Unformatted text preview: 5/4/11 Announcements Ligh3ng Homework: Propor3ons and Percents Must answer the ques3on being asked Last Five Lectures The Distribu3on of Sample Means The Logic of Hypothesis Tes3ng Change to Julian's Office Hours Wednesdays: 1:00 2:30 Thursdays: 10:30 12:00 1 5/4/11 Four Steps to Hypothesis Tes3ng 1. State a H0 and H1 F BE ROM FO RE Truth There is No Effect H0 is True There is an Effect H1 is True F BE ROM FO RE 2. Determine criteria for rejec3ng the null hypothesis Decisions 3. Obtain a random sample of size n from the popula3on and compute a test sta,s,c. 4. Made Decision: If value of test sta3s3c is more extreme than cri3cal values of test sta3s3c, reject H0; otherwise, fail to reject H0 Retain H0 CORRECT Reject H0 TYPE I ERROR, p = Retain H0 TYPE II ERROR, p = Reject H0 CORRECT Truth There is No Effect H0 is True There is an Effect H1 is True F BE ROM FO RE Single Sample Experiment Popula3on 0, 0 Decisions Retain H0 CORRECT Reject H0 TYPE I ERROR, p = Retain H0 TYPE II ERROR, p = Reject H0 CORRECT n Treatment Random Sample X 2 5/4/11 Single Sample Experiment Hypothe3cal Single Sample Experiment Hypothe3cal Popula3on 0, 0 Popula3on 0, 0 Treatment Treated Popula3on 1, 1 n n Treatment Random Sample Random Sample X n Treatment Random Sample X ZTest zX = X! "X F BE ROM FO RE ZTest zX = X! "X F BE ROM FO RE One Sample Test Requires knowledge of popula3on and One Sample Test Requires knowledge of popula3on and 3 5/4/11 ! null ! null !
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Xtreated 5 5/4/11 F BE ROM FO RE !? !2? !X = = n n Popula3on Sample !
"sigma" s
"s" Parameters and Sta3s3cs F BE ROM FO RE F BE ROM FO RE Popula3on Sample Standard Devia3on corrected for the purpose of es,ma,on Sample ? ? A ESTIM TION X s to correct for underes,ma,on STATISTICS (ESTIMATES) PARAMETERS 6 5/4/11 Calcula3ng the Standard Devia3on Popula,on SS = !(Xi " )2 F BE ROM FO RE Calcula3ng the Standard Devia3on Popula,on SS = !(Xi " )2 F BE ROM FO RE != SS N != SS N Sample SS = !(Xi " X)2 Sample s= SS n !1 SS = !(Xi " X)2 s= SS n !1 Calcula3ng the Standard Devia3on F BE ROM FO RE ! = !2
Standard Deviation = Variance Popula,on SS = !(Xi " )2 != SS N Sample ( Standard Deviation)2 = Variance SS = !(Xi " X)2 s= SS n !1 7 5/4/11 Calcula3ng the Variance Popula,on SS = !(Xi " )2 F BE ROM FO RE Parameters and Sta3s3cs F BE ROM FO RE !2 = SS N Popula3on Sample Sample SS = !(Xi " X)
2 SS s = n !1
2 ? 2? A ESTIM TION X s2 STATISTICS (ESTIMATES) PARAMETERS !? !2? !X = = n n s2 ! " 2 8 5/4/11 s !!
