352 CHAPTER 7 Hypothesis Testing with One Sample Truth about defendant Verdict Innocent Guilty Not guilty Justice Type II error Guilty Type I error Justice Hypothesis testing is sometimes compared to the legal system used in the United States. Under this system, these steps are used. 1. A carefully worded accusation is written. 2. The defendant is assumed innocent (H0) until proven guilty. The burden of proof lies with the prosecution. If the evidence is not strong enough, then there is no conviction. A “not guilty” verdict does not prove that a defendant is innocent. 3. The evidence needs to be conclusive beyond a reasonable doubt. The system assumes that more harm is done by convicting the innocent (type I error) than by not convicting the guilty (type II error). The table at the left shows the four possible outcomes. Identifying Type I and Type II Errors The USDA limit for salmonella contamination for ground beef is 7.5%. A meat inspector reports that the ground beef produced by a company exceeds the USDA limit. You perform a hypothesis test to determine whether the meat inspector’s claim is true. When will a type I or type II error occur? Which error is more serious? (Source: U.S. Department of Agriculture) SOLUTION Let p represent the proportion of the ground beef that is contaminated. The meat inspector’s claim is “more than 7.5% is contaminated.” You can write the null hypothesis as H0: p … 0.075 The proportion is less than or equal to 0.075. and the alternative hypothesis is Ha: p 7 0.075. (Claim) The proportion is greater than 0.075. You can visualize the null and alternative hypotheses using a number line, as shown below. 0.095 0.085 0.055 0.065 0.075 H0 p : p ≤ 0.075 Ha: p > 0.075 Ground beef meets USDA limits Ground beef exceeds USDA limits A type I error will occur when the actual proportion of contaminated ground beef is less than or equal to 0.075, but you reject H0. A type II error will occur when the actual proportion of contaminated ground beef is greater than 0.075, but you do not reject H0. With a type I error, you might create a health scare and hurt the sales of ground beef producers who were actually meeting the USDA limits. With a type II error, you could be allowing ground beef that exceeded the USDA contamination limit to be sold to consumers. A type II error is more serious because it could result in sickness or even death. TRY IT YOURSELF 2 A company specializing in parachute assembly states that its main parachute failure rate is not more than 1%. You perform a hypothesis test to determine whether the company’s claim is false. When will a type I or type II error occur? Which error is more serious? Answer: Page A40 EXAMPLE 2
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