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© 1997 Prentice-Hall, Inc. Importance of Normal Distribution n Describes many random processes or continuous phenomena n Can be used to approximate discrete probability distributions l l n Binomial Poisson Basis for classical statistical inference 5-1 Normal Distribution © 1997 Prentice-Hall, Inc. n n n n ‘Bell-shaped’ & symmetrical f(X ) Mean, median, mode are equal ‘Middle spread’ is 1.33 s Random variable has infinite range 5-2 X Mean Median Mode Standardize the Normal Distribution © 1997 Prentice-Hall, Inc. Z Normal Distribution X s Standardized Normal Distribution s s = 1 X = 0 One table! 5-3 Z Standardizing Example © 1997 Prentice-Hall, Inc. Normal Distribution s = 10 = 5 5-4 6 .2 X Standardizing Example © 1997 Prentice-Hall, Inc. Z X s Normal Distribution 6 .2 5 10 .1 2 Standardized Normal Distribution s = 10 = 5 5-5 s = 1 6 .2 X = 0 .1 2 Z Obtaining the Probability © 1997 Prentice-Hall, Inc. Standardized Normal Probability Table (Portion) Z .0 0 .0 1 .02 s = 1 0 .0 .0 0 0 0 .0 0 4 0 .0 0 8 0 0.1 .0 3 9 8 .0 4 3 8 .0478 .0478 0 .2 .0 7 9 3 .0 8 3 2 .0 8 7 1 = 0 0 .3 .1 1 7 9 .1 2 1 7 .1 2 5 5 Probabilities 5-6 .1 2 Z Shaded area exaggerated © 1997 Prentice-Hall, Inc. 5-7 Example P(3.8 X 5) Example P(3.8 X 5) © 1997 Prentice-Hall, Inc. Z X s Normal Distribution 3 .8 5 10 .1 2 Standardized Normal Distribution s = 10 s = 1 .0478 3 .8 = 5 5-8 X -.1 2 = 0 Shaded area exaggerated Z © 1997 Prentice-Hall, Inc. 5-9 Example P(2.9 X 7.1) © 1997 Prentice-Hall, Inc. Example P(2.9 X 7.1) Z Normal Distribution Z X s X s 2 .9 5 10 7 .1 5 10 .2 1 .2 1 Standardized Normal Distribution s = 10 s = 1 .1664 .0832 .0832 2 .9 5 7 .1 5 - 10 X -.2 1 0 .2 1 Shaded area exaggerated Z © 1997 Prentice-Hall, Inc. 5 - 11 Example P(X 8) Example P(X 8) © 1997 Prentice-Hall, Inc. Z X s Normal Distribution 8 5 10 .3 0 Standardized Normal Distribution s = 10 s = 1 .5000 .3821 .1179 = 5 5 - 12 8 X = 0 Shaded area exaggerated .3 0 Z Central Limit Theorem © 1997 Prentice-Hall, Inc. As sample size gets large enough ( 30) ... sampling distribution becomes almost normal. XX 5 - 13 © 1997 Prentice-Hall, Inc. Introduction to Estimation 5 - 14 Statistical Methods © 1997 Prentice-Hall, Inc. SSta tatis tistic ticaall M Meeth thooddss DDeessccrip riptiv tivee SSta tatis tistic ticss In Infe fere renntia tiall SSta tatis tistic ticss EEsstim timaatio tionn 5 - 15 HHyyppooth theessis is TTeesstin tingg Estimation Process © 1997 Prentice-Hall, Inc. Population J J Mean, , is unknown J J J Sample J J J J 5 - 16 Random Sample Mean J J`X = 50 I am 95% confident that is between 40 & 60. © 1997 Prentice-Hall, Inc. Population Parameters Are Estimated E Esstim tim aate te ppooppuula latio tionn ppaara ram m eete ter... r... M M eeaann P Pro roppoortio rtionn pp V Vaaria riannccee D Diffe iffere renncceess 5 - 17 ss 22 11 22 w with ith ssaam m pple le ssta tatis tistic tic `x `x ppss 22 ss `x `x11 -`x -`x22 Point Estimation © 1997 Prentice-Hall, Inc. n Provides single value l n n Based on observations from 1 sample Gives no information about how close value is to the unknown population parameter Example: Sample mean`X = 3 is point estimate of unknown population mean 5 - 18 Interval Estimation © 1997 Prentice-Hall, Inc. n Provides range of values l n Gives information about closeness to unknown population parameter l n Based on observations from 1 sample Stated in terms of probability Example: Unknown population mean lies between 50 & 70 with 95% confidence 5 - 19 © 1997 Prentice-Hall, Inc. Key Elements of Interval Estimation A probability that the population parameter falls somewhere within the interval. Confidence interval Confidence limit (lower) 5 - 20 Sample statistic (point