Math chapter 9.4
Exercises 1–6 are the Check Your Understanding exercises located within the section.Understanding the Concepts
In Exercises 7 and 8, fill in each blank with the appropriate word or phrase.
| 7. | To perform a t-test when the sample size is small, the sample must show no evidence of strong  and must contain no . |
| 8. | The number of degrees of freedom for the Student’s t-test of a population mean is always 1 less than the . |
In Exercises 9 and 10, determine whether the statement is true or false. If the statement is false, rewrite it as a true statement.
| 9. | A t-test is used when the population standard deviation is unknown. |
| 10. | A t-test is used when the number of degrees of freedom is unknown. |
Practicing the Skills
| 11. | Find the P-value for the following values of the test statistic t, sample size n, and alternate hypothesis H1. If you use Table A.3, you may specify that P is between two values.a.t = 2.336, n = 5, H1: μ > μ0b.t = 1.307, n = 18, H1: μ ≠ μ0c.t = −2.864, n = 51, H1: μ < μ0d.t = −2.031, n = 3, H1: μ ≠ μ0 |
| 12. | Find the P-value for the following values of the test statistic t, sample size n, and alternate hypothesis H1. If you use Table A.3, you may specify that P is between two values.a.t = −1.584, n = 19, H1: μ ≠ μ0b.t = −2.473, n = 41, H1: μ < μ0c.t = 1.491, n = 30, H1: μ ≠ μ0d.t = 3.635, n = 4, H1: μ > μ0 |
| 13. | Find the critical value or values for the following values of the significance level α, sample size n, and alternate hypothesis H1.a.α = 0.05, n = 27, H1: μ ≠ μ0b.α = 0.01, n = 61, H1: μ > μ0c.Page 456α = 0.10, n = 16, H1: μ ≠ μ0d.α = 0.05, n = 11, H1: μ < μ0 |
| 14. | Find the critical value or values for the following values of the significance level α, sample size n, and alternate hypothesis H1.a.α = 0.05, n = 39, H1: μ > μ0b.α = 0.01, n = 34, H1: μ < μ0c.μ = 0.10, n = 6, H1: μ ≠ μ0d.α = 0.05, n = 25, H1: μ ≠ μ0 |
Working with the Concepts
| 15. | Is there a doctor in the house? The market research firm Salary.com reported that the mean annual earnings of all family practitioners in the United States was $178,258. A random sample of 55 family practitioners in Los Angeles that month had mean earnings of = $192,340 with a standard deviation of $42,387. Do the data provide sufficient evidence to conclude that the mean salary for family practitioners in Los Angeles is greater than the national average?a.State the null and alternate hypotheses.b.Compute the value of the t statistic. How many degrees of freedom are there?c.State your conclusion. Use the α = 0.05 level of significance. |
| 16. | College tuition: The mean annual tuition and fees for a sample of 14 private colleges in California was $37,900 with a standard deviation of $7,200. A dotplot shows that it is reasonable to assume that the population is approximately normal. Can you conclude that the mean tuition and fees for private institutions in California differs from $35,000?a.State the null and alternate hypotheses.b.Compute the value of the t statistic. How many degrees of freedom are there?c.State your conclusion. Use the α = 0.01 level of significance.Based on data from collegeprowler.com |
| 17. | Big babies: The National Health Statistics Reports described a study in which a sample of 360 one-year-old baby boys were weighed. Their mean weight was 25.5 pounds with standard deviation 5.3 pounds. A pediatrician claims that the mean weight of one-year-old boys is greater than 25 pounds. Do the data provide convincing evidence that the pediatrician’s claim is true? Use the α = 0.01 level of significance. |
| 18. | Good credit: The Fair Isaac Corporation (FICO) credit score is used by banks and other lenders to determine whether someone is a good credit risk. Scores range from 300 to 850, with a score of 720 or more indicating that a person is a very good credit risk. An economist wants to determine whether the mean FICO score is lower than the cutoff of 720. She finds that a random sample of 100 people had a mean FICO score of 703 with a standard deviation of 92. Can the economist conclude that the mean FICO score is less than 720? Use the α = 0.05 level of significance. |
| 19. | Commuting to work: The American Community Survey sampled 1923 people in Colorado and asked them how long it took them to commute to work each day. The sample mean one-way commute time was 24.5 minutes with a standard deviation of 13.0 minutes. A transportation engineer claims that the mean commute time is less than 25 minutes. Do the data provide convincing evidence that the engineer’s claim is true? Use the α = 0.05 level of significance. |
