Power Section 9.7

Objectives Compute the power of a test

Compute the power of a test Objective 1 Compute the power of a test

Errors in Hypothesis Tests Recall that in a hypothesis test, there are two types of errors: Type I Error – when a true 𝐻 0 is rejected. Type II Error – when we fail to reject a false 𝐻 0 . The probability of making a Type I Error is denoted by 𝛼 and the probability of making a Type II Error is denoted by 𝛽.

Power The power of a test is the probability of not making a Type II Error. In other words, the power of a test is the probability that we reject 𝐻 0 when it is false. We would like to have a small probability of rejecting a true 𝐻 0 , and a large probability of rejecting a false 𝐻 0 . That is, we would like the probability of a Type I Error to be small, and the power to be large. The power of a test about a population mean depends on the true value of the population mean. To compute the power, we specify a value 𝜇 1 for the population mean that satisfies the alternate hypothesis. The power is the probability that the test statistic falls in the critical region when 𝜇 1 is the true value of the population mean.

Computing the Power of a Test Step 1: Find the critical value: 𝑧 𝛼 for a one-tailed test or 𝑧 𝛼 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: 𝑥 ∗ = 𝜇 0 − 𝑧 𝛼 ∙ 𝜎 𝑛 Right-tailed: 𝑥 ∗ = 𝜇 0 + 𝑧 𝛼 ∙ 𝜎 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 ∗ = 𝜇 0 − 𝑧 𝛼 ∙ 𝜎 𝑛 𝑥 𝑟𝑖𝑔ℎ𝑡 ∗ = 𝜇 0 + 𝑧 𝛼 ∙ 𝜎 𝑛 Step 3: Let 𝜇 1 be a specific value that satisfies the alternate hypothesis. Sketch a normal curve with mean 𝜇 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: Area to the left of 𝑥 ∗ Right-tailed: Area to the right of 𝑥 ∗ Two-tailed: Sum of the area to the left of 𝑥 𝑙𝑒𝑓𝑡 ∗ and the area to the right of 𝑥 𝑟𝑖𝑔ℎ𝑡 ∗

Example The 2008 General Social Survey indicates that Americans watch an average of 2.98 hours of television per day, with a standard deviation of 𝜎 = 2.66 hours. A sociologist believes that the mean number 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 𝐻 0 : 𝜇 = 2.98 𝐻 1 : 𝜇 < 2.98 at the 𝛼 = 0.05 level. Assume the population standard deviation for college students is also 𝜎 = 2.66. Find the power of the test against the alternate 𝜇 1 = 2. Solution: This is a one-tailed test; therefore, we use the critical value 𝑧 𝛼 = 1.645. We have 𝜇 0 = 2.98, 𝑧 𝛼 = 1.645, 𝜎 = 2.66, and 𝑛 = 75. This is a left-tailed test. Therefore, 𝑥 ∗ = 𝜇 0 − 𝑧 𝛼 ∙ 𝜎 𝑛 = 2.98 – (1.645) 2.66 75 = 2.475

Solution The graph below presents a normal curve with mean 2 and the value of 𝑥 ∗ = 2.475 Since this is a left-tailed test, the power is the area to the left of 𝑥 ∗ = 2.475. To find this area, we find the 𝑧-score for 2.475, using the value 𝜇 1 = 2. 𝑧= 2.475 −2 2.66 75 = 1.55 The power is the area under the normal curve to the left of 𝑧 = 1.55. This area is 0.9394.

Illustration of the Concept of Power The graph presents two distributions for 𝑥 . The curve on the right is the distribution under the assumption that 𝐻 0 is true. This distribution has mean 𝜇 0 = 2.98. The curve on the left is the distribution under the assumption that the mean is equal to the alternate value 𝜇 1 = 2. The critical region is the region to the left of 𝑥 ∗ = 2.475. The area of the critical region under the null hypothesis distribution is the significance level 𝛼 = 0.05. The area under the alternate distribution is the power, 0.9394.

You Should Know… The meaning of power of a test How to compute the power of a test