STAT 400 Lecture AL1 1 Spring 2015 Dalpiaz Answers for 7 2 8 2 Dr Statman claims that his new revolutionary study method Study While You Sleep is more effective than the traditional study methods In an experiment 250 students enrolled in the same section of STAT 100 at UIUC were divided into two groups One hundred students volunteered to study using SWYS method and the other 150 students did whatever students usually do At the end of the semester the averages of the total number of points out of 500 were compared for the two groups Note This is NOT a good experiment design a SWYS Traditional sample average total points 450 410 sample standard deviation 20 45 Construct a 95 confidence interval for the difference in the average total points for SWYS and traditional study methods n 1 100 and n 2 150 are large t 2 can be approximated by z 2 X Y z 2 s 12 s 22 n1 n 2 40 8 2 b 450 410 1 96 20 2 100 45 2 150 31 8 48 2 Perform the appropriate test at a 1 level of significance Claim New Old Test Statistic H 0 New Old 0 Z 20 2 45 2 100 150 Reject H 0 if Z z 0 01 2 326 Reject H 0 at 0 01 H 1 New Old 0 X Y 0 450 410 0 9 562 s 12 s 22 n1 n 2 Rejection Region vs n 1 and n 2 are large c Test H 0 S T 30 vs H 1 S T 30 at 0 05 X Y 0 450 410 30 2 39 Test Statistic Z Rejection Region Reject H 0 if Z z 0 05 1 645 s 12 s 22 n1 n 2 Reject H 0 at 0 05 2 20 2 45 2 100 150 p value 0 0084 Two work designs are being considered for possible adoption in an assembly plant A time study is conducted with 10 workers using design A and 12 workers using design B The sample means and sample standard deviations of their assembly times in minutes are Design A Design B Sample Mean 78 3 85 6 Sample Standard deviation 4 8 6 5 Construct a 90 confidence interval for the difference in the mean assembly times between design A and Design B Use Welch s T 2 s2 s2 1 2 n 1 n2 2 2 s2 s2 1 1 1 2 n 2 1 n 2 n1 1 n1 2 4 8 2 6 5 2 10 12 2 2 4 8 2 6 5 2 1 1 10 1 10 12 1 12 19 76324 19 degrees of freedom X Y t 2 7 3 4 173 s 12 s 22 n1 n 2 78 3 85 6 1 729 11 473 3 127 4 8 2 6 5 2 10 12 3 A national equal employment opportunities committee is conducting an investigation to determine if women employees are as well paid as their male counterparts in comparable jobs Random samples of 14 males and 11 females in junior academic positions are selected and the following calculations are obtained from their salary data Sample Mean Male Female 48 530 47 620 780 750 Sample Standard deviation Assume that the populations are normally distributed with equal variances a Construct a 95 confidence interval for the difference between the mean salaries of males and females in junior academic positions 2 spooled 14 1 780 2 11 1 750 2 14 11 2 588 443 47826 spooled 767 1 X Y t 2 spooled 1 n1 1 14 11 2 23 degrees of freedom n2 48 530 47 620 2 069 767 1 1 1 14 11 910 639 47 b 270 53 1 549 47 What is the p value of the test H 0 Male Female vs H 1 Male Female Test Statistic T X Y 0 s pooled 1 1 n1 n 2 48 530 47 620 0 2 944 767 1 1 1 14 11 t 0 005 n 1 n 2 2 23 d f 2 807 p value 2 tailed 0 005 2 0 01 p value 0 0073 4 A new revolutionary diet and exercise plan is introduced Eight participants were weighed in the beginning of the program and then again a week later The results were as follows Participant Weight Before Weight After 1 2 3 4 5 6 7 8 213 207 222 220 232 224 201 198 230 219 188 183 218 220 182 175 6 2 8 3 11 5 2 7 Pounds Lost a Construct a 90 confidence interval for the average number of pounds lost during one week on that plan D di 6 2 8 3 11 5 2 7 n 40 8 5 pounds 8 d d2 d d D d D 2 6 2 8 3 11 5 2 7 36 4 64 9 121 25 4 49 6 2 8 3 11 5 2 7 1 3 3 2 6 0 7 2 1 9 9 4 36 0 49 4 0 112 OR 312 d 2 d 40 2 312 2 s D2 n n 1 8 s D2 16 7 s D s D2 s D t 2 D 90 confidence level 0 10 5 1 895 8 n 5 2 68 2 n 1 112 16 7 16 4 pounds Confidence interval 4 d D 0 05 2 n 1 7 degrees of freedom t 1 895 2 2 32 7 68 b Is there enough evidence to conclude that the average weight loss is less than 7 pounds per week Use 0 05 What is the p value of this test Claim 7 H0 7 Test Statistic T D 0 sD n Rejection Region H1 7 5 7 1 4142 4 8 Reject H 0 if T t 0 05 8 1 7 d f 1 895 The value of the test statistic does not fall into the Rejection Region Do NOT Reject H 0 at 0 05 p value 0 10
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