࣐ உै Ƅ ྲ ႞ߦଐଌƄ ፥ ፑᚼ ዙዋ ଗଓƄ ࠦ‫࢖ݹ‬ஊै Ƅ Ƈ ܻ‫࢖ހݘ‬ଐଌƄ࣊ ࢵ ࣠ޯࡷ Fࠦީࣀ ჱࡩƄ MSA SSA c1 MSE SST (XiJ  X)2  SSA SSE rc – 1 SSE (r1)c –– MSA MSE –– –– Ơ Ž‫߾ޟݜ‬ (i XiJ) / r Ɔ߄ ܻࠧࣃ ࡷ rࡼ ࡷࠦ‫ ࡓ࣠ރ‬ᅚଌƄ Jࠦެ‫ࡩݘޢ‬Ƅ࣊ ࢵ ࣠ޯࡷ X̅J (i J XiJ )/ rcƇ ܻެ‫ࡩݘޢ‬Ƅᇯ ᇍ ࡩ౰ేީࡓ ᚻࡩƄ࣊ ࢵ ࣠ޯଐଌƄ X̅̅ A = r (XJ –X̅̅)2ࡣ ࡷܻ‫ࡩݘ‬Ƅܻࠧྲ ႞ߦ‫཯ݹ‬དྷࡩƄƇ ܻ‫࢖ހݘ‬ଐଌƄƖ ࣠ዙಣಝ SSA A =  (XiJ – X̅J ࡣ ࡷܻ‫ࡩݘ‬Ƅܻࠧྲ ႞ߦ‫ݹ‬ଘଓ཯དྷࡩƄƥܾ࣐ உै ƄƇ ܻ‫ ୹ހݘ‬୪Ɩ ࣠ዙಣಝ SSE  (XiJ – X̅̅ )2 = SSA + SSE =፥ ፑܻዙዋ ଗଓƄƇ ܻ‫࢖ހݘ‬ଐଌƄƖ ࣠ዙಣಝ =SST ᇯ ᇍ ࡩ ࡣ ߦࢵ ଛଓƄࡴ ߘ༟༎Ƅ  = 0.01ܻࡷ߄ ެ‫ݑ‬ƥ ે ઻ၮଛଓƄࡴ ߘ༟༎Ƅ  – 0.05ܻࡷ߄ ެ‫ݑ‬Fযওߘ࣯ ࣐ ‫ݹݘ‬ƥ ࠦ‫࢖ݹ‬ஊै ƄƇ ܻၴ Ǝƌࡼ ࡷƉ ƥƏ c – 1 = ࣊ ࣀ ხࡩƄࠦ‫ݹ‬༇໪ Ƈ ܻၴ Ǝƌ Ƈ ܻެ‫ࡩݘޢ‬Ƅƌ߄ ၮ = c ‫ࠁޢ ݟ‬ (r – 1) c = ƢܻߥଐଌƄࠦ‫ݹ‬༇໪ Ƈ ܻၴ Ǝƌ ‫ݑޢެࠦۑ‬ᇯ ᆝ ፥ ጿƇ Ƅ߄ ܻࠧࣃ ଐଌƄƌ߄ ၮ = r ‫ࠁޢ ݟ‬ Ƥܻ‫ރ‬ƌƀ ơ ƥ߄ எै Ƅ໠ ໄ୍෺‫ࡩ ࣀݘ࢖ۑ‬Ƅࢪ ߦ୐กƥࠦߦ࡯ޯ༧ீ ࠦߦ࡯ၩƀࠦ‫މ‬ൎ൉‫މ‬፥ ጿƣ ࣠‫ࡩ ࢧܻހ‬Ƅࢪ ߦ‫ރ‬በሼྲ ႖݊ ‫ޢ‬ხ჋ :( ‫ ێ‬ƞ ܸޯࡴ ࣊ ࢵ ࣠ޯࡷƣ ᇴᆝ ƄƍƂܻࡷ౰౎ޯࠀ ƀ‫ۑ‬ƥܻࣀ ჴࡷ௪ ௎୍ܻงܻಛಗƥƨܻ‫ ࣍ީޢݘޢ‬ƨܻ‫ݹݘ‬Ə࣠‫ࠦކ‬ၮƏ࣠ࡷƇ ܻ‫ީޢݘ‬ଐଌƄ‫ۑ‬Ǝ࣠ᐳ႖ 5Ƈ ܻ‫ࡷީޢݘ‬ %5ࠦ‫ࡷݘެ࣠ݹ‬Ʀ ࣠ޯࣀ ࡷ߄ ެ‫ݑ‬ଘଓƢƀƨܻ‫ݹ‬ƥܻࣀ ჴࡷƞ ൎ൉ၩᇯ ᇍ ࡩƇ ܻ‫ީޢݘ‬ଐଌƄ 161 ܸୃกܸ଍ଉŽ–ໝ ແࡦफउ –ࠣ‫ݶ‬ Ɖܸࢤ ެߕକଐƁƢࠣ‫ࡦ ࢽޟܸ ࢲޟ‬ƁƢࠣ‫ޟ‬පඓ ൷Ɓ౭౑ࢲ କଐƁƄ ܸࢲ ƁƋኃ ᒔᐺ። ጏ࢓‫ݕ‬ ࡦƁ࣬ ࣊ Ɓ࢓ߢೕಹኃ ቮƁ೾೥࢓଍ଉƁƄ ƁƋƁ߁ ࢟ ſ ። ፎକଐ߁ ެࢲ କଐƁƢ। ्࢟ ࣝࡦƁŻܸࢤ ߾ ଔଐƁ ƛ ܸߢࢽ ࡦƁ߁ ዖዄ Ž૊ સၫƋࣝެ࠾ኃ ቮƁƊܸެࢲ ଘଐƁ (1 ƞ ൎ൉ၩ (2 ƞ ൎ൉ၩ (3 ƞ ൎ൉ၩ 83 81 91 87 78 79 90 79 81 84 82 80 82 80 ___ 88 ___ 410 ___ 400 435 ࡣ ஊै Ƅ H0: 1 = 2 = 3 H1 ≠ 1 , 2 , 3 X1  X  410  82 , 5 X2  