430 Chapter 9 Hypothesis Testing Working with the Concepts 51. Facebook: A study by the Web metrics firm Experian showed that in August 2011, the mean time spent per visit to Facebook was 20.8 minutes. Assume the standard deviation is �� = 8 minutes. Suppose that a simple random sample of 100 visits in August 2013 has a sample mean of ̄x = 23 minutes. A social scientist is interested to know whether the mean time of Facebook visits has increased. a. State the appropriate null and alternate hypotheses. H0 : �� = 20.8, H1: �� > 20.8 b. Compute the value of the test statistic. 2.75 c. State a conclusion. Use the �� = 0.05 level of significance. Reject H0. 52. Are you smarter than a second-grader? A random sample of 60 second-graders in a certain school district are given a standardized mathematics skills test. The sample mean score is ̄x = 52. Assume the standard deviation of test scores is �� = 15. The nationwide average score on this test is 50. The school superintendent wants to know whether the second-graders in her school district have greater math skills than the nationwide average. a. State the appropriate null and alternate hypotheses. H0 : �� = 50, H1: �� > 50 b. Compute the value of the test statistic. 1.03 c. State a conclusion. Use the �� = 0.01 level of significance. Do not reject H0. 53. Height and age: Are older men shorter than younger men? According to the National Health Statistics Reports, the mean height for U.S. men is 69.4 inches. In a sample of 300 men between the ages of 60 and 69, the mean height was ̄x = 69.0 inches. Public health officials want to determine whether the mean height �� for older men is less than the mean height of all adult men. a. State the appropriate null and alternate hypotheses. H0 : �� = 69.4, H1: �� < 69.4 b. Assume the population standard deviation to be �� = 2.84 inches. Compute the value of the test statistic. −2.44 c. State a conclusion. Use the �� = 0.01 level of significance. Reject H0. 54. Calibrating a scale: Making sure that the scales used by businesses in the United States are accurate is the responsibility of the National Institute for Standards and Technology (NIST) in Washington, D.C. Suppose that NIST technicians are testing a scale by using a weight known to weigh exactly 1000 grams. They weigh this weight on the scale 50 times and read the result each time. The 50 scale readings have a sample mean of ̄x = 1000.6 grams. The scale is out of calibration if the mean scale reading differs from 1000 grams. The technicians want to perform a hypothesis test to determine whether the scale is out of calibration. a. State the appropriate null and alternate hypotheses. H0 : �� = 1000, H1: �� ≠ 1000 b. The standard deviation of scale reading is known to be �� = 2. Compute the value of the test statistic. 2.12 c. State a conclusion. Use the �� = 0.05 level of significance. Reject H0. 55. Measuring lung function: One of the measurements used to determine the health of a person’s lungs is the amount of air a person can exhale under force in one second. This is called the forced expiratory volume in one second, and is abbreviated FEV1. Assume the mean FEV1 for 10-year-old boys is 2.1 liters and that the population standard deviation is �� = 0.3. A random sample of 100 10-year-old boys who live in a community with high levels of ozone pollution are found to have a sample mean FEV1 of 1.95 liters. Can you conclude that the mean FEV1 in the high-pollution community is less than 2.1 liters? Use the �� = 0.05 level of significance. Yes, reject H0. 56. Heavy children: Are children heavier now than they were in the past? The National Health Examination and Nutrition Survey (NHANES) published in 2004 reported that the mean weight of six-year-old girls in the United States was 49.3 pounds. Another NHANES survey, published in 2012, reported that a sample of 177 six-year-old girls had an average weight of 51.9 pounds. Assume the population standard deviation is �� = 17 pounds. Can you conclude that the mean weight of six-year-old girls is higher in 2012 than in 2004? Use the �� = 0.01 level of significance. No, do not reject H0. 57. House prices: Data from the National Association of Realtors indicates that the mean price of a home in Denver, Colorado, during April through June of 2012 was 260.7 thousand dollars. A random sample of 50 homes sold in 2013 had a mean price of 290.5 thousand dollars. a. Assume the population standard deviation is �� = 150. Can you conclude that the mean price in 2013 differs from the mean price in April through June of 2008? Use the �� = 0.05 level of significance. No b. Following is a boxplot of the data. Explain why it is not reasonable to assume that the population is approximately normally distributed. Outliers 0 200 400 600 800 c. Explain why the assumptions for the hypothesis test are satisfied even though the population is not normal. n > 30 58. SAT scores: The College Board reports that in 2012 the mean score on the math SAT was 516 and the population standard deviation was �� = 117. A random sample of 20 students who took the test in 2013 had a mean score of 521. Following is a dotplot of the 20 scores. 350 400 450 500 550 600 650 700 a. Are the assumptions for a hypothesis test satisfied? Explain. Yes
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