Document 14335410

Section 9.1: Interval Estimation , is an iid sequence of RVs with common mean common standard deviation is the (sample) mean of these RVs Section 9.1: Interval Estimation RV as approaches a Normal then and if ✦ , ✦ , If Central Limit Theorem: ➤ 1 Sample Mean Distribution Choose one of the random variate generators in rvgs to generate a sequence of random variable samples with fixed sample size ➤ ) " , -, +&. #$ ! . ( ) >; = < <; : LL L L LLLL L L L LLLL L L L L LLLLL L L 5432 4532 4532 8796 8796 8796 A continuous-data histogram can be created using program cdh @= ? 3 ; 63 7 > ? ? :; >; 6 4 <; : BA6 >4 : <; 3 =? LLL LLL LLL LLL / 01 LLL LLL L L L LLL L L K GKK J EDC G CDI HGF EDC LLL ➤ +&* #$ *( '&% #$ %( ! " ) ! ) " ) With the -point samples indexed , the corresponding and sample standard deviation can be sample mean calculated using Algorithm 4.1.1 ➤ Section 9.1: Interval Estimation 2 Properties of Sample Mean Histogram Independent of , Section 9.1: Interval Estimation the histogram density approximates the Normal ✦ is sufficiently large, If ➤ ✦ the histogram mean is approximately the histogram standard deviation is approximately ✦ ➤ pdf 3 Section 9.1: Interval Estimation \ cZ ` ba B_^ db op cZ ` ba B_^ Z XY W b db j d XY Y e f h g i OP M M M M M M M M Exponential XY \ \\ \\ \\ \\ \\ \ \\ \\ \\ \\ ]\\[ nnn n n n n nn n n n n n n n nn n n n n n nn n nn n n n n n nn n n n n nn n n n n n n n n n nn n n n nn n n n nn n n n n n n n n nn n n n n n n nn n n n n n n nn n n n n n n nn n n n nn n n n n n T nn n n nn n n n nn n n n n n n n n nnn n n nn n n n n n n n n n nn n n n n n n n nnn n n nn n n n nn nn nn n nn n n nn n lkmj nn nn n nn n n n n nn Q SRQ Q SRQ UV rvgs was used to generate and samples of size \\ \\ [\ Z W N ➤ nnn n nn nn nn n n n nn n n n n n n n n n n n n n n n nn n n n n n n n n n nn nn n nn n n n n n n nn n n n n n n n n n n n n n n n n n n n n n nn n n n n n n n n n n n n n n n n n n nn n n n n n n n n nn T n n nn n n n n n n n n n n n n n nn nn n n n n n n n n n n n n n n n n n n n n n n n n n n n n n nn nn n n n n n n n n n n n n n n n n n n n n n n n nn n n n n n d XY nn n nn n n n n nn nn b f nnn nn lkmj i h g Y j RSQ Q Q SRQ UV Example 9.1.2 samples of size 4 Example 9.1.2 The histogram mean and standard deviation are approximately and ➤ The histogram density corresponding to the 36-point sample means RV is closely matched by the pdf of a Normal ➤ The histogram density corresponding to the 9-point sample means matches relatively well, but with a skew to the left ✦ N ➤ OP For Exponential samples, is large enough for the sample mean to be approximately Normal ✦ is not large enough Section 9.1: Interval Estimation 5 More on Example 9.1.2 Essentially all of the sample means are within an interval of width of centered about , if is large, all the sample means as In general: The accuracy of the Normal pdf approximation is dependent on the shape of a fixed population pdf If the samples are drawn from a population with ✴ a highly asymmetric pdf (like the Exponential pdf): may need to be as large as 30 or more for good fit ✴ a pdf symmetric about the mean (like the Uniform pdf): as small as 10 or less may produce a good fit t sr ✦ ✦ ➤ M Because will be close to ➤ q ➤ Section 9.1: Interval Estimation 6 Standardized Sample Mean Distribution ,8 We can standardize the sample means by subtracting and dividing the result by to form the standardized sample means defined by O u v ,u u u ➤ 5|{z |5{z |5{z 8~} 8~} 8~} Generate a continuous-data histogram for the standardized sample means by program cdh { }{ ~ } | B} | { w H H xy ➤ Section 9.1: Interval Estimation 7 Properties of Standardized Sample Mean Histogram Independent of , ✦ If is sufficiently large, the histogram density approximates the Normal ✦ Section 9.1: Interval Estimation M ➤ the histogram mean is approximately 0 the histogram standard deviation is approximately 1 ✦ ➤ pdf 8 Section 9.1: Interval Estimation ¨¨ ¨ ¨ ¨ ¨¨ ¨ ¨ ¨¨¨ ¨ ¨ ¨ ¨ ¨¨¨ ¨ ¨¨¨ ¨ ¨ ¨¨ ¨ ¨ ¨¨ ¨ © ¨¨ ¨ ¨¨ ¨¨ ¨ ¨ ¨ ¨ ¨ ¨ ¨¨ ¨¨ ¨ ¨ ¨ ¨ ¨ ¨ ¨ ¨ ¨ ¨ ¨ ¨ ¨ ¨ ¨ ¨ ¨ ¨¨ ¨ ¨ ¨ ¨ ¨¨ ¨¨ ¨ ¨ ¨ ¨ ¨ ¨¨ ¨ ¨ ¨ ¨ ¨ ¨¨¨ ¨ ¨ ¨ ¨ ¨ ¨ ¨ ¨ ¨ ¨ ¨ ¨¨¨ ¨¨¨¨ ¨ ¨ ¨ ¨ ¨ ¨¨ ¨ ¨ ¨ ¨ ¨ ¨¨¨ ¨ ¨ ¨ ¨¨ ¨ ¨ ¨¨ ¨ ¨ ¨ ¨¨ ¨ ¨ ¨ ¨¨¨ ¨¨¨ ¨ ¨ ¨¨¨ ¨ ¨¨ ¨ ¨ ¨¨¨ ¨ ¨ ¨¨ ¨ ¨ ¨ ¨ ¨ ¨ ¨¨¨ ¨ ¨ ¨ ¨ ¨ ¨ ¨ ¨ ¨ ¨ ¨ ¨ ¨ ¨¨ ¨¨¨ ¨ ¨ ¨ ¨ ¨ ¨¨ ¨ ¨ ¨ ¨ ¨ ¨ ¨ ¨ ¨¨ ¨ ¨¨¨ ¨ ¨ ¨ ¨ ¨ ¨ ¨ ¨ ¨ ¨ ¨ ¨ ¨¨¨ ¨¨ ¨ ¨ ¨ ¨ ¨¨¨ ¨ ¨¨ ¨¨ ¨ ¨ ¨ ¨ ¨ ¨ ¨¨ ¨ ¨ ¨ ¨ ¨ ¨¨ ¨ ¨¨ ¨ ¨ ¨ ¨¨ ¨¨ ¥ ¦ ¢£ ¨ ¨ ¨¨ ¨ § ¤ ¨¨ ¨ ¨ ¨¨ ¨ ¨ ¨ ¨¨ ¨ ¨ ¨ ¨ ¨ ¡ ¨¨ ¨ ¨ ¨ ¨ ¨ ¨ ¨ ¨ ¨ ¨ ¨ ¨ ¨¨ ¨ ¨ ¨ ¨ ¨ ¨ ¨ ¨ ¨¨ ¨ ¨ ¨ ¨ ¨ ¨ ¨ ¨ ¨ ¨ ¨ ¨ ¨¨ ¨ ¨ ¨¨ ¨¨ ¨ ¨ ¨ ¨ ¨ ¨ ¨ ¨ ¨¨ ¨ ¨ ¨ ¨ ¨ ¨¨ ¨ ¨ ¨ ¨ ¨ ¨ ¨ ¨ ¨ ¨ ¨¨ ¨ ¨ ¨ ¨ ¨ ¨ ¨ ¨ ¨ ¨ ¨ ¨ ¨ ¨¨ ¨ ¨ ¨ ¨ ¨ ¨ ¨ ¨ ¨ ¨¨ ¨ ¨ ¨ ¨ ¨ ¨ ¨ ¨ ¨¨ ¨ ¨ ¨ ¨ ¨¨ ¨ ¨ ¨ ¨ ¨ ¨ ¨ ¨ ¨¨ ¨ ¨ ¨ ¨¨¨ ¨ ¨¨¨ ¨¨ ¨ ¨ ¨ ¨¨ ¨ ¨ ¨ ¨ ¨¨ ¨¨ ¨ ¨¨¨ ¨ ¨ ¨¨ ¨ ¨ ¨ ¨¨¨¨ ¨ ¨¨¨ ¨ ¨ ¨ ¨ ¨ ¨ ¨ ¨¨¨ ¨ ¨ ¨ ¨ ¨ ¨¨ ¨¨ ¨ ¨ ¨ ¨ ¨ ¨ ¨ ¨ ¨ ¨ ¨ ¨¨¨¨ ¨¨ ¨¨ ¨ ¨ ¨ ¨ ¨ ¨ ¨ ¨ ¨ ¨ ¨ ¨¨ ¨ ¨ ¨ ¨ ¨ ¨ ¨ ¨¨ ¨ ¨¨ ¨¨ ¨ ¨ ¨ ¨ ¨¨ ¨ ¨ ¨ ¨ ¨¨ ¨ ¨ ¨¨ ¨ ¨ ¨¨ ¨¨ ¨ ¨ ¨¨¨ ¨ ¨ ¨¨ ¨¨¨¨ ¨¨ ¥ ¦ ¢£ § ¤ ➤ ¨¨¨ Example 9.1.4 The sample means from Example 9.1.2 were standardized 9 Properties of the Histogram in Example 9.1.4 ➤ The histogram density corresponding to the 36-point sample means random variable almost exactly matches the pdf of a Normal M ➤ The histogram mean and standard deviation are approximately 0.0 and 1.0 respectively The histogram density corresponding to the 9-point sample means random variable, but not as well matches the pdf of a Normal M ➤ Section 9.1: Interval Estimation 10 Want to replace population standard deviation standard deviation in standardization equation with sample O u v ➤ t-Statistic Distribution Definition 9.1.1 Section 9.1: Interval Estimation ✦ ✦ Each sample mean is a point estimate of Each sample variance is a point estimate of Each sample standard deviation is a point estimate of ✦ ➤ 11 , not v , ,ª will converge to , The mean of ✦ is v l¬ The point estimate of « ➤ v v To remove this bias, it is conventional to multiply the sample variance by a bias correction ➤ will converge to The sample variance is a biased point estimate of ➤ The mean of ✦ The sample mean is an unbiased point estimate of ª ª ➤ Removing Bias Section 9.1: Interval Estimation 12 Section 9.1: Interval Estimation ÇÇÇ ¯ °± ÇÇÇ B¿µ ¼»´ º¹ ¸ ³ ½ µ O v Generate a continuous-data histogram using cdh ½ ½ » ¼¹ º º¹ ¸ v ➤ Calculate the -statistic ® ® ➤ ¾» ½ ³ ¹ µ³ ¶ ¼ ¸¹ ¼¹ ´ º¹ ¸ ÇÇ ÇÇ ÇÇ Ç Ç Ç ÇÇÇÇ Ç Ç Ç Ç Ç ÇÇÇÇ Ç Ç Ç Ç Ç ÇÇÇÇ Ç Ç 5´³ ² ´5³ ² ´5³ ² 8¶·µ 8¶·µ 8¶·µ ÇÇÇ ÇÇÇ ÇÇÇ Ç Ç Ç ÇÇÇ Ç Ç ÆÆ +ÃÂÁÀ Ã ÂÄÀ Æ+ÃÂÅÀ Example 9.1.5 13 Properties of t-statistic Histogram v O O v If is sufficiently large, the histogram density approximates the pdf of a Student random variable v ➤ , the histogram standard deviation is approximately If ➤ , the histogram mean is approximately 0 If ➤ Section 9.1: Interval Estimation 14 Section 9.1: Interval Estimation Ó Ê Î Ë ÈÈ ÈÈ ÈÈ ÈÈ ÈÈ ÈÈ È ÕÓ ÈÈ ÈÈ ÈÈ ÈÈ ÈÈ ÈÈ ÈÔ Í ÊË Ì ÊË ÐÑ É ÊË ÊË Ò Î Ó Ê Î Ë ÈÈ ÈÈ ÈÈ ÈÈ ÈÈ ÈÈ È ÕÓ ÈÈ ÈÈ ÈÈ ÈÈ ÈÈ ÈÈ ÈÔ Í ÊË Ì ÊË ÐÑ É ÊË ÊË Ò ÊÈ É ß ß ßßßß ×Ö Ë ß ß ß ß ß ßß ßßß ß ß ß ß ß ß ß ß ßß Ø ß ß ß ßßß ß ß ß ß ß ßßß ß ß ß ßßß ÊÈ Ì ß ß ßß ßß ß ß ß ß ß ßß Ë ß ß ß ßß ßß ßß ßß ßß ß ß ß ßß ßß ßß ß ß ßß ß ß ß ß ßßß ß ß ß ß ß ß ß ß ß ß ÊÈ Í ßßß ß ßß ß ß ß ß ß ß Ë ßßß ß ß ß ß ß ß ß ß ß ß ß ß ßßß ß ß ß ß ß ß ß ß ß ßß ß ß ß ß ß ßß ß ß ß ß ß ß ß ß ß ß ß ß ß ß ßßß ßß ÊË ß ßß ßß Ë ß ß ßß ß ß ß ßß ßßß ß ß ß ß ß ßß ßßß ß ßß ß ß ß ß ß ßßß ß ß ß ß ß ß ß ß ß ß ßß ß ß ß ß ß ß ß ß ß ß Ê Í ß ß ß ß ßß ß ß ß ß ß ß ß Ë ßßß ß ß ß ß ß ß ß ß ß ß ß ßß ß ßßß ß ß ß ß ß ß ß ß ß ß ß ß ß ßßß ß ß ß ß ß ß ß ß ß ×Ò ßßß ß ß ß ß Ê Ì ßßß ß ßß ÜÝ Ë ß ßß ß ß ß ßß ß ß ß ÞÖ È ÙÚ ßßß ß ß ß ß È Û ßßß ßß ß ß ß ß Í Ê É ß Ë Ë ÊË Ï Î ➤ ÊÈ É ß ß ßßßß ×Ö Ë ß ß ß ß ßß ß ß ß ß ß ßß ß ß ß ßß ßß Éà ß ß ß ßß ß ß ß ß ß ß ß ßßß ß ß ß ß ß ßßß ÊÈ Ì