USES AND ABUSES Statistics in the Real World 404 CHAPTER 7 Hypothesis Testing with One Sample EXERCISES In Exercises 1–3, assume that you work for the Internal Revenue Service. You are asked to write a report about the claim that 50% of U.S. adults think the amount of federal income tax they pay is too high. (Source: Gallup) 1. What is the null hypothesis in this situation? Describe how your report could be incorrect by trying to prove the null hypothesis. 2. Describe how your report could make a type I error. 3. Describe how your report could make a type II error. Uses Hypothesis testing is important in many different fields because it gives a scientific procedure for assessing the validity of a claim about a population. Some of the concepts in hypothesis testing are intuitive, but some are not. For instance, the American Journal of Clinical Nutrition suggests that eating dark chocolate can help prevent heart disease. A random sample of healthy volunteers were assigned to eat 3.5 ounces of dark chocolate each day for 15 days. After 15 days, the mean systolic blood pressure of the volunteers was 6.4 millimeters of mercury lower. A hypothesis test could show whether this drop in systolic blood pressure is significant or simply due to sampling error. Careful inferences must be made concerning the results. The study only examined the effects of dark chocolate, so the inference of health benefits cannot be extended to all types of chocolate. You also would not infer that you should eat large quantities of chocolate because the benefits must be weighed against known risks, such as weight gain and acid reflux. Abuses Not Using a Random Sample The entire theory of hypothesis testing is based on the fact that the sample is randomly selected. If the sample is not random, then you cannot use it to infer anything about a population parameter. Attempting to Prove the Null Hypothesis When the P@value for a hypothesis test is greater than the level of significance, you have not proven the null hypothesis is true—only that there is not enough evidence to reject it. For instance, with a P@value higher than the level of significance, a researcher could not prove that there is no benefit to eating dark chocolate—only that there is not enough evidence to support the claim that there is a benefit. Making Type I or Type II Errors Remember that a type I error is rejecting a null hypothesis that is true and a type II error is failing to reject a null hypothesis that is false. You can decrease the probability of a type I error by lowering the level of significance a. Generally, when you decrease the probability of making a type I error, you increase the probability b of making a type II error. Which error is more serious? It depends on the situation. In a criminal trial, a type I error is considered worse, as explained on page 352. If you are testing a person for a disease and they are assumed to be disease-free 1H02, then a type II error is more serious because you would fail to detect the disease even though the person has it. You can decrease the chance of making both types of errors by increasing the sample size. Do You Consider the Amount of Federal Income Tax You Pay as Too High, About Right, or Too Low? Too low 4% No opinion 2% Too high 50% About right 44%
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