Rec 12 - Hypothesis Testing for Means

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STAT 1430 Recitation 12 Hypothesis Testing for Means PART 1: For this assignment, assume the significance level, α, is equal to 0.05 unless otherwise stated. 1. In 2007, the average daily commute time for an American worker was 46 minutes. Many believe that the current state of the economy has led to an increase in the average daily commute time. Suppose commute times in 2011 follow a Normal distribution with a standard deviation of 10 minutes. A random sample of 75 American workers is taken, and an average commute time in 2011 of 50 minutes is reported. Is this enough evidence that the average commute time is higher in 2011 than it was in 2007? 1. Check any necessary conditions 2. Set up your hypothesis test by specifying the null and alternative hypotheses 3. Calculate the test statistic 4. Find your p-value 5. State your decision to reject or fail to reject. 6. State your conclusion statistically and in the context of the problem. 2. The survey of Study Habits and Attitudes (SSHA) is a psychological test that measures the motivation, attitude toward school, and study habits of students. Scores range from 0-200. The mean score for US college students is about 115 and the standard deviation is about 30. A teacher who suspects that older students have better attitudes toward school gives the SSHA to 40 students who are at least 30 years of age. Their mean score is 120. Assuming that σ=30 for the older students, is there enough evidence that the teacher is correct in her suspicions? 1. Check any necessary conditions 2. Set up your hypothesis test by specifying the null and alternative hypotheses 3. Calculate the test statistic 4. Find your p-value 5. State your decision to reject or fail to reject. 6. State your conclusion statistically and in the context of the problem. Do steps 1-6 above for each problem. FOR THE TEST STATISTIC AND THE p-VALUE STEPS, USE STAT CRUNCH. Hypothesis testing in StatCrunch -Open Stat Crunch (MyStatlab/StatCrunch/Visit StatCrunch Website/Open StatCrunch) -STATS/ZSTATS/ONE SAMPLE/WITH SUMMARY -Enter the sample mean, Standard deviation, and sample size -Under “Perform”, click Hypothesis Test, and fill out the null and alternative hypothesis (choose Ha is <, >, or not equal to as well) -COMPUTE!

STAT 1430 Recitation 12 Hypothesis Testing for Means 3. The Oil Price Information Center reports the mean price per gallon of regular gasoline is $2.29 with a standard deviation of $0.22. I think that this report is inaccurate and take a sample of 58 randomly selected gas stations. I find that the average price per gallon is $2.24. Do I have enough evidence that the report is wrong? 4. A health report claims that the average American sleeps 6.9 hours per night. You believe it’s less than that. You take a random sample of 64 Americans and find their average sleep is 6.4 hours. Do you have enough evidence for your claim? Assume the population standard deviation is 0.9 hours. 5. You run a hypothesis test to see whether the average price of gas is higher than the reported value of $2.53. You find a p-value of .001 and reject Ho. Which type of error could you be committing here, Type 1 or Type 2? 6. In a jury trial, the jury makes a decision, and they could be right or wrong. Describe how a type 1 error could happen in this situation, and how a type 2 error could happen in this situation. Which error do you believe is worse to commit and why? (This is just your own opinion.) 7. If you reject Ho, what is the only type of error you could possibly make? 8. If you fail to reject Ho, what is the only type of error you could possibly make? 9. Bob’s local pizza place claims it delivers pizzas in 30 minutes on average. Bob is convinced it’s more than that. He does a hypothesis test and gets a p-value of .001. a. What does Bob conclude? b. If Bob made the wrong conclusion what error did he make? c. What would be the impact of his error? 10. Bob’s local pizza place claims it delivers pizzas in 30 minutes on average. Bob is convinced it’s more than that. He does a hypothesis test and gets a p-value of .10. a. What does Bob conclude? b. If Bob made the wrong conclusion what error did he make? c. What would be the impact of his error? 11. Which type of error is the same as the significance level of a hypothesis test? a. Type 1 error b. Type 2 error c. Both d. Neither 12. A yogurt company is supposed to fill cups with 8 oz of yogurt on average. A quality control person comes and tests a sample of yogurt cups to see if the average is different from 8 oz. a. She gets a p-value of .03. What does she conclude? What error could she be making? And what is the impact of the error?