400 435  80 ., X 3   87 5 5 410  400  435  83 ( 5 )( 3 ) SSA = 5[(82 – 83)2 + (80 – 83)2 + (87 – 83)2] = 130 SSE = (87 – 82)2 + (83 – 82)2 + (79 – 82)2 + (81 – 82)2 + (80 – 82)2 + (78 – 80)2 + (81 – 80)2 + (79 – 80)2 + (82 – 80)2 + (80 – 80)2 + (90 – 87)2 + (91 – 87)2 + (84 – 87)2 + (82 – 87)2 + (88 – 87)2= 110 SST = (87 – 83)2 + (83 – 83)2 + . . + (88 – 83)2 = SSA + SSE = 240 ANOVA௪ ௎ܻީޯࡩƄࡣ ‫࡯ޢ‬ႁ ޯࡩ ‫ ۑ‬ơ ƥ߄ ၴ ௪ ௎࣠ࡓ ޯࡩࠦߥ‫ࡩ ࣀܻހ‬ƄƇ ܻ‫ࡩީޢܻރ‬ƄƢ߄ႀ ޯఱ ఠƥ ࠦީఱ ఝ ࠦߦ࡯ၩଛଓƄƤྲ ႞ߦ‫ކ‬ Ɔ߄ ዙኸ ଛଓƄ໠ ໄ‫ހ‬ 162 ࣊ ࢵ ࣠ޯࡷ Ƈ ܻ‫࢖ހݘ‬ଐଌƄ SSA = 130 Ơ ࣝ‫ݽ‬ ܸࢤ ࡦƁࠣߣ࡬ၦଘଐ ANOVAƞ Ƣ߁ ၱ ‫ ێ‬ƞ Ƣ߁ ၱ ࠦ‫࢖ݹ‬ஊै ƄƇ ܻၴ Ǝƌ c–1=2 Ƈ ܻ‫࢖ހݘ‬ଐଌƄƖ ࣠ዙಣಝ MSA = 130/2 = 65 ౰ే‫ݓ‬ ޯࡩƄ MSA/MSE = 65/9.17 =7.09 ܸୃกܸ଍ଉŽ–ໝ ແࡦफउ –ࠣ‫ݶ‬ Ɖܸࢤ ެߕକଐƁƢࠣ‫ࡦ ࢽޟܸ ࢲޟ‬ƁƢࠣ‫ޟ‬පඓ ൷Ɓ౭౑ࢲ କଐƁƄ ܸࢲ ƁƋኃ ᒔᐺ። ጏ࢓‫ݕ‬ ࡦƁ࣬ ࣊ Ɓ࢓ߢೕಹኃ ቮƁ೾೥࢓଍ଉƁƄ ƁƋƁ߁ ࢟ ſ ። ፎକଐ߁ ެࢲ କଐƁƢ। ्࢟ ࣝࡦƁŻܸࢤ ߾ ଔଐƁ ƛ ܸߢࢽ ࡦƁ߁ ዖዄ Ž૊ સၫƋࣝެ࠾ኃ ቮƁƊܸެࢲ ଘଐƁ ౰ేၩƥƀܾ࣐ உै Ƅ ࡣ ၯ Ƅƌ ྲ ႞ߦଐଌƄ SSE = 110 Ɔ߄ ዙኸ ଛଓƄ ፥ ፑܻዙዋ ଗଓƄ (r – 1)c = 12 SST = 240 Rc – 1 = 14 MSE = 110/12 = 9.17 –– F = ࠦ‫ࡩޢ‬ƥ߄ எै Ƅࠦࡺ‫ࡩߥޢ‬Ƅࡼ ࡷ౰౎ࡁ ƀ ‫ ۑ‬ơ ƥ߄ ၴ ࡼ ࡷ F = 7.09ࠦ‫ ࣀ࣠ހ‬ഷവറƄࠦࡺ‫ࡩߥޢ‬Ƅƣ ƀ‫ ࠁޢ ݟ‬ƥ ƣ ܾ‫ݶ ࡣހ‬ ܻߥࡩƄƓ ࢖ߦࡩƄƧ ƀū H0࢛ ߙमएܻ୐คܺߙū 12 ‫ و‬2ࠦ‫ݹ‬༇໪ Ƈ ܻၴ Ǝƌƥ  = 0.05߄ ެ‫ݑ‬3.88 žƄ༇໮ ଗଓƄ፟ ፑƂƎܻఴ ఓƥ‫ޯ࡯ ߎۑ‬༧ဈ ܻᐱ᏶ܾ‫ހ‬ū H1ࡣ ީߥ‫ކ‬ƥūࠦ‫ݹ‬ƥܻࣀ ჴࡷࠦߦ࡯ޯശവറƄࠦߦ࡯ၩ൑ᑉᐽƇ ܻ‫ީޢݘ‬ଐଌƄ࣊ ࢵ ࣠ޯࡷ ‫߄ۑ‬ၰ Ƅƥࡣ ࡷܻ‫ࡩݘ‬ƥƀ߄ ၰ ƄƥƤܻ༬ဈ Ƅ፥ ጿ௪ ௎ܻީޯࡩƄࡣ ‫࡯ޢ‬༨ဈ ࠢ‫ࡩ ࣀܻހ ࢇހܾރ‬Ƅ ྯ ႓ܸ‫ݕ‬ ࡦƁࡠ ࢤ ߣࡦƁ௧ ோƋܸೕೆ ƣ ࣠‫ࡩ ࢧܻހ‬Ƅࢇ ‫ݹ‬ƌܻެࢢ ƣ ܾ‫ހ‬žܻၮƌଘଓƄܻᐱᐟࡓ ೘಼ƣ ᇴᆝ ƄƍƄܻࡷ %95ࠦߥ‫ࠦމ‬ၴ Ǝਔ ৹࢖‫ކݘ‬ƣ Ƅበሼྲ ႖‫ ݌ ݪ‬मऒ .1 በሼྲ ႡࡩƄƞ ࢖‫ކݘ‬ƥ‫ۑ‬ƣ ࣠‫ࡩ ࢧܻހ‬Ƅࡼ ࡷƢƄ༇໮ 500ࡼ ࡷ౰౗ࡁ Ƅે ઻ၮƧ ࣠ޯ༨࿻ ࠢ‫ޢݘ‬ხ჋Ƨ ኅ ቱƄƟ ࣠ࠒ ࣀ ଐଌƄ Ƈ ߃ ၯ ƀ߄ ߘƥūഭ ୨‫ࡩ ࣐ީޢ‬Ƅ݊ ‫ݹ‬Ə࣠ޯࡩƄ݊ ީჴ჆ࢇ ‫ݹ‬ƌܻެࢧ ࡩܻ‫ހ‬ƣ ࣠‫ࡩ ࢧܻހ‬Ƅƣ ƄƏƥƄƣ ଛଓࠦ‫ܻࢠޢ‬ଐଌƄƆ౰౎உै Ƅࡼ ࡷ ƢƄ༇໮ 520Ƨ ƥܻఱ ఓ፥ ጒܻࣀ ஊै Ƅ࣊ ࢵ ࣠ࡩƄƣ ƄƇ ߄ ၴ ƥƥƆƌ୹ ୪ 25ܻࠩࡺᄽ஛ ࠦ୆෷Ƅ࣠ࣃ ‫ݑޢެࠦݑ‬በሼྲ ႡࡩƄ ‫ۑ‬በሼྲ ႡࡩƄžܻၮƌƄƇ ୍ܻชƄƥࠦ‫ࡩߦ࢖ࢠޢ‬Ƅࠦၩܻ‫ ࡓ ࡼ ࢢޢ‬೘಼፥ ደ ‫ۑ‬75Ƨ Ǝܻ‫ݘޢ‬ଐଌƄƞ Ƅ࢖༨စ ଘଓƄƥ Ɔ࣠ߘܻᐱᏬኆ ቱܻࠩࠖ ޯჱႾ཯དྷࡩƄƇ ൎ൉‫ހ‬ᇴᇍ ࡩƄบ คᇴᆝ ƄƍƄܻࡷƎܻީޯࠀ Ƅ‫ࡩ ࢧ࡯ ݪ‬Ƅࡼ ࡷƇ ൎ൉‫ހ‬ᇴᆝ ߴ ޯެࡷ‫ ݌ ݪ‬मअ .2 ࡼ ࡩࡣ ࣍ Ǝ 5000ࡼ ࡷࡣ ߘƄྲ ႞ࡓ ᚻᑃᐽࠦࡷƥܻߥଐଌƄƆ࣠ߘ‫ ࣍ ࡣۑ‬Ǝ 5000ܻࠧƎ߄ ߘྲ ႞ࡓ ᚻᑃᐽࠦࡷƥܻߥࡷ ߃ ၯ ܾ‫ۑ ݹ‬Ǝ౰౎ࡷƣ ƥਔ ৹ߎ ‫ࡩޢ‬ᇴᇍ ޯࡩƄ݊ ߙमऒ 5000ࡼ ࡷ౰౎ࡁ ƀྲ ႞ࡓ ᚻᑃᐽࠦࡷƥܻߥଐଌƄƆ࣠ߘƥūࠦ೘ಹൎ൉ࡷƣ ࣠ࡓ ‫ކ‬ 5100࣠ࠧྲ ႞ࡓ ᚻᑃᐽࠦࡷƥܻߥଐଌƄƆ࣠ߘ࣊ ࢵ ࣠ޯࡷƣ Ƅ߄ ༬࿻ ƥࠦ‫ ߘ ࣐ݘ‬64ࡼ ࡷࠦ୆෷Ƅ࣠ࣃ ‫ެޯ ߴݑޢެࠦݑ‬ଐଌƄ ‫ࡩ ࢧ࡯ ݪ‬ƄƇ ൎ൉‫ހ‬ᇴᆝ ƣ ܾ‫ހ‬Ɠ ࢖ߦࡩƄߴ ޯެଐଌƄࡣ ީߥ‫ݹ‬ƣ Ƅ‫ ݪ‬༬࿻ ࡣ ࠧ‫ۑ‬480࣠ࠧƧ Ǝܻ‫ݘޢ‬ଐଌƄƞ Ƅ࢖༨စ ଘଓƄƥ ‫ۑ‬%5ࠦ‫ࡷݘެ࣠ݹ‬Ʀ ࣠ޯࣀ ࡷ߄ ެ‫ݑ‬5000ྲ ႞ࡓ ᚻᑃᐽࠦࡷƥܻߥࡷƆ࣠ߘܻಛಗ ࣊ ࢵ ࣠‫ހ‬ഭ ୨‫ࡩ ࣐ީޢ‬Ƅ݊ ‫ݹ‬Ə࣠ޯࡩƄ݊ ީჴႾ߄ ެ഻വറƄƣ Əƥƣ ƀࠦ‫ܻࢠޢ‬ଐଌܻ‫ހ‬Ɔ౰౎உै ܻ‫ ࢭހ‬ყஎै ܻ‫ެޢ ߄ހ‬༬ဈ ഁ೨୹ ୪ƞ ࢖‫ ݹݘ‬.3 Ʀ ࣠ޯࣀ ࡷ߄ ެ‫ݑ‬౰౎ޯ༧࿻ ƣ ƀഁ೨࢖ଐଌƄ‫ ݌ ݪ‬मअƥ‫ۑ‬10Ƨ Ǝܻ‫ࡷݘޢ‬ƞ Ƅ࢖༨စ Ƅƥ݅ ࡁ 80Ƨ ƥܻఱ ఓ፥ ጒܻࣀ ࠁ Ƅ߃ ࠧࡣ ࡺ‫ࡩݘ‬ƥū݅ ࡁ 80ࡼ ࡷ౰౎ࡁ ƀƢܻ‫ࡩݘ‬ƄƄ߃ ࠧƧ ߄ ެ༬ீ ƣ Əƥ࣊ ࢵ ࣠ޯࡷƣ ᇴᆝ ƄƍƄܻࡷ %1ࠦ‫ࡷݘެ࣠ݹ‬ ݅ ࡁ 85ࠦެ‫ࡩݘޢ‬Ƅ፥ ጿƣ Ə࣠ࡩƄ࣊ ࢵ ࣠ޯࡷƣ Ƅ߄ ၴ ƥ‫ ࠁޢ ݟ‬Ƅ߄ ެ༬ீ 25ࡼ ࡷࠦ୆෷Ƅ࣠ࣃ ‫ ߃ݑޢެࠦݑ‬ၯ ƀ߄ ߥߙ ‫ۑ‬Ǝܻީޯࠀ ଘଓƄƄ߃ ࠧžƄ༇໮ Ƅࡼ ࡓ ೘಼ߎ ‫ۑ ࡁޢ‬ ࡼ ࡷ݅ ࡯ࡷ 100ે ઻ၮƧ ࣠ޯ༨࿻ ࠢ‫ݹ‬౰౔ఴ ఓƎܻߥ‫ݑ‬Ƈ ܻၮ༇໮ ࡼ ࡷ %90ƣ Ƅ౰౎ޯ༧࿻ ƣ Ƅॡ ॐࣃ ჴࣀ ࡷਔ ৲मअ .4 ܻᐱᐟࡷ 85ƣ Ƅ߄ ༬࿻ ƥ ࠦၮ༇໮ Ɔƌ࢖ߦࡷ 100ࡼ ࡷࠦެ‫ݑޢ‬ॡ ॐࣃ ჴࣀ ଐଌƄ߃ ၯ ܾ‫ݹ‬Ƅ߃ ࠧࡣ ࡺ‫ۑࡩݘ‬Ǝܻߥ‫ࡩݘ‬Ƅ 163 ܸୃกܸ଍ଉŽ–ໝ ແࡦफउ –ࠣ‫ݶ‬ Ɖܸࢤ ެߕକଐƁƢࠣ‫ࡦ ࢽޟܸ ࢲޟ‬ƁƢࠣ‫ޟ‬පඓ ൷Ɓ౭౑ࢲ କଐƁƄ ܸࢲ ƁƋኃ ᒔᐺ። ጏ࢓‫ݕ‬ ࡦƁ࣬ ࣊ Ɓ࢓ߢೕಹኃ ቮƁ೾೥࢓଍ଉƁƄ ƁƋƁ߁ ࢟ ſ ። ፎକଐ߁ ެࢲ କଐƁƢ। ्࢟ ࣝࡦƁŻܸࢤ ߾ ଔଐƁ ƛ ܸߢࢽ ࡦƁ߁ ዖዄ Ž૊ સၫƋࣝެ࠾ኃ ቮƁƊܸެࢲ ଘଐƁ ࠦ‫ࡷݘެ࣠ݹ‬Ʀ ࣠ޯࣀ ࡷ߄ ެ‫ݑ‬Ƅ߃ ࠧ౰౎ޯ༧࿻ ƣ Ƅॡ ॐࣃ ჴࣀ ൞ᒑᐽߎ ‫ެܻ ࢵީࠦۑ ࡁޢ‬ଐଌƄࠦ‫ޢ‬ዙኒ ࡩƄે ઻ၮƧ ࣠ޯ༨࿻ ࣊ ߥߙ ‫ۑ‬%10 ‫ و‬%5 ၖ࿏ ܻޯެࡩƄบ คᇴᆝƥ્ ઻‫ݓޢ‬ ࣃ ჴࡩƄܻࠧ࢖ዙኸ ે ઻ၮƞ ࢖‫ޯݘ‬ᑃᐽܻࠧƎܻީޯࠀ ƄᎨ᎙ࠦ୆෷ܻ‫ ࡓࠩ࢖ހ‬ᚻࡩƄ߰ ୆กܻࢧ ଐଌƄࡼ ࡷ໠ ໄၮ࣠‫ ރ‬.5 ፥ ፑܻޯࡩᇴᆝ x̅1 = 1234 (Ɵ Ƅ౰౔ၰ Ƅࠦၮܻࢵ ) …. S1 = 36… n1 = 8 x̅2 = 1136 … S2 = 40 … n2 = 7 ‫ۑ‬%5ࠦ‫ࡷݘެ࣠ݹ‬Ʀ ࣠ޯࣀ ࡷ߄ ެ‫ݑ‬Ƨ ࢖ࠧ࣠ࠅ ໠ ໄ࣐ ࢵ ࣠ޯଐଌƄ໠ ໄ‫ހ‬Ɵ ࢖ߦࡩƄࡣ ࠧ ࠦ‫ࡩޯܻࡩޢ‬ƄƇ ܻ‫ࡩީޢܻރ‬Ƅܻޯ࣐ ‫ݑ‬Ƅ፥ ፑƄ࣠ޯࡩƄે ઻ၮƇ Ƅƌ࢖ߦࡷ 5 ‫ و‬6ܻዙ዗ܻࡺᄽ஛ ƣ ܻޯჱ‫ ݑޢ‬.6 x̅1 = 40…. S1 = 8 x̅2 = 50 … S2 = 10 %5ࠦ‫ࡷݘެ࣠ݹ‬Ʀ ࣠ޯࣀ ࡷ߄ ެ‫ݑ‬໠ ໄޯჱ‫ࡩݘޢ‬Ƅ࣯ ࣐ ࢵ ࣠ޯࡷ໠ ໄ‫ހ‬Ƨ ࢖ࠧ࣠ࠅ Ɵ ࢖ߦࡩƄࡣ ࠧ ‫ و‬، ‫ ࠀެ࣠ن‬ਔ ৲ࡴ ᐱᐟࡷ 450ƣ Ƅ߄ ၴ ƥūࠦެ‫୐ࠦࡷݘޢ‬෺߄ ࡷࡼ ࡷ࢝ ༧ೞ 1000ࡼ ࡷࠦ‫ݑޢެࠦࡷ ࡓ࣠ރ‬፥ ጿ .7 ࡣ ࠧ‫ۑ‬ƣ ࣠ެࠀ ਔ ৲ࡴ ᐱᐟࡷ 400ƣ Ƅ߄ ၴ ƥƦ ༇໩ Ƅࠦ୐෺߄ ࡷࡼ ࡷ࢝ ༧ೞ 800ࡼ ࡷࠦ‫ࡷ ࡼݑޢެࠦࡷ ࡓ࣠ރ‬ ߄ ެ‫ݑ‬ᑑ ᐽƍƥ໠ ໄၯ ߄ ޯࡩƄƆƌܻၮள ஭‫୓ݘ‬෺ܻࡺ‫ ࠅ࣠ࠧ࢖ݹܻߙޢ‬ƣ ܻߦ࡯ޯ༧ဈ ໠ ໄޯჱႾ߄ ଐଌƄƣ ܾ‫ހ‬ቧሼ࣓‫ކ‬Ɔܻ࣐ ‫ݘ‬ଐଌƄƇ ܻ‫ࡩީޢܻރ‬Ƅ ‫ۑ‬%5ࠦ‫ࡷݘެ࣠ݹ‬Ʀ ࣠ޯࣀ ࡷ ࠦႅ ޯެଐଌƄቿቱ࣠ީߥଐଌƄ౰ేၩƥቿቱ࣠ީߥଐଌƄࠦ୆คƥ౰౔ࡓ ᚻࡩଘଓƄžƄ༅໮ ଘଓƄƌ߄ ၮūƤܻ‫ރ‬ƌƀƣ Ƅ౰౔ߘଘଓƄơ ƥ߄ ၴ ࣯ ࣐ ‫ ݹݘ‬.8 ƥƀơ ࣠ީߘ‫ ݪ‬༬࿻ ࡣ ࠧ‫ ࢧެ ݊ۑ‬ଐଌƄƉ ܻ୓คƀࡼ ࡷࠦ୆෷Ƅ࣠ࣃ ‫ݑޢެࠦݑ‬፥ ጿࠦߦ࡯ޯശവറƄƊ ܻީࢧ ࡩƄƇ ܻၮܻࢵ ơ ൎ൉ၯ ࠦၮܻࢵ ࡼ ‫ࡷ ࣀޯߥ ࡣݑ‬ቿቱ࣠ީߥଐଌƄƇ Ƅ߄ ၰ ࣠ࡩƄƉ ܻ୓คƀƣ ܾ‫ ހ‬%5ࠦ‫ࡷݘެ࣠ݹ‬Ʀ ࣠ޯࣀ ࡷ߄ ެ‫ݑ‬Ɠ ࢖ߦࡩƄ࢛ ߙƎ ቿቱ࣠ީߥࡷ ቿቱ࣠ީߥࡷ౰ేၩ ፥ ፑܻዙዋ ƀ 9- 8 10- 9 11- 10 60 75 80 ܻၰ ܻީࢢ 30 90 ܻၰ ܻީࢢ 25 100 ܻၰ ܻީࢢ 30 110 ‫ܻۑ‬ಛಗൎ൉ၯ ߴ ୓คƄ཯དྷࡩƄƊ ܻީࢧ ࡩƄ 12 – 11 ܻၰ ܻީࢢ 65 35 100 ፥ ፑܻዙዋ ଘଓƄ 280 120 400 ơ ƥ߄ ၴ ፥ ጿƆƌƎƄ࣠ࡩƄၖ࿏ ܻޯެࡩƄܻࡷࠦ୐෺߄ ࡷ፥ ጿ 65ࡼ ࢵ Ɵ ࣠ߙൎ൉ࡷܻၮ 37ࡼ ࡷࠦ୆෷Ƅ࣠ࣃ ‫ݑ ࣐ ݝݑޢެࠦݑ‬ƀ .9 ƈ ܻ‫ރ‬ଘଓƄƌ߄ ၮƣ ܾ‫ހ‬Ɠ ࢖ߦࡩƄ౰౎ޯࠀ ƀ %10ࠦ‫ݘެ࣠ݹ‬ଐଌƄƦ ࣠ޯࣀ ࡷƢƄ߄ ႀ ޯࢵ ܻ‫ހ‬ūƤܻ‫ރ‬ƌƀƣ Ƅ౰౔ߘଘଓƄ 164 ܸୃกܸ଍ଉŽ–ໝ ແࡦफउ –ࠣ‫ݶ‬ Ɖܸࢤ ެߕକଐƁƢࠣ‫ࡦ ࢽޟܸ ࢲޟ‬ƁƢࠣ‫ޟ‬පඓ ൷Ɓ౭౑ࢲ କଐƁƄ ܸࢲ ƁƋኃ ᒔᐺ። ጏ࢓‫ݕ‬ ࡦƁ࣬ ࣊ Ɓ࢓ߢೕಹኃ ቮƁ೾೥࢓଍ଉƁƄ ƁƋƁ߁ ࢟ ſ ። ፎକଐ߁ ެࢲ କଐƁƢ। ्࢟ ࣝࡦƁŻܸࢤ ߾ ଔଐƁ ƛ ܸߢࢽ ࡦƁ߁ ዖዄ Ž૊ સၫƋࣝެ࠾ኃ ቮƁƊܸެࢲ ଘଐƁ ࡼ ‫୐ࠦࡷ ࣀޯߥ ࡣݑ‬෺߄ ଐଌƄ፥ ጿū 71 ، 70-66ࡼ ࣀ ࡩƄƇ ܻၮ࣠ዙಣಝ፥ ጿ໠ ໄ࡯ࡷܻ‫ࡩݘ‬Ƅࡼ ࡷƎ࣠ࡁኅ ቱƄƥ ࢖ࡺ‫ࡩݘ‬Ƅࠦ‫ߙޟ‬ ƈ ܻ‫ރ‬ƀ Ǝ࣠ࡁƍ 71+ 3 8 70-66 ፥ ፑܻዙዋ Ƅ 17 20 9 17 ‫ ࢪۑ‬ჱஎै Ƅ ፥ ፑܻዙዋ ƀ 26 11 37 ‫ۑ‬Ƣܻ‫ݹ‬Ƅ 5Ɔ߄ ଐଌ௪ ௎౮౑ީࡩƄࡼ ࡷƖ Ƅ࣠‫ރ‬Ƅࠦ‫ހݘ‬Ǝଘଓƣ ࣠ࡩܻஎै Ƅ፥ ጿơ ܻ‫ࡷޢ‬ଘଓƄƌ߄ ၮ໠ ໄ୍෺Ƥܻ‫ރ‬ƌƄơ ƥ߄ எै Ƅ.10 ࡣ ࠧ‫ۑ‬௪ ௎ܻީޯࡩ‫ا‬Ƨ ƥܻఱ ఠ݊ ࡷܻ‫ݹݘܻ ࣍ީޢݘޢ‬Ə࣠‫ކ‬Ɩ Ə࣠ࡷƖ ࣠‫ރ‬ᇯ ᇍ ࡩƣ ࣠ࡩܻॺॱ५ơ ܻ‫ࡷޢ‬ଘଓƄƌ߄ ၮƣ ƄƓ ౰౔ߙƀ ࠦ‫ ࡷݘެ࣠ݹ‬Ʀ ࣠ޯࣀ ࡷ߄ ެ‫ݹࠦݑ‬ƥܻࣀ ჴࡷ݊ ࡺޯࠖ ൞ᒑᐽƇ ܻ࣐ ࢵ ࣠ޯଐଌƄƣ ܾ‫ހ‬Ɠ ࢖ߦࡩƄ࢛ ߙƎƥƄơ ࣠ީߘ‫ ݪ‬༬࿻ ‫ۑ‬%5 ơ ƥଘଓƄƖ ࣠ެࡩƄ ፥ ጕܻ޲ࡩƄƖ ࣠ެࡩƄ ‫ࡩ޲ܻࡩ ݟ‬ƄƖ ࣠ެࡩƄ ݊ ‫ހ‬Ƅ༟༎ƄƖ ࣠ެࡩƄ 11 14 14 15 12 12 13 11 165 Ƣܻ‫ݹ‬Ƅ 5Ɔ߄ ଐଌ௪ ௎౮౑ީࡩƄࡼ ࡷƖ Ƅ࣠‫ރ‬ƀࠦ‫ހݘ‬Ǝଘଓƣ ࣠ࡩܻॺॱ५ơ ܻ‫ࡷޢ‬ଘଓƄƌ߄ ၮ 12 13 15 14 16 15 13 14 17 17 16 18 