The power of a test is the of. by using a particular and a. value of the that is an to the value
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1 DEFINITION The power of a test is the of a hypothesis. The of the is by using a particular and a value of the that is an to the value assumed in the. POWER AND THE DESIGN OF EXPERIMENTS Just as is a common choice for a level, a power of at least is a common requirement for that a test is. When an, a goal of having a value of can often be used in the. Example 6: Chantix tablets are used as an aid to help people stop smoking. In a clinical trial, 129 subjects were treated with Chantix twice a day for 12 weeks, and 16 subjects experienced abdominal pain. If someone claims that more than 8% of Chantix users experience abdominal pain, that claim is supported with a hypothesis test conducted with a 0.05 significance level. Using 0.18 as an alternative value of p, the power of the test is Interpret this value of the power of the test. CREATED BY SHANNON MARTIN GRACEY 157
2 8.3 TESTING A CLAIM ABOUT A PROPORTION Key Concept In section 8.2 we presented the individual of a. In this section we present for a (or ) made about a. We illustrate testing with the method, the method, and the use of. In addition to testing about population proportions, we can use the procedure for testing claims about or the of. Two common methods for testing a claim about a proportion are (1) to use a as an to the distribution, and (2) to use an method based on the. PART 1: BASIC METHODS OF TESTING CLAIMS ABOUT A POPULATION PROPORTION p OBJECTIVE NOTATION n p pˆ q CREATED BY SHANNON MARTIN GRACEY 158
3 REQUIREMENTS 1. The observations are a sample. 2. The for a are satisfied. 3. The conditions and are satisfied so the of proportions can be by a with and. Note that is the used in the. TEST STATISTIC FOR TESTING A CLAIM ABOUT A PROPORTION z P values: Critical values: FINDING THE NUMBER OF SUCCESSES x Computer software and designed for tests of usually require consisting of the and the number of, but the is often given instead of. CREATED BY SHANNON MARTIN GRACEY 159
4 Example 1: Identify the indicated values. Use the normal distribution as an approximation to the binomial distribution. In a survey, 1864 out of 2246 randomly selected adults in the United States said that texting while driving should be illegal (based on data from Zogby International). Consider a hypothesis test that uses a 0.05 significance level to test the claim that more than 80% of adults believe that texting while driving should be illegal. a. What is the test statistic? b. What is the critical value? c. What is the P-value? d. What is the conclusion? CREATED BY SHANNON MARTIN GRACEY 160
5 Example 2: The company Drug Test Success provides a 1-Panel-THC test for marijuana usage. Among 300 tested subjects, results from 27 subjects were wrong (either a false positive or a false negative). Use a 0.05 significance level to test the claim that less than 10% of the test results are wrong. Does the test appear to be good for most purposes? a. Identify the null hypothesis b. Identify the alternative hypothesis c. Identify the test statistic d. Identify the P-value or critical value(s) e. What is your final conclusion? CREATED BY SHANNON MARTIN GRACEY 161
6 Example 3: In recent years, the town of Newport experienced an arrest rate of 25% for robberies (based on FBI data). The new sheriff compiles records showing that among 30 recent robberies, the arrest rate is 30%, so she claims that her arrest rate is greater than the 25% rate in the past. Is there sufficient evidence to support her claim that the arrest rate is greater than 25%? a. Identify the null hypothesis b. Identify the alternative hypothesis c. Identify the test statistic d. Identify the P-value or critical value(s) e. What is your final conclusion? CREATED BY SHANNON MARTIN GRACEY 162
7 8.4 TESTING A CLAIM ABOUT A MEAN: SIGMA KNOWN Key Concept In this section, we discuss methods for made about a, assuming the is a value. Here we use the with the components of that were introduced in Section 8.2. TESTING CLAIMS ABOUT A POPULATION MEAN (WITH OBJECTIVE KNOWN) NOTATION n x x REQUIREMENTS 1. The is a ( ). 2. The of the is. 3. The is and/or. CREATED BY SHANNON MARTIN GRACEY 163
8 TEST STATISTIC FOR TESTING A CLAIM ABOUT A MEAN (WITH KNOWN) z P values: Critical values: Example 1: When a fair die is rolled many times, the outcomes of 1, 2, 3, 4, 5, and 6 are equally likely, so the mean of the outcomes should be 3.5. The author drilled holes into a die and loaded it by inserting lead weights, then rolled it 40 times to obtain a mean of Assume that the standard deviation of the outcomes is , which is the standard deviation for a fair die. Use a 0.05 significance level to test the claim that outcomes from the loaded die have a mean different from the value of 3.5 expected with a fair die. a. Identify the null hypothesis b. Identify the alternative hypothesis c. Identify the test statistic d. Identify the P-value or critical value(s) e. What is your final conclusion? CREATED BY SHANNON MARTIN GRACEY 164
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