t test Lecture 22 6 10 10 1 Last class t distribution helps us do statistical inference when population SD is unknown Computed confidence intervals Today Testing hypothesis using t distribution Lecture 22 6 10 10 2 For CI and hypothesis testing about a normal mean when is unknown the sample standard deviation s is used to estimate Note X z n replaced by X t s n When X is the mean of a random sample of size n from a normal distribution with mean then t has a t distribution with n 1 degrees of freedom df Lecture 22 6 10 10 3 To determine number of workers required to meet demand the productivity of new trainees is studied It is believed trainees can process more than 450 packages per hour within one week of hiring A sample of productivity of 50 trainees is observed and summarized as x 460 38 s 38 83 Can we conclude that this belief is correct based on the sample Lecture 22 6 10 10 4 Null hypothesis H0 0 Test statistic x 0 t s n Ha 0 0 0 p value 2P T t P T t P T t Lecture 22 6 10 10 5 The problem objective is to describe the population of the number of packages processed in one hour The hypotheses are H0 450 vs Ha 450 1 sided The t statistic x 460 38 450 t 1 89 s n 38 83 50 d f n 1 49 025 P value 05 There is sufficient evidence to infer that the mean productivity of trainees one week after being hired is greater than 450 packages at 05 significance level Lecture 22 6 10 10 6 Compare two population means based on two samples Matched pairs study turn two samples to one sample by taking differences Subjects are matched in pairs and outcomes compared within each pair Also common when observations are taken on same subject under different conditions Lecture 22 6 10 10 7 To compare mean reaction times to two types of traffic signs prohibitive I e g No Left Turn permissive II e g Left Turn Only 15 subjects were chosen and each subject was presented with 40 traffic signs 20 prohibitive and 20 permissive in random order The mean reaction times of the 15 subjects are as follows 1 2 3 4 5 6 7 8 9 10 11 12 13 14 I 7 6 10 2 9 5 1 3 3 0 6 3 5 3 6 2 2 2 4 8 11 3 12 1 6 9 7 6 II 7 3 9 1 8 4 1 5 2 7 5 8 4 9 5 3 2 0 4 2 11 0 11 0 6 1 6 7 d 0 3 1 1 1 1 0 2 0 3 0 5 0 4 0 9 0 2 0 6 0 3 1 1 0 8 0 9 Lecture 22 6 10 10 15 8 4 7 5 0 9 8 There is noticeable variability between subjects Some people react more quickly to any type of sign than other people The correlation within subjects is large If a person reacts quickly to one type of signs he she is more likely to react quickly to another type of signs as well Lecture 22 6 10 10 9 Null hypothesis H0 d 1 2 0 Test statistic d f n 1 Ha d 0 d 0 d 0 d 0 t sd n p value 2P T t P T t P T t Lecture 22 6 10 10 10 To compare mean reaction times to two types of traffic signs prohibitive I e g No Left Turn and permissive II e g Left Turn Only 15 subjects were chosen and each subject was presented with 40 traffic signs 20 prohibitive and 20 permissive in random order The mean reaction times of the 15 subjects are as follows 1 2 3 4 5 6 7 8 9 10 11 12 13 14 I 7 6 10 2 9 5 1 3 3 0 6 3 5 3 6 2 2 2 4 8 11 3 12 1 6 9 7 6 II 7 3 9 1 8 4 1 5 2 7 5 8 4 9 5 3 2 0 4 2 11 0 11 0 6 1 6 7 d 0 3 1 1 1 1 0 2 0 3 0 5 0 4 0 9 0 2 0 6 0 3 1 1 0 8 0 9 15 8 4 7 5 0 9 Use the paired t test to test the appropriate hypotheses at 05 Lecture 22 6 10 10 11 n 15 d 0 61 sd 0 394 Parameter of interest d 1 2 the difference between the average reaction times Hypotheses H0 d 0 vs Ha d 0 So here 0 0 Test statistic d 0 61 t 5 98 sd n 0 394 15 P value 2 P T 5 98 2 0 0005 Conclusion p value 05 so H0 is rejected Therefore the mean reaction times to the two types of traffic signs are significantly different Lecture 22 6 10 10 12 With paired samples we can derive a confidence interval for d 1 2 in the same way as we construct t intervals in one sample problem d d The statistics T sd n has a t distribution with df n 1 Manipulation of this t variable yields the following 100 1 CI d t 2 n 1 sd n Lecture 22 6 10 10 13 Construct a 95 confidence interval for the mean difference in reaction times d 1 2 d t 025 14 sd 15 0 61 2 145 0 394 15 0 61 0 218 0 392 0 828 The 95 CI does NOT contain 0 as expected Why Lecture 22 6 10 10 14 1 sample t test and CI Assumptions normal population unknown SD small samples Compare two population means via matched pairs turn 2 samples to just 1 sample by focusing on differences Exercises don t turn in 7 32 7 39 a c Lecture 22 6 10 10 15
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