ß ß ß ß ßß Ë ßßßß ßßßß ß ß ß ß ßß ß ß ßß ß ß ß ß ß ß ß ßß ß ß ß ßß ß ß ßß ß ß ßß ß ß ß ß ß ÈÊ Í ßßß ß ß ß ß ß ß ß ß ß ß ß ß ß ß ß ßß ß ß ß ß ß ß Ë ß ß ß ß ß ß ß ß ß ß ß ßß ß ßß ß ßß ßß ß ß ßß ßß ßß ß ß ß ß ßß ßßß ÊË ßßß ß ß Ë ß ß ß ß ßß ßß ß ß ßßß ß ß ß ßß ßßßß ß ß ß ß ß ß ß ß ß ß ß ß ß ß ßßßß ß ß ß ß ß ß ß ß ß ß Ê Í ß ß ßß ß ß ß ßß ß ß ß ß Ë ßßß ß ß ß ß ß ß ß ß ßß ß ß ßß ß ß ßßß ß ß ß ß ß ß ßß ßß ßß ß ß ß ßßß ß ß ß ß ß ß ß ß ß ×Ò ßßß ß ß ß ß ß Ê Ì ßß ß ß ß ß ß ÜÝ Ë ß ßß ß ß ß ßß ß ß ßßß ß ß ß ÞÖ È ÙÚ ßßß ß ß ß È Û ß ß ßß Í Ê É ßßß ß Ë Ë ÊË Ï ® Example 9.1.6 Generate -statistics from Example 9.1.2 15 Properties of the Histogram in Example 9.1.6 M Oâ The histogram density corresponding to the 36-point sample means RV relatively well matches the pdf of a Student ➤ v v á O The histogram mean and standard deviation are approximately 0.0 respectively and ➤ The histogram density corresponding to the 9-point sample means RV, but not as well matches the pdf of a Student ã ➤ Section 9.1: Interval Estimation 16 Interval Estimation v random variate Theorem 9.1.2 provides the justification for estimating an interval that is likely to contain the mean v distribution becomes M As , the Student indistinguishable from Normal ➤ ➤ is a Student v ® v Theorem 9.1.2: If is an (independent) random sample from a “source” of data with unknown mean , if and are the mean and standard deviation of this sample, and if is large, it is approximately true that ➤ Section 9.1: Interval Estimation 17 Section 9.1: Interval Estimation ðð ðððð ðð ðð ððð ôñ õ ÷ö ó òñ ððð ððð ððð ì ð ø ûüú ù þÿý ú ððð ððð ðð ðð ð ð ððð ð ðð ððð ðð ððð ððð ðð ððð ðð ðð ð ðððð ððð ðððð ðð ðð ð ð ðð ð ð ð ðð ð ð ð ð ð ð ð ð õ ø ððð v v å ® ® çë ëä êéè ç ® ç Then there exists a corresponding positive real number í ðð ðð ð ð ð ðð ððð ð ð ð ð ð ð ð ðð ððð ð ð ðð ð ì ó òñ ððð ôñ ððð ð ððð ððð ÷ö M v ä æ M å æ M is a Student random variable is a “confidence parameter” with íî ðð ðð ðð ððð ðð ððð ððð õ å ✦ ðð ðð ððð ð ð ðð ð ð ð ðð ð ð ð ðð ð ð ð ð ðð ð ððð ðð ð ðð ð ððð ðð ð ððð ✦ ï ìí î ððð ó ➤ ôñ òñ Interval Estimation (2) Suppose 18 Interval Estimation (3) ® ë v v ® çë ç v v ® , Student is unknown. Since Suppose ➤ Right inequality: ➤ Left inequality: ➤ So, with probability ë ë v ® ç v v ® v ë ç v ➤ ® ç v å will be approximately true with probability v v ç ® v v ë ë v Section 9.1: Interval Estimation çë ® v ® ç v å (approximately), 19 Theorem 9.1.3 If æ å æ ç , such that v v å Section 9.1: Interval Estimation ë ë v v êéè ® ® ç ç ® there exists an associated positive real number M with M Then, given a confidence