| 20. | Watching TV: The General Social Survey asked a sample of 1298 people how much time they spent watching TV each day. The mean number of hours was 3.09 with a standard deviation of 2.87. A sociologist claims that people watch a mean of 3 hours of TV per day. Do the data provide sufficient evidence to disprove the claim? Use the α = 0.01 level of significance. |
| 21. | Weight loss: In a study to determine whether counseling could help people lose weight, a sample of people experienced a group-based behavioral intervention, which involved weekly meetings with a trained interventionist for a period of six months. The following data are the numbers of pounds lost for 14 people, based on means and standard deviations given in the article.18.224.8 3.920.017.1 8.813.417.333.829.7 8.531.219.315.1Source: Journal of the American Medical Association 299:1139–1148a.Following is a boxplot for these data. Is it reasonable to assume that the conditions for performing a hypothesis test are satisfied? Explain. b.If appropriate, perform a hypothesis test to determine whether the mean weight loss is greater than 10 pounds. Use the α = 0.05 level of significance. What do you conclude? ©TRBfoto/Getty Images |
| 22. | How much is in that can? A machine that fills beverage cans is supposed to put 12 ounces of beverage in each can. Following are the amounts measured in a simple random sample of eight cans.11.96 12.10 12.04 12.13 11.98 12.05 11.91 12.03a.Following is a dotplot for these data. Is it reasonable to assume that the conditions for performing a hypothesis test are satisfied? Explain. b.Page 457If appropriate, perform a hypothesis test to determine whether the mean volume differs from 12 ounces. Use the α = 0.05 level of significance. What do you conclude? |
| 23. | Credit card debt: Following are outstanding credit card balances for a sample of 16 college seniors at a large university. 8704191021723152387 3353342618529 5937695024851213347a.Following is a dotplot for these data. Is it reasonable to assume that the conditions for performing a hypothesis test are satisfied? Explain. b.According to the report How America Pays for College, by Sallie Mae, the mean outstanding balance for college seniors in 2012 was $515. If appropriate, perform a hypothesis test to determine whether the mean debt for seniors at this university differs from $515. |
| 24. | Rats: A psychologist is designing an experiment in which rats will navigate a maze. Ten rats run the maze, and the time it takes for each to complete the maze is recorded. The results are as follows.66.368.152.568.362.655.642.160.969.269.3a.Following is a boxplot for these data. Is it reasonable to assume that the conditions for performing a hypothesis test are satisfied? Explain. b.The psychologist hopes that the mean time for a rat to run the maze will be greater than 60 seconds. If appropriate, perform a hypothesis test to determine whether the mean time is greater than 60 seconds. |
| 25. | Keep cool: Following are prices, in dollars, of a random sample of ten 7.5-cubic-foot refrigerators.314377330285319274332350299306a.Following is a dotplot for these data. Is it reasonable to assume that the conditions for performing a hypothesis test are satisfied? Explain. b.A consumer organization reports that the mean price of 7.5-cubic-foot refrigerators is greater than $300. Do the data provide convincing evidence of this claim? Use the α = 0.01 level of significance.Exercise Video – Hypothesis Test for Mean with Sigma Unknown – P-Value Method (TI-84 PLUS) |
| 26. | Free dessert: In an attempt to increase business on Monday nights, a restaurant offers a free dessert with every dinner order. Before the offer, the mean number of dinner customers on Monday was 150. Following are the numbers of diners on a random sample of 12 days while the offer was in effect.206169191152212139142151174220192153a.Following is a boxplot for these data. Is it reasonable to assume that the conditions for performing a hypothesis test are satisfied? Explain. b.Can you conclude that the mean number of diners increased while the free dessert offer was in effect? Use the α = 0.01 level of significance. |
