8-1 Basics of Hypothesis Testing 387 Because the calculations of power are quite complicated, the use of technology is strongly recommended. (In this section, only Exercises 29, 30, and 31 involve power.) Power and the Design of Experiments Just as 0.05 is a common choice for a significance level, a power of at least 0.80 is a common requirement for determining that a hypothesis test is effective. (Some statisticians argue that the power should be higher, such as 0.85 or 0.90.) When designing an experiment, we might consider how much of a difference between the claimed value of a parameter and its true value is an important amount of difference. If testing the effectiveness of the XSORT gender selection method, a change in the proportion of girls from 0.5 to 0.501 is not very important, whereas a change in the proportion of girls from 0.5 to 0.9 would be very important. Such magnitudes of differences affect power. When designing an experiment, a goal of having a power value of at least 0.80 can often be used to determine the minimum required sample size, as in the following example. There is a 0.436 probability of rejecting p = 0.5 when the true value of p is actually 0.7. It makes sense that this test is more effective in rejecting the claim of p = 0.5 when the population proportion is actually 0.7 than when the population proportion is actually 0.6. (When identifying animals assumed to be horses, there’s a better chance of rejecting an elephant as a horse—because of the greater difference—than rejecting a mule as a horse.) In general, increasing the difference between the assumed parameter value and the actual parameter value results in an increase in power, as shown in the table above. YOUR TURN. Do Exercise 30 “Calculating Power.” Finding the Sample Size Required to Achieve 80% Power EXAMPLE 4 Here is a statement similar to one in an article from the Journal of the American Medical Association: “The trial design assumed that with a 0.05 significance level, 153 randomly selected subjects would be needed to achieve 80% power to detect a reduction in the coronary heart disease rate from 0.5 to 0.4.” From that statement, we know the following: ■ Before conducting the experiment, the researchers selected a significance level of 0.05 and a power of at least 0.80. ■ The researchers decided that a reduction in the proportion of coronary heart disease from 0.5 to 0.4 is an important difference that they wanted to detect (by correctly rejecting the false null hypothesis). ■ Using a significance level of 0.05, power of 0.80, and the alternative proportion of 0.4, technology such as Minitab is used to find that the required minimum sample size is 153. The researchers can then proceed by obtaining a sample of at least 153 randomly selected subjects. Because of factors such as dropout rates, the researchers are likely to need somewhat more than 153 subjects. (See Exercise 31.) YOUR TURN. Do Exercise 31 “Finding Sample Size to Achieve Power.” Process of Drug Approval Gaining Food and Drug Administration (FDA) approval for a new drug is expensive and timeconsuming. Here are the different stages of getting approval for a new drug: • Phase I study: The safety of the drug is tested with a small (20–100) group of volunteers. • Phase II: The drug is tested for effectiveness in randomized trials involving a larger (100–300) group of subjects. This phase often has subjects randomly assigned to either a treatment group or a placebo group. • Phase III: The goal is to better understand the effectiveness of the drug as well as its adverse reactions. This phase typically involves 1,000–3,000 subjects, and it might require several years of testing. Lisa Gibbs wrote in Money magazine that “the (drug) industry points out that for every 5,000 treatments tested, only 5 make it to clinical trials and only 1 ends up in drugstores.” Total cost estimates vary from a low of $40 million to as much as $1.5 billion.
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