SECTION 7.2 Hypothesis Testing for the Mean (s Known) 371 Hypothesis Testing Using a Rejection Region Employees at a construction and mining company claim that the mean salary of the company’s mechanical engineers is less than that of one of its competitors, which is $95,600. A random sample of 20 of the company’s mechanical engineers has a mean salary of $93,300. Assume the population standard deviation is $9500 and the population is normally distributed. At a = 0.05, test the employees’ claim. SOLUTION Because s is known 1s = $95002, the sample is random, and the population is normally distributed, you can use the z@test. The claim is “the mean salary is less than $95,600.” So, the null and alternative hypotheses can be written as H0: m Ú $95,600 and Ha: m 6 $95,600. (Claim) Because the test is a left-tailed test and the level of significance is a = 0.05, the critical value is z0 = -1.645 and the rejection region is z 6 -1.645. The standardized test statistic is z = x - m s 1n Because s is known and the population is normally distributed, use the z@test. = 93,300 - 95,600 9500 220 Assume m = $95,600. ≈ -1.08. Round to two decimal places. The figure shows the location of the rejection region and the standardized test statistic z. Because z is not in the rejection region, you fail to reject the null hypothesis. z 2 1 −1 −2 0 z0 = −1.645 z ≈−1.08 α= 0.05 α 1 − = 0.95 5% Level of Significance Interpretation There is not enough evidence at the 5% level of significance to support the employees’ claim that the mean salary is less than $95,600. Be sure you understand the decision made in this example. Even though your sample has a mean of $93,300, you cannot (at a 5% level of significance) support the claim that the mean of all the mechanical engineers’ salaries is less than $95,600. For instance, the difference between your test statistic 1x = $93,3002 and the hypothesized mean 1m = $95,6002 could be due to sampling error. TRY IT YOURSELF 9 The CEO of the company in Example 9 claims that the mean workday of the company’s mechanical engineers is less than 8.5 hours. A random sample of 25 of the company’s mechanical engineers has a mean workday of 8.2 hours. Assume the population standard deviation is 0.5 hour and the population is normally distributed. At a = 0.01, test the CEO’s claim. Answer: Page A41 Picturing the World Each year, the Environmental Protection Agency (EPA) publishes reports of gas mileage for all makes and models of passenger vehicles. In a recent year, the subcompact car with the best mileage had a mean combined city/highway mileage of 113 miles per gallon. An auto manufacturer claims its subcompact cars have a combined city/highway mileage that exceeds the population mean of 25.7 miles per gallon. To support its claim, it tests 36 vehicles and obtains a sample mean of 28.5 miles per gallon. Assume the population standard deviation is 14.7 miles per gallon. (Source: U.S. Department of Energy) Is the evidence strong enough to support the claim that the subcompact car’s combined city/highway mileage exceeds 25.7 miles per gallon? Use a z-test with A = 0.10. See TI-84 Plus steps on page 415. EXAMPLE 9
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