| 27. | Effective drugs: When testing a new drug, scientists measure the amount of the active ingredient that is absorbed by the body. In a study done at the Colorado School of Mines, a new antifungal medication that was designed to be applied to the skin was tested. The medication was applied to the skin of eight adult subjects. One hour later, the amount of active ingredient that had been absorbed into the skin was measured for each subject. The results, in micrograms, were1.28 1.81 2.71 3.13 1.55 2.55 3.36 3.86a.Construct a boxplot for these data. Is it appropriate to perform a hypothesis test? do you conclude?b.If appropriate, perform a hypothesis test to determine whether the mean amount absorbed is greater than 2 micrograms. Use the α = 0.05 level of significance. What do you conclude? |
| 28. | More effective drugs: An antifungal medication was applied to the skin of eight adult subjects. One hour later, the amount of active ingredient that had been absorbed into the skin was measured for each subject. The results, in micrograms, were2.13 1.88 2.07 1.19 2.51 5.61 2.81 3.05a.Construct a boxplot for these data. Is it appropriate to perform a hypothesis test?b.If appropriate, perform a hypothesis test to determine whether the mean amount absorbed is less than 3 micrograms. Use the α = 0.05 level of significance. What do you conclude? |
| 29. | Interpret calculator display: A sample of adults was asked how many hours per day they spend on social media. The following display from a TI-84 Plus calculator presents the results of a hypothesis test regarding the mean number of hours per day spent on social media. a.State the null and alternate hypotheses.b.What is the value of ?c.What is the value of s?d.How many degrees of freedom are there?e.Do you reject H0 at the 0.05 level? State a conclusion.f.Someone wants to test the hypothesis H0: μ = 1.8 versus H1: μ > 1.8. Use the information in the display to compute the t statistic for this test.g.Page 458Compute the P-value for the test in part (f).h.Can the null hypothesis in part (f) be rejected at the 0.05 level? State a conclusion. |
| 30. | Interpret calculator display: A sample of adults was asked how many hours per week they spend on watching television. The following display from a TI-84 Plus calculator presents the results of a hypothesis test regarding the mean number of hours per week spent watching television. a.State the null and alternate hypotheses.b.What is the value of ?c.What is the value of s?d.How many degrees of freedom are there?e.Do you reject H0 at the 0.05 level? State a conclusion.f.Someone wants to test the hypothesis H0: μ = 22.5 versus H1: μ ≠ 22.5. Use the information in the display to compute the t statistic for this test.g.Compute the P-value for the test in part (f).h.Can the null hypothesis in part (f) be rejected at the 0.05 level? State a conclusion. |
| 31. | Interpret computer output: A veterinarian recorded the weights, in grams, for a sample of hamsters. The following MINITAB output presents the results of a hypothesis test regarding the mean weight of hamsters. a.State the null and alternate hypotheses.b.What is the value of ?c.What is the value of s?d.How many degrees of freedom are there?e.Do you reject H0 at the 0.05 level? State a conclusion.f.Someone wants to test the hypothesis H0: μ = 6.5 versus H1: μ < 6.5. Use the information in the output to compute the t statistic for this test.g.Compute the P-value for the test in part (f).h.Can the null hypothesis in part (f) be rejected at the 0.05 level? State a conclusion. |
| 32. | Interpret computer output: A sample of adults was asked how many miles per day they commute to work. The following MINITAB output presents the results of a hypothesis test regarding the mean number of miles commuted to work. a.State the null and alternate hypotheses.b.What is the value of ?c.What is the value of s?d.How many degrees of freedom are there?e.Do you reject H0 at the 0.05 level? State a conclusion.f.Someone wants to test the hypothesis H0: μ = 9 versus H1: μ > 9. Use the information in the output to compute the t statistic for this test.g.Compute the P-value for the test in part (f).h.Can the null hypothesis in part (f) be rejected at the 0.05 level? State a conclusion. |
| 33. | Does this diet work? In a study of the effectiveness of a certain diet, 100 subjects went on the diet for a period of six months. The sample mean weight loss was 0.5 pound, with a sample standard deviation of 4 pounds.a.Find the t statistic for testing H0: μ = 0 versus H1: μ > 0.b.Find the P-value for testing H0: μ = 0 versus H1: μ > 0.c.Can you conclude that the diet produces a mean weight loss that is greater than 0? Use the α = 0.05 level of significance. |