parameter M ✦ å ➤ ✦ is an independent random sample from a “source” of data with unknown mean and are the sample mean and sample standard deviation is large ✦ ➤ 20 lies somewhere between ® å v v v â M M , å and å OM ® For example, if then , use rvms Section 9.1: Interval Estimation âq M á â N M N ® ç ➤ v and level of confidence v For a fixed sample size to determine ç ➤ v and ® ç confident that Nâ , we are ç â M If M ➤ å Example 9.1.7 21 Definition 9.1.2 The interval defined by the two endpoints M M confidence interval estimate for å is the level of confidence associated with this interval estimate and is the critical value of Section 9.1: Interval Estimation ® ® ç v ➤ v is a å v ® ç ➤ 22 Algorithm 9.1.1 , â M å å v v å v v M M confident that ➤ The midpoint of the interval is Section 9.1: Interval Estimation If is sufficiently large, then you are the mean lies within the interval ➤ å v ® ç ✦ ® ✦ ç ✦ M Pick a level of confidence (typically ) Calculate the sample mean and standard deviation (use Algorithm 4.1.1) Calculate the critical value idfStudent Calculate the interval endpoints ✦ To calculate an interval estimate for the unknown mean of was the population from which a random sample drawn: ➤ 23 OM q q M O O ã á O M Nã N NM P O O ã Nâ M N ç Interval: ® ✦ Nã Determine confidence interval estimate: NM ✦ ã To calculate a We are approximately confident that is between 0.950 and 3.014 NM ➤ ã NM P and Nã ➤ is drawn from a population with unknown mean Pq â M â : M â â M ã M P Oã P â M Pq The random sample of size q ➤ M Example 9.1.8 ➤ Section 9.1: Interval Estimation 24 P q Nã N NM P P á â N M N Interval: ç ✦ confidence interval estimate: ® Determine Nã ✦ Nâ To calculate a We are approximately confident that is between 0.708 and 3.256 Nâ ➤ Example 9.1.8, ctd. N M â O O ã Nã N NM P M â O á Nâ N M N Interval: ç ✦ confidence interval estimate: ® Determine Nã ✦ N To calculate a We are approximately confident that is between 0.150 and 3.814 N N ➤ ➤ Note: ➤ M ➤ is not large Section 9.1: Interval Estimation 25 Tradeoff - Confidence Versus Sample Size ➤ For a fixed sample size ✦ ✦ ➤ More confidence can be achieved only at the expense of a larger interval A smaller interval can be achieved only at the expense of less confidence The only way to make the interval smaller without lessening the level of confidence is to increase the sample size ➤ Bad news: interval size decreases with Section 9.1: Interval Estimation , not Good news: with simulation, we can collect more data ➤ 26 is essentially independent of ç is large then ® Note: if 8 7 4 6 8 " 8 1 : ; " &6 9 9 9 9 9 9 8 1 : " &6 9 9 9 9 9 9 9 9 Section 9.1: Interval Estimation ! 