STAT 400 Lecture AL1 Spring 2015 Dalpiaz Examples for 7 2 8 2 Two independent samples X1 X2 Xn Y1 Y2 Yn 1 from population 1 mean 1 std dev 1 2 from population 2 mean 2 std dev 2 If n 1 and n 2 are large or population 1 and population 2 are approximately normal then X Y is approximately normal with mean 1 2 and standard deviation A confidence interval for 1 2 is 12 22 X Y z 2 n n 1 2 Test statistic for testing H 0 1 2 0 is Z If 1 and 2 are unknown use X Y 0 12 22 n1 n 2 X Y t 2 s 12 s 22 n1 n 2 Conservative approach the number of degrees of freedom the smaller of n 1 1 and n 2 1 Welch s T 2 s 12 s 22 n1 n 2 the number of degrees of freedom 2 2 2 2 1 s1 1 s 2 n1 1 n1 n 2 1 n 2 If n 1 and n 2 are large t 2 can be approximated by z 2 12 22 n1 n 2 1 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 SWYS Traditional sample average total points 450 410 sample standard deviation 20 45 a Construct a 95 confidence interval for the difference in the average total points for SWYS and traditional study methods b Perform the appropriate test at a 1 level of significance c Test H 0 S T 30 vs H 1 S T 30 at 0 05 2 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 If we can assume that population 1 and population 2 standard deviations are equal i e 1 2 then we can use X Y t 2 spooled where 2 spooled 1 n1 1 n2 n 1 1 s 12 n 2 1 s 22 n1 n 2 2 Then the number of degrees of freedom n 1 n 2 2 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 Sample Standard deviation Male Female 48 530 47 620 780 750 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 b What is the p value of the test H 0 Male Female vs H 1 Male Female The t Distribution r t 0 40 t 0 25 t 0 10 t 0 05 t 0 025 t 0 01 t 0 005 7 0 263 0 711 1 415 1 895 2 365 2 998 3 499 19 0 257 0 688 1 328 1 729 2 093 2 539 2 861 23 0 256 0 685 1 319 1 714 2 069 2 500 2 807 0 253 0 674 1 282 1 645 1 960 2 326 2 576 Matched Pair Comparison Pair Difference 1 X1 Y1 D1 X1 Y1 2 X2 Y2 D2 X2 Y2 Xn Yn Dn Xn Yn n Assume that the differences D i X i Y i are a random sample from normal distribution with mean and standard deviation D s A confidence interval for is D t 2 D n The number of degrees of freedom n 1 D 0 Test statistic for testing H 0 0 is T sD n 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 Pounds Lost 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 a Construct a 90 confidence interval for the average number of pounds lost during one week on that plan 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
View Full Document
Unlocking...