2 2 true standard error = ! X true standard error = ! X
estimated standard error = s X s sX = = n s2 n 9 5/4/11 t Sta3s3c tX = X! sX Check Your Understanding Which of the following is a fundamental difference between the tsta3s3c and a zsta3s3c? A B C D The t sta3s3c uses
the sample mean in place of the popula3on mean. The t sta3s3c uses the sample variance in place of the popula3on variance. The t sta3s3c computes the standard error by dividing the standard devia3on by n  1 instead of dividing by n. All of the above are differences between t and z. sX = s2 n Check Your Understanding Which of the following is a fundamental difference between the tsta3s3c and a zsta3s3c? A B C D The t sta3s3c uses the sample mean in place of the popula3on mean. The t sta3s3c uses the sample variance in place of the popula3on variance. The t sta3s3c computes the standard error by dividing the standard devia3on by n  1 instead of dividing by n. All of the above are differences between t and z. t Sta3s3c tX = X! sX sX = s2 n 10 5/4/11 = 100 2 = 225 Sampling Distribu3on of tx = 100 2 = 225 = 100 2 = 225 X 11 5/4/11 = 100 2 = 225 = 100 2 = 225 X = 89 X = 89 z X = X! "X = 100 2 = 225 = 100 2 = 225 X = 89 z X = X! X! = "X "2 n X = 89 z X = 89 ! 100 X! X! = = "X 225 "2 2 n 12 5/4/11 = 100 2 = 225 = 100 2 = 225 X = 89 zX = !11 10.61 X = 89 zX = !11 = !1.04 10.61 = 100 2 = 225 = 100 2 = 225 X = 89 tX = X! sX 13 5/4/11 = 100 2 = 225 = 100 2 = 225 X = 89 X = 89 tX = X! X! = sX s2 n tX = X! X! 89 ! 100 = = sX s2 s2 n 2 Calcula3ng the Variance Popula,on SS = !(Xi " )2 F BE ROM FO RE Calcula3ng the Variance Popula,on SS = !(Xi " )2 F BE ROM FO RE !2 = SS N !2 = SS N Sample SS = !(Xi " X)2 Sample s2 = SS n !1 SS = !(Xi " X)2 s2 = SS n !1 14 5/4/11 SS = !(Xi " X)2
i 1 2 X 87 91 i 1 2 X 87 91 SS = !(Xi " X)2
SS = (87 ! 89)2 + (91 ! 89)2 X = 89 X = 89 SS = !(Xi " X)2
i 1 2 X 87 91 SS = !(Xi " X)2
i 1 2 X 87 91 SS = (87 ! 89)2 + (91 ! 89)2 SS = (2) + (2)
2 2 SS = (87 ! 89)2 + (91 ! 89)2 SS = (2)2 + (2)2
SS = 8 X = 89 X = 89 15 5/4/11 s2 = SS n !1 s2 = s2 = SS n !1 8 2 !1 SS s = n !1
2 = 100 2 = 225 s2 = 8 2 !1 X = 89 s2 = 8
tX = 89 ! 100 s2 2 16 5/4/11 = 100 2 = 225 = 100 2 = 225 X = 89 X = 89 tX = 89 ! 100 s2 2 = 89 ! 100 8 2 tX = 89 ! 100 s2 2 = 89 ! 100 89 ! 100 = 8 4 2 = 100 2 = 225 = 100 2 = 225 X = 89 X = 89 tX = 89 ! 100 2 tX = 89 ! 100 !11 = 2 2 17 5/4/11 = 100 2 = 225 = 100 2 = 225 X = 89 tX = 89 ! 100 !11 = = 5.5 2 2 _ Distribu3on of ZX Check Your Understanding Under which of the following circumstances is it most likely to get a sample with a _ zx > 1.96 or z_ < 1.96? x 2.5% High Probability Samples if H0 is True 2 1.5 1 .5 .5 1 1.5 2 2.5% 2.5 2.5 zcri3cal = 1.96 zcri3cal = +1.96 A B C With a sample of size 2 With a sample of size 60 The likelihood of gepng an extreme z_ does not x depend on sample size 18 5/4/11 Check Your Understanding Under which of the following circumstances is it most likely to get a sample with a _ zx > 1.96 or z_ < 1.96? x _ Distribu3on of ZX 2.5% High Probability Samples if H0 is True 2 1.5 1 .5 .5 1 1.5 2 2.5% A B C With a sample of size 2 With a sample of size 60 The likelihood of gepng an extreme zx does not depend on sample size _ 2.5 2.5 zcri3cal = 1.96 zcri3cal = +1.96 Check Your Understanding Under which of the following circumstances is it most likely to get a sample with a _ zx > 1.96 or z_ < 1.96? x Check