AP Statistics: Section 6.2 TESTS OF SIGNIFICANCE USES/PURPOSES: 1. What are these tests used for? - To assess… 2. What do these tests compare? 3. What we calculating in these tests? 4. What are the 4 components of a Test of Significance? a. b. c. d. **In Ch. 6, we are testing a claim about the population mean ()** Example: Over his career, a pitcher for a baseball team has thrown his fastball at an average speed of 90 mph and a standard deviation of 2.2 mph. Some ESPN analysts have been saying recently that his average fastball speed is getting slower because he is getting older. The pitcher doesn’t believe them and wants to test and see if this is really the case. He takes a sample of 10 of his pitches, recorded below: 85, 87, 92, 90, 82, 85, 86, 86, 88, 81 Test to see if his average fastball speed is getting slower if = 0.05. 1. Hypotheses: Ho: Ha: 2. Test Statistic: 3. P-Value: 4. Conclusion: COMPONENTS: (page 454 – 462 in the book) 1. Hypotheses always describe… What types of symbols used? Null Hypothesis Symbol: What is it? Alternative Hypothesis Symbol: What is it? Generic Form of Ha: Two types of alternative hypotheses: 2. Tests Statistic Formula GENERIC: SPECIFIC (means): Z = __________________ 3. P-Value Definition: Calculation: (in Ch. 6) The smaller the p-value… But how small is small enough to say that our claim (Ho) is not true? GIVEN IN PROBLEM or ______________ Common significance levels: 4. Conclusion TWO CONCLUSIONS: REJECTING Ho: FAILING TO REJECT Ho: Our conclusion must always be a _______________________ in terms of ________. *Tests of Significance Summary: bottom of page 459 to top of page 450 in the book* Example 2: The average weekly earnings for men in managerial positions at a specific company is reported to be approximately normal with a mean of $725 and a standard deviation of $125. However you believe that the mean is too high (you suspect the company actually pays its employees less). To test this claim, you randomly select 40 managers and find their average weekly salary is $670. Use α=0.01 to test the claim. 1. Hypotheses: Ho: Ha: 2. Test Statistic: 3. P-Value: 4. Conclusion: 2- SIDED TEST OF SIGNIFICANCE Example 3: The state reports that the mean age of registered voters in Bucks County is 39.5 years old. As a curious researcher, you believe that the mean age is actually different. You take an SRS of 150 registered Bucks County voters and come up with a mean of 42 years old. The population standard deviation is known to be 10 years. Perform a test of significance at the 0.05 significance level. 1. Hypotheses: Ho: Ha: 2. Test Statistic: 3. P-Value: 4. Conclusion: AP Statistics Worksheet 6.2A - Tests on the Mean 1. State the appropriate null and alternative hypotheses for each situation. a. A tenant group thinks that the apartments are smaller than the advertised 1250 square feet. b. Larry switches to new motor oil that is advertised to increase his current gas mileage, which is 32 mpg. He wants to determine if his mileage has increased. c. A manufacturer measures the diameter in a sample of motors to determine whether the mean diameter has changed from the goal of 5 mm. d. A researcher thinks that a loud noise will cause mice to complete a maze faster than the previous average of 18 seconds. e. A sociologist suspects that at least one third of male students will name math as their favorite subject. 2. A commercial aircraft manufacturer buys rivets. A rivet is considered bad if its shear strength is less than 925 lbs. A random sample of 50 rivets is selected and the mean strength is calculated to be 921.18 with = 18. Test the hypothesis if = .05. 3. The following question was asked of the Student Affairs Office at a college: “How far does the average community-college student commute to college daily?” The office claims that it is on average 9 miles. We think it is actually more. If we take a random sample of 50 students and find a mean of 10.22 miles, test the hypothesis at a significance level of 0.05 with = 5. 4. It has been claimed that the mean weight of women students at a college is 54.4 kg. A professor sets out to show the mean is not 54.4 kg. In a random sample of 100 weights, the mean is 53.75. Is this sufficient evidence for the professor to reject the statement? Use = .01 and = 5.4. 5. The manager at Air Express feels that the weights of packages shipped recently are less than in the past. Records show that in the past packages have had a mean weight of 36.7 lb. and a standard deviation of 14.2 lb. A random sample of last month’s shipping records yielded a mean weight of 32.1 lb for 64 packages. Is this sufficient evidence to reject the null hypothesis in favor of the manager’s claim? Use = 0.01. 6. The estimated U.S. intake of trans-fatty acids is 8 g per day. Consider a research project involving 150 individuals in which their daily intake of trans-fatty acids was measured. Suppose the sample mean was 12.5 g. Assuming that = 8.0, test the research hypothesis that > 8 at = 0.05. 7. The average stay in days for nongovernment not-for-profit hospitals is given to be 7.0 days. A sample of 40 such hospitals was selected to test the hypothesis that the average stay different than the national average. The sample mean equals 6.1 days. Is this sufficient evidence to reject the null hypothesis? Use = 0.05 and = 1.5 days. AP Statistics Worksheet 6.2B –Tests on the Mean 1. A manufacturer claims that a new brand of air-conditioning unit uses only 6.5 kilowatts of electricity per day. A consumer agency believes the true figure is higher and runs a test on a sample of size 50. If the sample mean is 7.0 kilowatts and we know the population standard deviation is .4, should the manufacturer’s claim be rejected at a significance level of 5%? Of 1%? 2. A local chamber of commerce claims that the mean family income level in a city is $12,250. An economist runs a hypothesis test, using a sample of 135 families, and find s mean of $11,500 with a standard deviation of $3180. Should the $12,250 claim be rejected at a 5% level of significance? 3. Someone comments: “Because only a minority of high school students take the SAT, the scores overestimate the ability of typical high school seniors.” The mean SAT math score is claimed to be 475. You test the person’s claim by giving the test to an SRS of 500 seniors from California, and fin the mean score to be 461. Assuming the standard deviation is 100, is there evidence at the .01 level of significance to support the person’s claim? 4. Do middle-aged male executives have different average blood pressure than the general population? The National Center for Health Statistics reports that the mean blood pressure for males 35 to 44 years of age is 128 and the standard deviation in this population is 15. The medical director of a company looks at the medical records of 72 executives in this age group and finds that the mean blood pressure is 126.07. Is this evidence that executive blood pressures differ from the national average?