Statistics homework, answer questions only on word doc attached below. Use a TI 84 whenever possible. Show work whenever possible. Use the answer sheet to double check work.
Math 12 Section 9.1 In Exercises 7 and 8, fill in each blank with the appropriate word or phrase. 7. The __________ hypothesis states that a parameter is equal to a certain value while the ___________ hypothesis states that the parameter differs from this value. 8. Rejecting H0 when it is true is called a _________ error, and failing to reject H0 when it is false is called a ____________ error. In Exercises 9–12, determine whether the statement is true or false. If the statement is false, rewrite it as a true statement. 9. H1: μ > 50 is an example of a left-tailed alternate hypothesis. 10. If we reject H0, we conclude that H0 is false. 11. If we do not reject H0, then we conclude that H1 is false. 12. If we do not reject H0, we conclude that H0 is true. In Exercises 13–16, determine whether the alternate hypothesis is left-tailed, right-tailed, or two-tailed. 15. H0: μ = 1 H1: μ ≠ 1 Section 9.1 Cont. 25. Big dogs: A veterinarian claims that the mean weight of adult German shepherd dogs is 75 pounds. A test is made of H0 : μ = 75 versus H1: μ ≠ 75. The null hypothesis is not rejected. State an appropriate conclusion. 29. Scales: It is desired to check the calibration of a scale by weighing a standard 10-gram weight 100 times. Let μ be the population mean reading on the scale, so that the scale is in calibration if μ = 10 and out of calibration if μ ≠ 10. A test is made of the hypotheses H0: μ = 10 versus H1: μ ≠ 10. Consider three possible conclusions: (i) The scale is in calibration. (ii) The scale is not in calibration. (iii) The scale might be in calibration. a. Which of the three conclusions is best if H0 is rejected? b. Which of the three conclusions is best if H0 is not rejected? c. Assume that the scale is in calibration, but the conclusion is reached that the scale is not in calibration. Which type of error is this? d. Assume that the scale is not in calibration. Is it possible to make a Type I error? Explain. e. Assume that the scale is not in calibration. Is it possible to make a Type II error? Explain. Math 12 Section 9.2 In Exercises 23–28, fill in each blank with the appropriate word or phrase. 23. The ___________ is the probability, assuming H0 is true, of observing a value for the test statistic that is as extreme as or more extreme than the value actually observed. 24. The smaller the P-value is, the stronger the evidence against the ______ hypothesis becomes. 26. If we decrease the value of the significance level α, we the __________ probability of a Type I error. 27. If we decrease the value of the significance level α, we __________ the probability of a Type II error. 51. Netflix: A study conducted in 2015 by the technology company Rovi (now TiVo) showed that the mean time spent per day browsing the video streaming service Netflix for something to watch was 19.3 minutes. Assume the standard deviation is σ = 8. Suppose a simple random sample of 100 visits taken this year has a sample mean of = 21.5 minutes. A social scientist is interested to know whether the mean time browsing Netflix has increased. a. State the appropriate null and alternate hypotheses. b. Compute the value of the test statistic. c. State a conclusion. Use the α = 0.05 level of significance. Section 9.2 Cont. 57. House prices: Data from the National Association of Realtors indicate that the mean price of a home in Denver, Colorado, in December 2016 was 366.5 thousand dollars. A random sample of 50 homes sold in 2017 had a mean price of 396.3 thousand dollars. a. Assume the population standard deviation is σ = 150. Can you conclude that the mean price in 2017 differs from the mean price in December 2016? Use the α = 0.05 level of significance. b. Following is a boxplot of the data. Explain why it is not reasonable to assume that the population is approximately normally distributed. c. Explain why the assumptions for the hypothesis test are satisfied even though the population is not normal. Section 9.2 Cont. 59. . What are you drinking? Environmental Protection Agency standards require that the amount of lead in drinking water be less than 15 micrograms per liter. Twelve samples of water from a particular source have the following concentrations, in units of micrograms per liter: 11.4 13.9 11.2 14.5 15.2 8.1 12.4 8.6 10.5 17.1 9.8 15.9 a. Explain why it is necessary to check that the population is approximately normal before performing a hypothesis test. b. Following is a dotplot of the data. Is it reasonable to assume that the population is approximately normal? c. Assume that the population standard deviation is σ = 3. If appropriate, perform a hypothesis test at the α = 0.01 level to determine whether you can conclude that the mean concentration of lead meets the EPA standard. What do you conclude? Section 9.2 Cont. 61. Interpret calculator display: The age in years was recorded for a sample of books in a college library. The following display from a TI-84 Plus calculator presents the results of a hypothesis test regarding the mean age of books in this library. a. What are the null and alternate hypotheses? b. What is the value of the test statistic? c. What is the P-value? d. Do you reject H0 at the α = 0.05 level? State a conclusion. e. Do you reject H0 at the α = 0.01 level? State a conclusion. Section 9.2 Cont. 65. Statistical or practical significance: A new method of teaching arithmetic to elementary school students was evaluated. The students who were taught by the new method were given a standardized test with a maximum score of 100 points. They scored an average of one point higher than students taught by the old method. A hypothesis test was performed in which the null hypothesis stated that there was no difference between the two groups, and the alternate hypothesis stated that the mean score for the new method was higher. The P-value was 0.001. True or false: a. Because the P-value is very small, we can conclude that the mean score for students taught by the new method is higher than for students taught by the old method. b. Because the P-value is very small, we can conclude that the new method represents an important improvement over the old method Math 12 Section 9.3 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. 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 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. a. 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? Section 9.3 Cont. 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, were 1.28 1.81 2.71 3.13 1.55 2.55 3.36 3.86 a. 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? 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. 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. 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. 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. Section 9.3 Cont. 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.