STAT 1430 Recitation 12 Hypothesis Testing for Means b. What if her p-value was .30 instead of .03? Answer the same questions. 13. Suppose it’s reported that the average gas price in Ohio is $2.10 per gallon. You believe it’s lower than that. Your random sample of 100 gas stations in Ohio has an average gas price of $1.98 per gallon. What is your null hypothesis? a. Ho: μ x = $ 2.10 b. Ho: μ x > $ 2.10 c. Ho: μ x = $ 2.20 d. None of the above 14. Suppose it’s reported that the average gas price in Ohio is $2.10 per gallon. You believe it’s lower than that. Your random sample of 100 gas stations in Ohio has an average gas price of $1.98 per gallon. What is your alternative hypothesis? a. Ha: μ x = $ 2.10 b. Ha: μ x > $ 2.10 c. Ha: μ x < $ 2.10 d. Ha: μ x < $ 1.98 15. Suppose it’s reported that the average gas price in Ohio is $2.10 per gallon. You believe it’s lower than that. Your random sample of 100 gas stations in Ohio has an average gas price of $1.98 per gallon. Assume the standard deviation of all gas prices is .50. What is the test statistic? a. 1.98 b. 2.40 c. -2.40 d. None of the above. 16. Suppose you are doing a hypothesis test where your hypotheses are Ho: u = 10 vs Ha: u > 10, and your test statistic is equal to 2.87. What is the p-value? a. .4979 b. .0021 c. .9979 d. None of the above. PART 2 For this assignment, assume the significance level, α, is equal to 0.05 unless otherwise stated. 17. If you have enough evidence against H 0 and you are able to reject it, your p-value must have been: a. Small b. Large

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STAT 1430 Recitation 12 Hypothesis Testing for Means 18. Suppose your p-value is .10. What is your conclusion? a. Reject Ho b. Fail to Reject Ho 19. The p-value is pre-set and known before the hypothesis test is conducted, but the significance level is unknown until the data has been collected. a. True b. False 20. The p-value changes if you obtain a new data set, but the significance level stays the same. a. True b. False 21. Yogurt containers are supposed to have a mean fill amount of 8 ounces. You believe the containers are under filled. What are the null and alternative hypotheses? a. H 0 : µ = 8 and H A : µ < 8 b. H 0 : µ < 8 and H A : µ ≥ 8 c. H 0 : ¯ x = 8 and H A : x < 8 d. None of the above 22. For a two-tailed hypothesis test (meaning Ha is µ ≠µ o ) the further away your test statistic gets from 0 the more likely it is that you will reject Ho. a. True b. False 23. For a two-tailed hypothesis test (meaning Ha is µ≠µ o ) the further away your test statistic gets from 0 the larger your p-value gets. a. True b. False 24. A two-tailed hypothesis test (meaning Ha is µ ≠ µ o ) is related to a confidence interval. If the value in H 0 is not in the confidence interval, it means you ____________ H 0 . a. Reject b. Fail to reject 25. The U.S. Census Bureau data show that the mean household income in the area served by a shopping mall is $42,500 per year with a standard deviation of $7,000. Some market research firm questions shoppers at the mall. The researchers suspect the mean household income of mall shoppers is higher than that of the general population. A random sample of 40 shoppers had a mean household income of $43,000. a. What are the null and alternative hypotheses? b. What is the sample statistic? c. What is the test statistic? d. What is the p-value? e. What is the conclusion?

STAT 1430 Recitation 12 Hypothesis Testing for Means 26. A test of the null hypothesis H 0 : µ = µ 0 gives test statistic z = 1.90. Our significance level is α = 0.05. a. What is the p-value if the alternative is H A : µ > µ 0 ? Do you reject the null hypothesis? b. What is the p-value if the alternative is H A : µ ≠ µ 0 ? Do you reject the null hypothesis? 27. If your 95% confidence interval contains the value in your null hypothesis of your hypothesis test, do you reject Ho or fail to reject Ho? Explain. 28. Explain why you double the probability when you have a two-tailed (not equal to) alternative hypothesis. 29. Different people have different ideas about what to do if alpha equals 0.05 and your p-value is exactly equal to 0.05 as well (this is a rare event). What would YOU do, reject Ho or fail to reject Ho? 30. If you have two p-values, one smaller than the other, which one is more likely to reject Ho? a. The one with the smaller p-value b. The one with the larger p-value c. Not enough information to tell

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