386 CHAPTER 8 Hypothesis Testing Because determination of power requires a particular value that is an alternative to the value assumed in the null hypothesis, a hypothesis test can have many different values of power, depending on the particular values of the population parameter chosen as alternatives to the null hypothesis. PART 3 Power of a Hypothesis Test We use b to denote the probability of failing to reject a false null hypothesis, so P(type II error) = b. It follows that 1 - b is the probability of rejecting a false null hypothesis, so 1 - b is a probability that is one measure of the effectiveness of a hypothesis test. DEFINITION The power of a hypothesis test is the probability 1 - b of rejecting a false null hypothesis. The value of the power is computed by using a particular significance level a and a particular value of the population parameter that is an alternative to the value assumed true in the null hypothesis. Power of a Hypothesis Test EXAMPLE 3 Consider these preliminary results from the XSORT method of gender selection: There were 13 girls among the 14 babies born to couples using the XSORT method. If we want to test the claim that girls are more likely 1p 7 0.52 with the XSORT method, we have the following null and alternative hypotheses: H0: p = 0.5 H1: p 7 0.5 Let’s use a significance level of a = 0.05. In addition to all of the given test components, finding power requires that we select a particular value of p that is an alternative to the value assumed in the null hypothesis H0: p = 0.5. Find the values of power corresponding to these alternative values of p: 0.6, 0.7, 0.8, and 0.9. SOLUTION The values of power in the following table were found by using Minitab, and exact calculations are used instead of a normal approximation to the binomial distribution. Specific Alternative Value of p B Power of Test = 1 − B 0.6 0.820 0.180 0.7 0.564 0.436 0.8 0.227 0.773 0.9 0.012 0.988 INTERPRETATION On the basis of the power values listed above, we see that this hypothesis test has power of 0.180 (or 18.0%) of rejecting H0: p = 0.5 when the population proportion p is actually 0.6. That is, if the true population proportion is actually equal to 0.6, there is an 18.0% chance of making the correct conclusion of rejecting the false null hypothesis that p = 0.5. That low power of 18.0% is not so good.
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