ܸୃกܸ଍ଉŽ–ໝ ແࡦफउ –ࠣ‫ݶ‬ Ɖܸࢤ ެߕକଐƁƢࠣ‫ࡦ ࢽޟܸ ࢲޟ‬ƁƢࠣ‫ޟ‬පඓ ൷Ɓ౭౑ࢲ କଐƁƄ ܸࢲ ƁƋኃ ᒔᐺ። ጏ࢓‫ݕ‬ ࡦƁ࣬ ࣊ Ɓ࢓ߢೕಹኃ ቮƁ೾೥࢓଍ଉƁƄ ƁƋƁ߁ ࢟ ſ ƛ ܸߢࢽ ࡦƁ߁ ዖዄ Ž૊ સၫƋࣝެ࠾ኃ ቮƁƊܸެࢲ ଘଐƁ 166 ። ፎକଐ߁ ެࢲ କଐƁƢ। ्࢟ ࣝࡦƁŻܸࢤ ߾ ଔଐƁ ܸୃกܸ଍ଉŽˍ ໝ ແࡦफउˍ ࠣ‫ݶ‬ Ɖܸࢤ ެߕକଐƁƢࠣ‫ࡦ ࢽޟܸ ࢲޟ‬ƁƢࠣ‫ޟ‬පඓ ൷Ɓ౭౑ࢲ କଐƁƄ ܸࢲ ƁƋኃ ᒔᐺ። ጏ࢓‫ݕ‬ ࡦƁ࣬ ࣊ Ɓ࢓ߢೕಹኃ ቮƁ೾೥࢓଍ଉƁƄ ƁƋƁ߁ ࢟ ſ ƛ ܸߢࢽ ࡦƁ߁ ዖዄ Ž૊ સၫƋࣝެ࠾ኃ ቮƁƊܸެࢲ ଘଐƁ ። ፎକଐ߁ ެࢲ କଐƁƢ। ्࢟ ࣝࡦƁŻܸࢤ ߾ ଔଐƁ ( Z) ‫( ﺟدول اﻟﺗوزﯾﻊ اﻟطﺑﯾﻌﻲ اﻟﻣﻌﯾﺎري‬1) ‫ﻣﻠﺣﻖ‬ 167 ܸୃกܸ଍ଉŽˍ ໝ ແࡦफउˍ ࠣ‫ݶ‬ Ɖܸࢤ ެߕକଐƁƢࠣ‫ࡦ ࢽޟܸ ࢲޟ‬ƁƢࠣ‫ޟ‬පඓ ൷Ɓ౭౑ࢲ କଐƁƄ ܸࢲ ƁƋኃ ᒔᐺ። ጏ࢓‫ݕ‬ ࡦƁ࣬ ࣊ Ɓ࢓ߢೕಹኃ ቮƁ೾೥࢓଍ଉƁƄ ƁƋƁ߁ ࢟ ſ ƛ ܸߢࢽ ࡦƁ߁ ዖዄ Ž૊ સၫƋࣝެ࠾ኃ ቮƁƊܸެࢲ ଘଐƁ ። ፎକଐ߁ ެࢲ କଐƁƢ। ्࢟ ࣝࡦƁŻܸࢤ ߾ ଔଐƁ t݇ ‫ݶ‬ƌࣝ‫ރ‬ƞ Ƣ߁ ၱ 2 ࢄ ࠏ ࡬ࡴ 168 ܸୃกܸ଍ଉŽ–ໝ ແࡦफउ –ࠣ‫ݶ‬ Ɖܸࢤ ެߕକଐƁƢࠣ‫ࡦ ࢽޟܸ ࢲޟ‬ƁƢࠣ‫ޟ‬පඓ ൷Ɓ౭౑ࢲ କଐƁƄ ܸࢲ ƁƋኃ ᒔᐺ። ጏ࢓‫ݕ‬ ࡦƁ࣬ ࣊ Ɓ࢓ߢೕಹኃ ቮƁ೾೥࢓଍ଉƁƄ ƁƋƁ߁ ࢟ ſ ƛ ܸߢࢽ ࡦƁ߁ ዖዄ Ž૊ સၫƋࣝެ࠾ኃ ቮƁƊܸެࢲ ଘଐƁ ። ፎକଐ߁ ެࢲ କଐƁƢ। ्࢟ ࣝࡦƁŻܸࢤ ߾ ଔଐƁ Ƥ ᇱᆚ ݇ ‫ݽ‬ ୶ ୧ƞ Ƣ߁ ၱ ࢄ ࠏ ࡬ࡴ 169 ܸୃกܸ଍ଉŽˍ ໝ ແࡦफउˍ ࠣ‫ݶ‬ Ɖܸࢤ ެߕକଐƁƢࠣ‫ࡦ ࢽޟܸ ࢲޟ‬ƁƢࠣ‫ޟ‬පඓ ൷Ɓ౭౑ࢲ କଐƁƄ ܸࢲ ƁƋኃ ᒔᐺ። ጏ࢓‫ݕ‬ ࡦƁ࣬ ࣊ Ɓ࢓ߢೕಹኃ ቮƁ೾೥࢓଍ଉƁƄ ƁƋƁ߁ ࢟ ſ ƛ ܸߢࢽ ࡦƁ߁ ዖዄ Ž૊ સၫƋࣝެ࠾ኃ ቮƁƊܸެࢲ ଘଐƁ ። ፎକଐ߁ ެࢲ କଐƁƢ। ्࢟ ࣝࡦƁŻܸࢤ ߾ ଔଐƁ ƌࣝ‫ރ‬ƞ Ƣ߁ ၱ ࢄ ࠏ ࡬ࡴ 170 ܸୃกܸ଍ଉŽˍ ໝ ແࡦफउˍ ࠣ‫ݶ‬ Ɖܸࢤ ެߕକଐƁƢࠣ‫ࡦ ࢽޟܸ ࢲޟ‬ƁƢࠣ‫ޟ‬පඓ ൷Ɓ౭౑ࢲ କଐƁƄ ܸࢲ ƁƋኃ ᒔᐺ። ጏ࢓‫ݕ‬ ࡦƁ࣬ ࣊ Ɓ࢓ߢೕಹኃ ቮƁ೾೥࢓଍ଉƁƄ ƁƋƁ߁ ࢟ ſ ƛ ܸߢࢽ ࡦƁ߁ ዖዄ Ž૊ સၫƋࣝެ࠾ኃ ቮƁƊܸެࢲ ଘଐƁ 171 ። ፎକଐ߁ ެࢲ କଐƁƢ। ्࢟ ࣝࡦƁŻܸࢤ ߾ ଔଐƁ ܸୃกܸ଍ଉŽˍ ໝ ແࡦफउˍ ࠣ‫ݶ‬ Ɖܸࢤ ެߕକଐƁƢࠣ‫ࡦ ࢽޟܸ ࢲޟ‬ƁƢࠣ‫ޟ‬පඓ ൷Ɓ౭౑ࢲ କଐƁƄ ܸࢲ ƁƋኃ ᒔᐺ። ጏ࢓‫ݕ‬ ࡦƁ࣬ ࣊ Ɓ࢓ߢೕಹኃ ቮƁ೾೥࢓଍ଉƁƄ ƁƋƁ߁ ࢟ ſ ƛ ܸߢࢽ ࡦƁ߁ ዖዄ Ž૊ સၫƋࣝެ࠾ኃ ቮƁƊܸެࢲ ଘଐƁ ። ፎକଐ߁ ެࢲ କଐƁƢ। ्࢟ ࣝࡦƁŻܸࢤ ߾ ଔଐƁ ࠣୃ෾࢓‫ݕ‬ ࡦƁ݇ ࠂ Ɓ࢓଍ଉƁ ࠦ‫ޢ‬൞൙‫ࡩݘ‬ƄƧ ƎƥƏܻ‫ࡩޢ‬ƄƎƄƌūࠦ୆෷ܻࢧ ࠁ ଘଓƄƟ ࢖࣐ ࡩƄūᅶै ܻࢢ ࣠‫ހ‬Ƅ࣯ ࠒ ީࢢ ߄ ዙಟಝƎ࣠ޯࡁኆ ቱƄƍܻޯࢵ ଘଓƄ .1 2009ūƣ ƌƎଘଓƄū݊ ‫ݹ‬Ə࣠ޯࡩƄƥྲ Ⴁჱᑃᐽ ƣ ܻެީࡩūƇ ƥ౰ే‫ހ‬ūਔ ৲߄ எै Ƅƅ ܻຨ຀ࡩƄƎƄƌūžܻࢧ ࠁ ଘଓƄࣷࣳၮƃ ƌܻީࡷūƅ ܻ‫ݹ‬ƍযও࡯ࢵ ūƧ ߄ ‫ࡩ ࣀݘ‬Ƅ .2 2001 ، ‫ࡩ࣠راق‬ƄūƎƄ࢖ߥࡩƄࡴ ‫ݑ‬ƌ፥ ጿƢ߄ ߥޯଐଌƄžܻࢧ ࠁ ଘଓƄūਔ ৲࣓ࡷࡣ ࢥ ߦࡩƄƥơ ܻ‫ࡩݘ‬Ƅ߄ ީ‫ ߄ݑ‬ዙಟಝū࣯ ࡺ‫ࡩެݘޢ‬Ƅ .3 ‫ۑ‬2007ūƣ ƌƎଘଓƄūƣ ܻዙኸ ū݊ ‫ݹ‬Ə࣠ޯࡩƄƥྲ Ⴁჱᑃᐽ በ቎ᚻ࡯ࡺଐଌƄūྲ Ⴁჱᑃᐽၑ࿒ ࢖ଐଌƄƎƄƌū፥ ገܻࢧ ࠁ ଘଓƄƉ ܻޯެჴࢵ ଘଓƄƇ ܻ‫ ࢵܻ ࢵޢ‬Ƅū߄ ‫ ࢸޢ‬ƎƆܻ༬စ ūܻ‫ ࡓޢ ࠑޢ‬ᚻࡩƄ .4 ‫ۑ‬2006ūࠦ‫ݹ‬ƌ࣠‫ࡩ ࣀݘ‬Ƅࠦ୆ก࢖‫ࡩݘ‬Ƅ በ቎ᚻ࡯ࡺଐଌƄūƆ߄ ၴ ūྲ Ⴁჱᑃᐽၑ࿒ ࢖ଐଌƄƎƄƌū፥ ገܻࢧ ࠁ ଘଓƄơ ଘଓ߄ ޯࢵ ଘଓƄūॡ ॐ࣐ ࢧ ࡷơ ൎ൉ၴ ūƌܻ‫ࡩ ࢧޢ‬Ƅ .5 ‫ۑ‬1993ūࠦ‫ݹ‬ƌ࣠‫ࡩ ࣀݘ‬Ƅࠦ୆ก࢖‫ࡩݘ‬Ƅ ƎƄƌūࠦ୆෷ܻࢧ ࠁ ଘଓƄƟ ࢖࣐ ࡩƄ፥ ጿࠦࡷ߄ ߥࡷū፥ ጾ࣠ࢵ ኆ ቱƄ߄ ዙಟಝ‫ ݪ‬ყީࠁ ƥॡ ॐ࣐ ࢧ ࡷơ ൎ൉ၴ ūƌܻ‫ࡩ ࢧޢ‬Ƅ .6 ‫ݹࠦۑ‬ƌ࣠‫ࡩ ࣀݘ‬Ƅࠦ୆ก࢖‫ࡩݘ‬Ƅበ቎ᚻ࡯ࡺ ଐଌƄū݊ ‫ݹ‬Ə࣠ޯࡩƄƥྲ Ⴁჱᑃᐽࣉ ߙܻၰ ū݊ ‫ݹ‬Ə࣠ޯࡩƄƥྲ ႡჱᑃᐽƊ ൎ൉ߦࡩƄƎƄƌū፥ ፑଘଓ߄ ޯࢵ ଘଓƄƥ१ ॐࢢ ࣠ࡩƄžܻࢧ ࠁ ଘଓƄūƆƌ࣠‫ ߄ݑ‬ዙ዇ ƀūƆƌ࣠‫ ݑ‬.7 ‫ۑ‬2014ūƣ ƌƎଘଓƄƣ ܻዙኸ ᑻ ᑧଐଌƄࠦ‫ܻࡷݘ‬ၴ ū፥ ገܻࢧ ࠁ ଘଓƄơ ଘଓ߄ ޯࢵ ଘଓƄƃ ƌܻީࡷūƣ ƥ༇໩ Ƅƥࡣ ‫ܻݑޢ‬ያዜ ƀࢪ ყ჈ƀū࣠ࠖ ެࡁ .8 2005ūƌ࣠‫ࢵݘ‬ 2007ūƌ࣠‫ ࢵݘ‬ᑻ ᑧଐଌƄࠦ‫ܻࡷݘ‬ၴ ūमअ߄ ߥޯࡩƄࠦ‫ ࢵ ࢪރ ࣏࢖ݹ‬ƀūᎨᎌ ಒ౷ᚻࡩƄ߄ ީ‫ݑ‬ቻᒗᐽƄ߄ ީ‫ࠦݑ‬ႁ ‫ࡩ ࣃޢ‬Ƅ .9 ࠦ୆ก࢖‫ࡩݘ‬Ƅࡣ ‫ࡩެޢ‬Ƅࠦၮ࣠ዙಣಝūमअ߄ ߥޯࡩƄࠦ‫ ރ ࣏࢖ݹ‬1 ፥ ገܻࢧ ࠁ ଘଓƄơ ଘଓ߄ ޯࢵ ଘଓƄū߄ ‫ޢ‬ସନଣ Ƅ߄ ީ‫ݑ‬ūॡ ॐ࣐ ࢧ ࡷ .10 2000 ، ࠣ‫ޟ‬ძީࠂ କଐƁ݇ ࠂ Ɓ࢓଍ଉƁ 1. Gupta S.P Statistical