2 " # ! 9 9 9 9 9 9 9 9 9 9 9 9 " # 9 9 9 9 9 9 9 9 " 9 9 9 9 9 9 9 9 9 9 9 9 9 9 9 9 " # 9 9 9 9 9 9 9 9 9 9 9 9 9 9 9 9 9 9 9 9 9 " 9 &6 : < 1 9 9 9 9 9 9 9 9 9 9 9 9 9 9 9 9 9 9 9 9 9 9 9 " 9 9 9 9 9 9 9 $ % 9 9 9 9 9 9 9 " 9 ! 9 4 5 # 2 1 3 0, . ) / - ,+ )* ( 9 $ &'% " ➤ where Answer: Use Algorithm 4.1.1 with Algorithm 9.1.1 to iteratively collect data until a specified interval width is achieved ➤ How large should be to achieve an interval estimate is user-specified? ➤ How Much More Data Is Enough? 27 > = v å M v M M å v ç ? E , the asymptotic value ® Mq , if can and wish to construct a confidence interval estimate, can be used in Algorithm 9.1.1 Nâ Mq M NP ® ç If M å M Unless is very close to be used ? F ➤ idfNormal ? D C BA 3 ç ® ? @ ➤ is idfStudent ➤ ® The asymptotic (large ) value of ç Asymptotic Value of Section 9.1: Interval Estimation 28 Example 9.1.9 ç ® for : ç N O Nâ OM confidence Section 9.1: Interval Estimation â For example, if and want to estimate with to within , a value of should be used M ➤ provided Mq ® ? E v can be determined by solving ® ç ® ç Given a reasonable guess for and a user-specified half-width parameter , if is used in place of , ? E ➤ 29 Example 9.1.10 If a reasonable guess for is not available, can be specified as a proportion of thereby eliminating from the previous equation â Oã Nâ Section 9.1: Interval Estimation M For example, if is of and confidence is desired, should be used to estimate to within ➤ ➤ 30 Program Estimate A typical application: estimate the value of an unknown population mean by using replications to generate an independent random variate sample Function Generate() represents a discrete-event or Monte Carlo simulation program that returns a random variate output ➤ ➤ Program estimate automates the interval estimation process ➤ G G G for ( = 1; <= ; ++) = Generate(); return ; v å Given a level of confidence , program estimate can be used to compute an interval estimate for with ➤ Section 9.1: Interval Estimation 31 v , the å Given an interval half-width and level of confidence algorithm computes the interval estimate ➤ Algorithm 9.1.2 ç ➤ H I I H H I I ® H å ® ® ? E = idfNormal(0.0, 1.0, 1- /2); /* */ = Generate(); = 1; = 0.0; = ; * sqrt( -1)){ while (( <40) or ( *sqrt( / ) > = Generate(); ++; = - ; = + * * ( - 1) / ; = + / ; } return , ; It is important to appreciate the need for sample independence in Algorithms 9.1.1 and 9.1.2 Section 9.1: Interval Estimation 32

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