Section 9.7 Power 465 Computing the Power of a Test Let ��0 be the value of �� specified by H0, let �� be the significance level, let n be the sample size, and let �� be the population standard deviation. Step 1: Find the critical value: z�� for a one-tailed test or z��∕2 for a two-tailed test. Step 2: ∙ For a one-tailed test, find the value of ̄x whose z-score is equal to the critical value. We call this value ̄x∗. Find the value of ̄x∗ as follows: Left-tailed: H1: �� < ��0 ̄x∗ = ��0 − z�� ⋅ �� √ n Right-tailed: H1: �� > ��0 ̄x∗ = ��0 + z�� ⋅ �� √ n ∙ For a two-tailed test, there are two values of ̄x∗. We call them ̄x∗ left and ̄x∗ right. They are computed as follows: ̄x∗ left = ��0 − z��∕2 ⋅ �� √ n ̄x∗ right = ��0 + z��∕2 ⋅ �� √ n Step 3: Let ��1 be a specific value that satisfies the alternate hypothesis. Sketch a normal curve with mean ��1. Step 4: The power is an area under the normal curve sketched in Step 3. The area depends on the form of the alternate hypothesis, as follows: Left-tailed: H1: �� < ��0 Area to the left of ̄x∗ Right-tailed: H1: �� > ��0 Area to the right of ̄x∗ Two-tailed: H1: �� ≠ ��0 Sum of the area to the left of ̄x∗ left and the area to the right of ̄x∗ right EXAMPLE 9.24 Compute the power of a test The 2008 General Social Survey indicates that Americans watch an average of 2.98 hours of television per day, with a standard deviation of �� = 2.66 hours. A sociologist believes that the mean viewing time for college students is less, because students spend more time on the Internet and playing video games. The sociologist will sample 75 college students and test the hypotheses H0 : �� = 2.98 H1: �� < 2.98 at the �� = 0.05 level. Assume the population standard deviation for college students is also �� = 2.66. Find the power of the test against the alternative ��1 = 2. Solution Step 1: This is a one-tailed test, with significance level �� = 0.05. Therefore, we use the critical value z�� = 1.645. Step 2: We have ��0 = 2.98, z�� = 1.645, �� = 2.66, and n = 75. This is a left-tailed test. Therefore, ̄x∗ = 2.98 − (1.645) 2.66 √ 75 = 2.475 Step 3: The following figure presents a normal curve with mean 2. The value of ̄x∗ =2.475 is indicated as well. 2 2.475
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