| 34. | Effect of larger sample size: The study described in Exercise 33 is repeated with a larger sample of 1000 subjects. Assume that the sample mean is once again 0.5 pound and the sample standard deviation is once again 4 pounds.a.Find the t statistic for testing H0: μ = 0 versus H1: μ > 0. Is the value of the t statistic greater than or less than the value obtained with a smaller sample of 100?b.Find the P-value for testing H0: μ = 0 versus H1: μ > 0.c.Can you conclude that the diet produces a mean weight loss that is greater than 0? Use the α = 0.05 level of significance.d.Explain why the mean weight loss is not of practical significance, even though the results are statistically significant at the 0.05 level. |
| 35. | Perform a hypothesis test? A sociologist wants to test the null hypothesis that the mean number of people per household in a given city is equal to 3. He surveys 50 households on a certain block in the city and finds that the sample mean number of people is 3.4 with a standard deviation of 1.2. Should these data be used to perform a hypothesis test? Explain why or why not. |
| 36. | Perform a hypothesis test? A health professional wants to test the null hypothesis that the mean length of hospital stay for a certain surgical procedure is 4 days. She obtains records for all the patients who have undergone the procedure at a certain hospital during a given year, and finds that the mean length of stay is 4.7 days with a standard deviation of 1.1 days. Should these data be used to perform a hypothesis test? Explain why or why not. |
| 37. | Larger or smaller P-value? In a study of sleeping habits, a researcher wants to test the null hypothesis that adults in a certain community get a mean of 8 hours of sleep versus the alternative that the mean is not equal to 8. In a sample of 250 adults, the mean number of hours of sleep was 8.2. A second researcher repeated the study with a different sample of 250, and obtained a sample mean of 7.5. Both researchers obtained the same standard deviation. Will the P-value of the second researcher be greater than or less than that of the first researcher? Explain. |
| 38. | Page 459Larger or smaller P-value? Juan and Mary want to test the null hypothesis that the mean length of text messages sent by students at their school is 10 characters versus the alternative that it is less. Juan samples 100 text messages and finds the mean length to be 8.4 characters. Mary samples 100 messages and finds the mean length to be 7.3 characters. Both Juan and Mary obtained the same standard deviation. Will Juan’s P-value be greater than, less than, or the same as Mary’s P-value? Explain. |
| 39. | Interpret a P-value: A real estate agent believes that the mean size of houses in a certain city is greater than 1500 square feet. He samples 100 houses, and performs a test of H0: μ = 1500 versus H1: μ > 1500. He obtains a P-value of 0.0002.a.The real estate agent concludes that because the P-value is very small, the mean house size must be much greater than 1500. Is this conclusion justified?b.Another real estate agent says that because the P-value is very small, we can be fairly certain that the mean size is greater than 1500, but we cannot conclude that it is a lot greater. Is this conclusion justified? |
| 40. | Interpret a P-value: The manufacturer of a medication designed to lower blood pressure claims that the mean systolic blood pressure for people taking their medication is less than 135. To test this claim, blood pressure is measured for a sample of 500 people who are taking the medication. The P-value for testing H0: μ = 135 versus H1: μ < 135 is P = 0.001.a.The manufacturer concludes that because the P-value is very small, we can be fairly certain that the mean pressure is less than 135, but we cannot conclude that it is a lot smaller. Is this conclusion justified?b.Someone else says that because the P-value is very small, we can conclude that the mean pressure is a lot less than 135. Is this conclusion justified? |
Extending the Concepts
| 41. | Using z instead of t: When the sample size is large, some people treat the sample standard deviation s as if it were the population standard deviation σ, and use the standard normal distribution rather than the Student’s t distribution, to find a critical value. Assume that a right-tailed test will be made with a sample of size 100 from a normal population, using the α = 0.05 significance level.a.Find the critical value under the assumption that σ is known.b.In fact, σ is unknown. How many degrees of freedom should be used for the Student’s t distribution?c.What is the probability of rejecting H0 when it is true if the critical value in part (a) is used? You will need technology to find the answer. |
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