Your Understanding Under which of the following circumstances is it most likely to get a sample with a _ _ tx > 1.96 o r tx < 1.96? A B C With a sample of size 2 With a sample of size 60 The likelihood of gepng an extreme z_ does not x depend on sample size A B C With a sample of size 2 With a sample of size 60 _ The likelihood of gepng an extreme tx does not depend on sample size 19 5/4/11 each of the 5000 samples has 2 individuals n = 2 each of the 5000 samples has 4 individuals n = 4 20 5/4/11 Comparison of n = 2 vs n = 4 n = 2 n = 4 n = 2 n = 2 n = 4 n = 4 21 5/4/11 n = 2 n = 2 n = 4 n = 4 each of the 5000 samples has 10 individuals n = 10 22 5/4/11 n = 4 Comparison of n = 4 and n = 10 n = 10 n = 4 n = 4 n = 10 n = 10 23 5/4/11 n = 4 n = 4 n = 10 n = 10 each of the 5000 samples has 60 individuals n = 60 24 5/4/11 Comparison of n = 10 and n = 60 n = 10 n = 60 n = 10 n = 10 n = 60 n = 60 25 5/4/11 n = 10 n = 10 n = 60 n = 60 n = 2 n = 4 n = 10 n = 60 26 5/4/11 n = 2 n = 2 n = 4 n = 4 n = 10 n = 10 n = 60 n = 60 any n z_ x n = 2 n = 2 n = 4 n = 4 n = 10 n = 10 n = 60 n = 60 27 5/4/11 Check Your Understanding Under which of the following circumstances is it most likely to get a sample with a _ _ tx > 1.96 or tx < 1.96? Check Your Understanding Under which of the following circumstances is it most likely to get a sample with a _ _ tx > 1.96 or tx < 1.96? A B C With a sample of size 2 With a sample of size 60 _ The likelihood of gepng an extreme tx does not depend on sample size A B C With a sample of size 2 With a sample of size 60 _ The likelihood of gepng an extreme tx does not depend on sample size n = 2 n = 2 n = 4 n = 4 n = 10 n = 10 n = 60 n = 60 p(zx> 1.96) = 0.05 28 5/4/11 n = 2 n = 2 n = 4 n = 4 n = 10 n = 10 p(tx> 1.96) = 0.0816 n = 60 p(tx> 1.96) = 0.0548 n = 60 p(tx> 1.96) = 0.0548 p(zx> 1.96) = 0.05 p(zx> 1.96) = 0.05 n = 2 n = 2 p(tx > 1.96) = 0.3004 n = 4 p(tx > 1.96) = 0.1448 n = 4 p(tx > 1.96) = 0.1448 n = 10 p(tx> 1.96) = 0.0816 n = 10 p(tx> 1.96) = 0.0816 n = 60 p(tx> 1.96) = 0.0548 n = 60 p(tx> 1.96) = 0.0548 p(zx> 1.96) = 0.05 p(zx> 1.96) = 0.05 29 5/4/11 zX = X ! "X tX = X! sX zX = X ! "X tX = X! sX zX = X ! "X Distribu3on of t when n = 2 30 5/4/11 tX = X! sX zX = X ! "X tX = X! sX zX = X ! "X Distribu3on of t when n = 4 Distribu3on of t when n = 10 tX = X! sX zX = X ! "X tX = X! sX zX = X ! "X Distribu3on of t when n = 60 Distribu3on of t when n = 31 5/4/11 Distribu3on of t With small sample sizes, distribu3on of t has more extreme values than the distribu3on of z As n increases, distribu3on of t looks more like distribu3on of z When sample size = , distribu3on of t is iden3cal to distribu3on of z (Asympto3cally unitnormal) t Sta3s3c tX = X! sX sX = s2 n t Sta3s3c X! tX = sX t Sta3s3c sX = s2 n X! tX = sX sX = s2 n s2 ! ! 2 s2 ! ! 2 32 5/4/11 POPULATION N = 20,000 = 100 = 15 F BE ROM FO RE Distribu3on of the Sample Variance how does s2 es,mate 2? Standard devia3ons for 10 samples calculated using Popula3on Formula (formula for ) Sample 01: = 13.43 Sample 02: = 11.86 Sample 03: = 11.87 Sample 04: = 13.12 Sample 05: = 13.96 Sample 06: = 11.97 Sample 07: = 13.05 Sample 08: = 9.65 Sample 09: = 15.25 Sample 10: = 12.34 Standard devia3ons for 10 samples calculated using corrected Formula (formula for s) Sample 01: s = 16.12 Sample 02: s = 14.23 Sample 03: s = 14.24 Sample 04: s = 15.75 Average s Sample 05: s = 16.75 = 151.83 / 10 Sample 06: s = 14.39 Sample 07: s = 15.66 = 15.18 Sample 08: s = 11.58 Sample 09: s = 18.30 Sample 10: s = 14.81 Average = 126.5 / 10 = 12.65 33 5/4/11 Calcula3ng the Variance Popula,on SS = !