414 Chapter 9 Hypothesis Testing Step 2: Choose a significance level and find the critical value. The significance level is �� = 0.05. Since the alternate hypothesis is �� < 4.04, this is a left-tailed test. The critical value corresponding to �� = 0.05 is −1.645. Step 3: Compute the test statistic. The test statistic is the z-score of the sample mean ̄x. The population standard deviation is �� = 0.15. Since we assume H0 to be true, the population mean is ��0 = 4.04. The sample size is n = 50. Therefore, the test statistic is z = ̄x − ��0 ��∕ √ n = 3.99 − 4.04 √ 0.15∕ 50 = −2.36 Step 4: Determine whether to reject H0. This is a left-tailed test, so we reject H0 if z < −1.645. Since −2.36 < −1.645, we reject H0 at the �� = 0.05 level. See Figure 9.3. Critical region: Area = 0.05 z = −2.36 −1.645 Critical value Figure 9.3 The value of the test statistic, z = −2.36, is in the level �� = 0.05 critical region. Therefore, we reject H0 at the �� = 0.05 level. Step 5: State a conclusion. We conclude that the mean price of a gallon of regular gasoline in Los Angeles is less than $4.04. Check Your Understanding 4. A test is made of H0 : �� = 15 versus H1: �� > 15. The sample mean is ̄x = 16.5, the sample size is n = 50, and the population standard deviation is �� = 5. a. Find the value of the test statistic z. 2.12 b. Find the critical region for a level �� = 0.05 test. z ≥ 1.645 c. Do you reject H0 at the �� = 0.05 level? Yes 5. A test is made of H0 : �� = 125 versus H1: �� < 125. The sample mean is ̄x = 123, the sample size is n = 100, and the population standard deviation is �� = 20. a. Find the value of the test statistic z. −1.00 b. Find the critical region for a level �� = 0.02 test. z ≤ −2.054 c. Do you reject H0 at the �� = 0.02 level? No 6. A test is made of H0 : �� = 100 versus H1: �� ≠ 100. The sample mean is ̄x = 97, the sample size is n = 75, and the population standard deviation is �� = 8. a. Find the value of the test statistic z. −3.25 b. Find the critical region for a level �� = 0.01 test. z ≤ −2.576 or z ≥ 2.576 c. Do you reject H0 at the �� = 0.01 level? Yes Answers are on page 432. With the critical value method, the value of the test statistic is considered to be unusual if it is in the critical region, and not unusual if it is not in the critical region. We will now describe the P-value method, which provides more information than the critical value method. Whereas the critical value method tells us only whether the test statistic was unusual or not, the P-value method tells us exactly how unusual the test statistic is. For this reason, the P-value method is the one more often used in practice. In particular, almost all forms of technology use the P-value method.
navidi_monk_elementary_statistics_2e_ch7-9
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