Section 9.2 Hypothesis Tests for a Population Mean, Standard Deviation Known 421 Step 1: State H0 and H1. The null hypothesis, H0, says that there is no difference between the mean level of satisfaction before and after telecommuting. Therefore, we have H0 : �� = 74 We are interested in knowing whether the mean level has changed. We are not specifically interested in whether it went up or down. Therefore, the alternate hypothesis is H1: �� ≠ 74 At this point, we assume that H0 is true. Step 2: Choose a level of significance. The level of significance is �� = 0.05. Step 3: Compute the test statistic. Since the sample size, n = 80, is large, ̄x is approximately normally distributed. The test statistic is the z-score for the sample mean ̄x. The population standard deviation is �� = 8. Because we assume H0 to be true, the population mean is �� = 74. Therefore, ̄x is normally distributed with mean 74 and standard error �� √ n = 8 √ 80 = 0.8944 We observed a value of ̄x = 76. The z-score is z = 76 − 74 0.8944 = 2.24 Step 4: Compute the P-value. The alternate hypothesis is �� ≠ 74, so this is a two-tailed test. The P-value is thus the sum of two areas: the area to the right of z = 2.24 and an equal area to the left of z = −2.24. Using Table A.2, we see that the area to the left of z = −2.24 is 0.0125. Therefore, the area to the right of z = 2.24 is also 0.0125. The P-value is therefore 0.0125 + 0.0125 = 0.0250. See Figure 9.9. The P-value is the sum of the tail areas: 0.0125 + 0.0125 = 0.0250 Area = 0.0125 Area = 0.0125 z = −2.24 z = 2.24 Figure 9.9 Step 5: Interpret the P-value. The P-value says that if H0 is true, then the probability of observing a test statistic as extreme as the one we actually observed is only 0.0250. In practice, this would generally be considered fairly strong evidence against H0. In particular, P < 0.05, so we reject H0 at the �� = 0.05 level. Step 6: State a conclusion. We conclude that the mean score among employees has changed since the adoption of telecommuting. Check Your Understanding 14. A social scientist suspects that the mean number of years of education �� for adults in a certain large city is greater than 12 years. She will test the null hypothesis H0 : �� =12 against the alternate hypothesis H1: �� > 12. She surveys a random sample of 100 adults and finds that the sample mean number of years is ̄x = 12.98. Assume that the standard deviation for the number of years of education is �� = 3 years. a. Compute the value of the test statistic. 3.27 b. Compute the P-value. 0.0005
navidi_monk_elementary_statistics_2e_ch7-9
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