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409Hw07ans - STAT 409 Fall 2011 Homework#7(due Friday...

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STAT 409 Homework #7 Fall 2011 (due Friday, October 21, by 4:00 p.m.) 1. A researcher wishes to determine whether the starting salaries of high-school math teachers in private schools are higher than those of high-school math teachers in public schools. She selects a sample of new math teachers from each type of school and calculates the sample means and sample standard deviations of their salaries. Private Public sample size 10 7 sample mean $36,800 $36,300 sample standard deviation $600 $546 Assume that the populations are normally distributed and the population variances are equal. a) Construct a 95% confidence interval for the difference in average starting salaries of high-school math teachers in private and public schools. ( ) ( ) 2 7 10 546 1 7 600 1 10 2 2 2 pooled - + - + - = s = 335246.4 pooled s 579 ( ) 2 1 pooled 2 1 1 Y X n n s t + α ± - 10 + 7 – 2 = 15 degrees of freedom α = 0.05 t 0.025 ( 15 ) = 2.131 ( ) 7 1 10 1 579 131 . 2 300 , 36 800 , 36 + ± - 500 ± 608 ( 108 , 1108 ) b) Perform the appropriate test at a 1% level of significance. That is, test H 0 : μ Private = μ Public vs. H 1 : μ Private > μ Public . Test Statistic: T = ( ) 2 1 pooled 1 1 0 Y X n n s + - - = ( ) 7 1 10 1 579 0 300 , 36 800 , 36 + - - 1.752 .
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Rejection Region: T > t 0.01 ( 10 + 7 – 2 = 15 degrees of freedom ) = 2.602 . Decision: Do NOT Reject H 0 at α = 0.01. c) Find the p-value of the test in part (b). t 0.05 ( 15 ) = 1.753 . one tail 0.05 . p-value = right tail 0.05 . d) Suppose instead we wish to test H 0 : μ Private = μ Public vs. H 1 : μ Private μ Public . ( We wish to determine whether the starting salaries of high-school math teachers in private schools are different from those of high-school math teachers in public schools. ) What is the p-value for this test? p-value = two tails 0.10 .
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From the textbook: 8th edition ( ) 2. 7.3-8 ( ) use the test proposed by Welch Recall: n = 7, m = 10, x = 1511.714, y = 1118.400, s x 2 = 49669.905, s y 2 = 15297.600. a) H 0 : μ X = μ Y vs. H 1 : μ X μ Y Two – tailed. - + - + 1 1 1 1 2 2 2 2 2 2 1 2 1 1 2 2 2 2 1 2 1 n n n n n n s s s s = - + - + 10 6 . 15297 1 10 1 7 905 . 49669 1 7 1 10 6 . 15297 7 905 . 49669 2 2 2 = 8.5995 = 8 degrees of freedom The observed value of t (test statistic) ( ) t 10 6 . 15297 7 905 . 49669 0 400 . 1118 714 . 1511 + - - = 4.235 . The rejection (critical) region is: Reject H 0 if t < – t 0.025 ( 8 ) = – 2.306 or t > t 0.025 ( 8 ) = 2.306. The test statistic does fall into the rejection region. Reject H 0 at α = 0.05. b) No, the 95% confidence interval for μ X μ Y in 6.3-4 ( Homework 4 ) did not cover 0?
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