estimate) Confidence limit (upper) Confidence Limits for Population Mean © 1997 Prentice-Hall, Inc. Parameter = Statistic ± Error (1) X E rro r (2 ) E rro r X o r X (3 ) © 1984-1994 T/Maker Co. 5 - 21 Z X s xx (4 ) E rro r Z s xx (5 ) X Zs xx E rro r s xx © 1997 Prentice-Hall, Inc. Many Samples Have Same Interval `X = ± Zs`x sx_ -1.65s`x +1.65s`x 90% Samples 5 - 22 `X © 1997 Prentice-Hall, Inc. Many Samples Have Same Interval `X = ± Zs`x sx_ -1.65s`x -1.96s`x +1.65s`x +1.96s`x 90% Samples 95% Samples 5 - 23 `X © 1997 Prentice-Hall, Inc. Many Samples Have Same Interval `X = ± Zs`x sx_ -2.58s`x -1.65s`x -1.96s`x +1.65s`x +2.58s`x +1.96s`x 90% Samples 95% Samples 99% Samples 5 - 24 `X Level of Confidence © 1997 Prentice-Hall, Inc. n n Probability that the unknown population parameter falls within interval Denoted (1 - a)% l n ais probability that parameter is not within interval Typical values are 99%, 95%, 90% 5 - 25 © 1997 Prentice-Hall, Inc. Sampling Distribution of Mean Intervals & Level of Confidence _ a /2 s x_ x 1 -a a /2 ``xx = X (1 - a) % of intervals contain . Intervals extend from `X - Zs`X to `X + Zs`X a % do not. Large number of intervals 5 - 26 _ © 1997 Prentice-Hall, Inc. n Data dispersion l n Measured by s Intervals extend from `X - Zs`X to`X + Zs`X Sample size l n Factors Affecting Interval Width s`X = s / n Level of confidence (1 - a) l Affects Z © 1984-1994 T/Maker Co. 5 - 27 Confidence Interval Estimates © 1997 Prentice-Hall, Inc. CCoonnfid fideennccee In ls rvaals terv Inte M Meeaann ssK Knnoowwnn 5 - 28 PPro rtionn roppoortio ss UUnnkknnoowwnn VVaaria riannccee FFin ite inite PPooppuula tionn latio © 1997 Prentice-Hall, Inc. Confidence Interval Estimate Mean (s Known) 5 - 29 Confidence Interval Estimates © 1997 Prentice-Hall, Inc. CCoonnfid fideennccee In ls rvaals terv Inte M Meeaann ss KKnnoowwnn 5 - 30 PPro rtionn roppoortio ss UUnnkknnoowwnn VVaaria riannccee FFin ite inite PPooppuula tionn latio © 1997 Prentice-Hall, Inc. n Confidence Interval Mean (s Known) Assumptions l l l Population standard deviation is known Population is normally distributed If not normal, can be approximated by normal distribution (n 30) 5 - 31 Confidence Interval Mean (s Known) © 1997 Prentice-Hall, Inc. n Assumptions l l l n Population standard deviation is known Population is normally distributed If not normal, can be approximated by normal distribution (n 30) Confidence interval estimate X Z aa //22 5 - 32 s n X Z aa //22 s n © 1997 Prentice-Hall, Inc. Estimation Example Mean (s Known) The mean of a random sample of n = 25 is`X = 50. Set up a 95% confidence interval estimate for if s = 10. 5 - 33 © 1997 Prentice-Hall, Inc. Estimation Example Mean (s Known) The mean of a random sample of n = 25 is`X = 50. Set up a 95% confidence interval estimate for if s = 10. X Z aa //22 5 0 1.9 6 s n 10 X Z aa //22 5 0 1.9 6 25 4 6 .0 8 5 3 .9 2 5 - 34 s n 10 25 Confidence Interval Solution* © 1997 Prentice-Hall, Inc. X Z aa //22 1.9 9 1.6 4 5 s n .0 5 100 X Z aa //22 n 1.9 9 1.6 4 5 1.9 8 2 1.9 9 8 5 - 35 s .0 5 100 © 1997 Prentice-Hall, Inc. Confidence Interval Estimate Mean (s Unknown) 5 - 36 Confidence Interval Estimates © 1997 Prentice-Hall, Inc. CCoonnfid fideennccee In ls rvaals terv Inte M Meeaann ss KKnnoowwnn 5 - 37 PPro rtionn roppoortio ssU Unnkknnoowwnn VVaaria riannccee FFin ite inite PPooppuula tionn latio © 1997 Prentice-Hall, Inc. n Confidence Interval Mean (s Unknown) Assumptions l l Population standard deviation is unknown Population must be normally distributed n Use Student’s t distribution n Confidence interval estimate X t aa //22,, nn 11 5 - 38 S n X t aa //22,, nn 11 S n Student’s t Distribution © 1997 Prentice-Hall, Inc. Standard Bellnormal shaped Symmetric t (df = 13) ‘Fatter’ tails t (df = 5) 0 5 - 39 Z t Student’s t Table © 1997 Prentice-Hall, Inc. Assume: n=3 df =n-1 =2 a = .10 a/2 =.05 a/2 U p p e r T a il A re a df .2 5 .1 0 .0 5 1 1 .0 0 0 3 .0 7 8 6 .3 1 4 2 0 .8 1 7 1 .8 8 6 2 .9 2 0 3 0 .7 6 5 1 .6 3 8 2 .3 5 3 .05 0 t values 5 - 40 t Student’s t Table © 1997 Prentice-Hall, Inc. Assume: n=3 df =n-1 =2 a = .10 a/2 =.05 a/2 U p p e r T a il A re a df .2 5 .1 0 .0 5 1 1 .0 0 0 3 .0 7 8 6 .3 1 4 2 0 .8 1 7 1 .8 8 6 2 .9 2 0 3 0 .7 6 5 1 .6 3 8 2 .3 5 3 .05 0 t values 5 - 41 2.920 t © 1997 Prentice-Hall, Inc. Estimation Example Mean (s Unknown) A random sample of n = 25 has`X = 50 & S = 8. Set up a 95% confidence interval estimate for . X t aa //22,, nn 11 5 0 2 .0 6 3 9 S n 8 X t aa //22,, nn 11 5 0 2 .0 6 3 9 25 4 6 .6 9 5 3 .3 0 5 - 42 S n 8 25 Thinking Challenge © 1997 Prentice-Hall, Inc. You’re a time study analyst in manufacturing. You’ve recorded the following task times (min.): 3.6, 4.2, 4.0, 3.5, 3.8, 3.1. What is the 90% confidence interval estimate of the population mean task time? 5 - 43 Alone Group Class © 1997 Prentice-Hall, Inc. Confidence Interval Solution* `X = 3.7 S = 3.8987 n = 6, df = n - 1 = 6 - 1 = 5 S / n = 3.8987 / 6 = 1.592 t.05,5 = 2.0150 3.7 - (2.015)(1.592) 3.7 + (2.015)(1.592) 0.492 6.908 5 - 44 © 1997 Prentice-Hall, Inc. Estimation of Mean for Finite Populations 5 - 45 Confidence Interval Estimates © 1997 Prentice-Hall, Inc. CCoonnfid fideennccee In ls rvaals terv Inte M Meeaann ssK Knnoowwnn 5 - 46 PPro rtionn roppoortio ss UUnnkknnoowwnn VVaaria riannccee FFin ite inite PPooppuula tionn latio Estimation for Finite Populations © 1997 Prentice-Hall, Inc. n Assumptions l Sample is large relative to population s n / N > .05 n Use finite population correction factor n Confidence interval (mean, s unknown) X t aa //22,, nn 11 5 - 47 S n N n N 1 X t aa //22,, nn 11 S n N n N 1 © 1997 Prentice-Hall, Inc. Confidence Interval Estimate of Proportion 5 - 48 Confidence Interval Estimates © 1997 Prentice-Hall, Inc. CCoonnfid fideennccee In ls rvaals terv Inte M Meeaann ssK Knnoowwnn 5 - 49 PPro rtionn roppoortio ss UUnnkknnoowwnn VVaaria riannccee FFin ite inite PPooppuula tionn latio © 1997 Prentice-Hall, Inc. n Assumptions l l l n Confidence Interval Proportion Two categorical outcomes Population follows binomial distribution Normal approximation can be used s n·p 5 & n·(1 - p) 5 Confidence interval estimate p ss Z 5 - 50 p ss (1 p ss ) n p p ss Z p ss (1 p ss ) n © 1997 Prentice-Hall, Inc. Estimation Example Proportion A random sample of 400 graduates showed 32 went to grad school. Set up a 95% confidence interval estimate for p. 5 - 51 © 1997 Prentice-Hall, Inc. Estimation Example Proportion A random sample of 400 graduates showed 32 went to grad school. Set up a 95% confidence interval estimate for p. p ss Z aa //22 .0 8 1.9 6 p ss (1 p ss ) n .0 8 ( 1 .0 8 ) 400 p p ss Z aa //22 p .0 8 1.9 6 .0 5 3 p .1 0 7 5 - 52 p ss (1 p ss ) n .0 8 ( 1 .0 8 ) 400 Thinking Challenge © 1997 Prentice-Hall, Inc. You’re a production manager for a newspaper. You want to find the % defective. Of 200 newspapers, 35 had defects. What is the 90% confidence interval estimate of the population proportion defective? 5 - 53 Alone Group Class © 1997 Prentice-Hall, Inc. Confidence Interval Solution* n·p 5 n·(1 - p) 5 p ss Z aa //22 .1 7 5 1.6 4 5 p ss (1 p ss ) n .1 7 5 (.8 2 5 ) 200 p p ss Z aa //22 p .1 7 5 1.6 4 5 .1 3 0 8 p .2 1 9 2 5 - 54 p ss (1 p ss ) n .1 7 5 (.8 2 5 ) 200 This Class... © 1997 Prentice-Hall, Inc. Please take a moment to answer the following questions in writing: n What was the most important thing you learned in class today? n What do you still have questions about? n How can today’s class be improved? 5 - 55
