SECTION 8.1 Testing the Difference Between Means (Independent Samples, s1 and s2 Known) 427 25. Getting at the Concept Explain why the null hypothesis H0: m1 = m2 is equivalent to the null hypothesis H0: m1 - m2 = 0. 26. Getting at the Concept Explain why the null hypothesis H0: m1 Ú m2 is equivalent to the null hypothesis H0: m1 - m2 Ú 0. Extending Concepts Testing a Difference Other Than Zero Sometimes a researcher is interested in testing a difference in means other than zero. In Exercises 27 and 28, you will test the difference between two means using a null hypothesis of H0: m1 - m2 = k, H0: m1 - m2 Ú k, or H0: m1 - m2 … k. The standardized test statistic is still z = 1x1 - x22 - 1m1 - m22 sx 1 -x2 where sx 1 -x2 = Bs 2 1 n1 + s 2 2 n2 . 27. Software Engineer Salaries Is the difference between the mean annual salaries of entry level software engineers in Santa Clara, California, and Greenwich, Connecticut, more than $4000? To decide, you select a random sample of entry level software engineers from each city. The results of each survey are shown in the figure at the left. Assume the population standard deviations are s1 = $14,060 and s2 = $13,050. At a = 0.05, what should you conclude? (Adapted from Salary.com) 28. Architect Salaries Is the difference between the mean annual salaries of entry level architects in Denver, Colorado, and Lincoln, Nebraska, equal to $9000? To decide, you select a random sample of entry level architects from each city. The results of each survey are shown in the figure. Assume the population standard deviations are s1 = $6560 and s2 = $6100. At a = 0.01, what should you conclude? (Adapted from Salary.com) x1 = $58,300 n1 = 32 x2 = $54,240 n2 = 30 Entry level architects in Denver, CO Entry level architects in Lincoln, NE Denver Lincoln Constructing Confidence Intervals for M1 − M2 You can construct a confidence interval for the difference between two population means m1 - m2, as shown below, when both population standard deviations are known, and either both populations are normally distributed or both n1 Ú 30 and n2 Ú 30. Also, the samples must be randomly selected and independent. 1x1 - x22 - zcBs 2 1 n1 + s 2 2 n2 6 m1 - m2 6 1x1 - x22 + zcBs 2 1 n1 + s 2 2 n2 In Exercises 29 and 30, construct the indicated confidence interval for m1 - m2. 29. Software Engineer Salaries Construct a 95% confidence interval for the difference between the mean annual salaries of entry level software engineers in Santa Clara, California, and Greenwich, CT, using the data from Exercise 27. 30. Architect Salaries Construct a 99% confidence interval for the difference between the mean annual salaries of entry level architects in Denver, Colorado, and Lincoln, Nebraska, using the data from Exercise 28. x1 = $88,900 n1 = 42 x2 = $81,600 n2 = 38 Entry level software engineers in Santa Clara, CA Entry level software engineers in Greenwich, CT Greenwich Santa Clara FIGURE FOR EXERCISE 27
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