(Xi " )2 F BE ROM FO RE !2 = SS N s2 = !(Xi " X)2 (90 " 100)2 + (110 " 100)2 200 = = = 200 n "1 2 "1 1 Sample SS = !(Xi " X)2 s2 = SS n !1 s2 = !(Xi " X)2 (90 " 100)2 + (110 " 100)2 200 = = = 200 n "1 2 "1 1 s2 = !(Xi " X)2 (90 " 100)2 + (110 " 100)2 200 = = = 200 n "1 2 "1 1 34 5/4/11 s2 = !(Xi " X)2 (90 " 100)2 + (110 " 100)2 200 = = = 200 n "1 2 "1 1 s2 = !(Xi " X)2 (90 " 100)2 + (110 " 100)2 200 = = = 200 n "1 2 "1 1 s2 = !(Xi " X)2 (90 " 100)2 + (110 " 100)2 200 = = = 200 n "1 2 "1 1 35 5/4/11 each of the 5000 samples has 2 individuals n = 2 each of the 5000 samples has 20 individuals n = 20 36 5/4/11 each of the 5000 samples has 120 individuals n = 120 n = 2 Comparison of n=2, n=20, n=120 n = 20 n = 120 37 5/4/11 n = 2 2 Distribu3on of t With small sample sizes, distribu3on of t has more extreme values than the distribu3on of z n = 20 2 As n increases, distribu3on of t looks more like distribu3on of z When sample size = , distribu3on of t is iden3cal to distribu3on of z (Asympto3cally unitnormal) n = 120 2 Distribu3on of t With small sample sizes, distribu3on of t has more extreme values than the distribu3on of z As n increases, distribu3on of t looks more like distribu3on of z When sample size = , distribu3on of t is iden3cal to distribu3on of z (Asympto3cally unitnormal) tX = X! X! > zX = sX "X when s X < " X 38 5/4/11 = 100 2 = 225 = 100 2 = 225 zX = 89 ! 100 !11 = = 1.04 "X 10.61 zX = 89 ! 100 !11 = = 1.04 "X 10.61 tX = 89 ! 100 11 = = 5.5 sX 2 tX = 89 ! 100 11 = = 5.5 sX 2 = 100 2 = 225 tX = X! X! > zX = sX "X when s X < " X zX = 89 ! 100 !11 = = 1.04 "X 10.61 tX = 89 ! 100 11 = = 5.5 sX 2 39 5/4/11 tX = X! X! > zX = sX "X when s X < " X
s ! < !X = n n
2 n = 2 2 n = 20 2 sX = 2 when s 2 < ! 2
2 n = 120 n = 2 2 2 n = 2 n = 20 2 2 n = 20 n = 120 2 2 n = 120 40 5/4/11 n = 2 2 3438/5000 = .69 69% with s2 < 2 2 n = 2 n = 20 2 2 n = 20 n = 120 2 2 n = 120 3438/5000 = .69 69% with s2 < 2 2 n = 2 3438/5000 = .69 69% with s2 < 2 2 n = 2 2750/5000 = .55 55% with s2 < 2 2 n = 20 2750/5000 = .55 55% with s2 < 2 2 n = 20 n = 120 2 2552/5000 = .51 51% with s2 < 2 2 n = 120 41 5/4/11 tX = X! X! > zX = sX "X when s X < " X
s2 !2 < !X = n n when s 2 < ! 2 tX = X! X! > zX = sX "X when s X < " X
s2 !2 < !X = n n when s 2 < ! 2 sX = sX = n = 2 tX = X! X! > zX = sX "X when s X < " X
s2 !2 < !X = n n when s 2 < ! 2
Will occur less frequently and less dramaKcally with larger samples p(tx > 1.96) = 0.3004 n = 4 p(tx > 1.96) = 0.1448 n = 10 p(tx> 1.96) = 0.0816 sX = n = 60 p(tx> 1.96) = 0.0548 p(zx> 1.96) = 0.05 42 5/4/11 Distribu3on of t With small sample sizes, distribu3on of t is more variable than distribu3on of z As n increases, distribu3on of t looks more like distribu3on of z When sample size = , distribu3on of t is iden3cal to distribu3on of z (Asympto3cally unitnormal) t Sta3s3c tX = X! sX sX = s2 n s2 ! ! 2 For Next Time Read: Chapter 9 Do Review Homework 5 solu3ons 43 ...
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 Winter '09
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 Standard Deviation, Standard Error, Variance, Harshad number

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