Methods ,Sultan Chand & Sons Publisher , New Delhi , India , 1994 2. Harry Frank and Steven C. Althoen , Statistics , Concepts and applications , Cambridge University Press , London , 1994 172 ܸୃกܸ଍ଉŽ–ໝ ແࡦफउ –ࠣ‫ݶ‬ Ɖܸࢤ ެߕକଐƁƢࠣ‫ࡦ ࢽޟܸ ࢲޟ‬ƁƢࠣ‫ޟ‬පඓ ൷Ɓ౭౑ࢲ କଐƁƄ ܸࢲ ƁƋኃ ᒔᐺ። ጏ࢓‫ݕ‬ ࡦƁ࣬ ࣊ Ɓ࢓ߢೕಹኃ ቮƁ೾೥࢓଍ଉƁƄ ƁƋƁ߁ ࢟ ſ ƛ ܸߢࢽ ࡦƁ߁ ዖዄ Ž૊ સၫƋࣝެ࠾ኃ ቮƁƊܸެࢲ ଘଐƁ ። ፎକଐ߁ ެࢲ କଐƁƢ। ्࢟ ࣝࡦƁŻܸࢤ ߾ ଔଐƁ 3. Gupta S.P , Practical Statistics , S. Chand and company LTD , New Delhi , 1998 4. Richard I .Levin & David S. Rubin , Statistics for Management , prentice ,Hall of India , New Delhi , 1992 5. Jane Miller , Statistics for Advance level , Cambridge University Press, Great Britain , 1994 173 ܸୃกܸ଍ଉŽ–ໝ ແࡦफउ –ࠣ‫ݶ‬ Ɖܸࢤ ެߕକଐƁƢࠣ‫ࡦ ࢽޟܸ ࢲޟ‬ƁƢࠣ‫ޟ‬පඓ ൷Ɓ౭౑ࢲ କଐƁƄ ܸࢲ ƁƋኃ ᒔᐺ። ጏ࢓‫ݕ‬ ࡦƁ࣬ ࣊ Ɓ࢓ߢೕಹኃ ቮƁ೾೥࢓଍ଉƁƄ ƁƋƁ߁ ࢟ ſ ƛ ܸߢࢽ ࡦƁ߁ ዖዄ Ž૊ સၫƋࣝެ࠾ኃ ቮƁƊܸެࢲ ଘଐƁ 174 ። ፎକଐ߁ ެࢲ କଐƁƢ। ्࢟ ࣝࡦƁŻܸࢤ ߾ ଔଐƁ ܸୃกܸ଍ଉŽ–ໝ ແࡦफउ –ࠣ‫ݶ‬ Ɖܸࢤ ެߕକଐƁƢࠣ‫ࡦ ࢽޟܸ ࢲޟ‬ƁƢࠣ‫ޟ‬පඓ ൷Ɓ౭౑ࢲ କଐƁƄ ܸࢲ ƁƋኃ ᒔᐺ። ጏ࢓‫ݕ‬ ࡦƁ࣬ ࣊ Ɓ࢓ߢೕಹኃ ቮƁ೾೥࢓଍ଉƁƄ ƁƋƁ߁ ࢟ ſ ‫لكا لحق ا ـوقا مـحفــ ا وظةا لـلـا نـاشـر‬ ‫لملـركـزا لدلميقــرلطـيا للعـريب ا برلنيا –ا أملانيا‬ ‫‪© Democratic Arabic Center‬‬ ‫‪Berlin10315 Gensingerstr :112‬‬ ‫‪Tel :0049-code Germany‬‬ ‫‪54884375-030‬‬ ‫‪91499898-030‬‬ ‫‪86450098-030‬‬ ‫‪book@democratica.de‬‬

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