1. –/1.29 pointsBBUnderStat12 8.1.003. If we fail to reject ...

1. –/1.29 pointsBBUnderStat12 8.1.003. If we fail to reject (i.e., "accept") the null hypothesis, does this mean that we have proved it to be true beyond all doubt? Explain your answer. Yes, if we fail to reject the null we have found evidence that the null is true beyond all doubt. No, it suggests that the evidence is not sufficient to merit rejecting the null hypothesis. No, it suggests that the null hypothesis is true only some of the time. Yes, it suggests that the evidence is sufficient to merit rejecting the alternative hypothesis beyond all doubt. 2. –/1.29 pointsBBUnderStat12 8.1.004. If we reject the null hypothesis, does this mean that we have proved it to be false beyond all doubt? Explain your answer. No, the test was conducted with a risk of a type I error. Yes, if we reject the null that suggests that it is false beyond all doubt. Yes, the test was conducted with a risk of a type I error. No, the test was conducted with a risk of a type II error. 3. –/1.29 pointsBBUnderStat12 8.1.011.MI. Suppose you want to test the claim that a population mean equals 44. (a) State the null hypothesis. H0: μ > 44 H0: μ < 44 H0: μ = 44 H0: μ ≠ 44 H0: μ ≥ 44 (b) State the alternate hypothesis if you have no information regarding how the population mean might differ from 44. H1: μ > 44 H1: μ < 44 H1: μ = 44 H1: μ ≠ 44 H1: μ ≥ 44 (c) State the alternative hypothesis if you believe (based on experience or past studies) that the population mean may exceed 44. H1: μ > 44 H1: μ < 44 H1: μ = 44 H1: μ ≠ 44 H1: μ ≥ 44 (d) State the alternative hypothesis if you believe (based on experience or past studies) that the population mean may be less than 44. H1: μ > 44 H1: μ < 44 H1: μ = 44 H1: μ ≠ 44 H1: μ ≥ 44 4. –/1.29 pointsBBUnderStat12 8.1.015.MI. The body weight of a healthy 3-month-old colt should be about μ = 66 kg. (a) If you want to set up a statistical test to challenge the claim that μ = 66 kg, what would you use for the null hypothesis H0? μ > 66 kg μ < 66 kg μ = 66 kg μ ≠ 66 kg (b) In Nevada, there are many herds of wild horses. Suppose you want to test the claim that the average weight of a wild Nevada colt (3 months old) is less than 66 kg. What would you use for the alternate hypothesis H1? μ > 66 kg μ < 66 kg μ = 66 kg μ ≠ 66 kg (c) Suppose you want to test the claim that the average weight of such a wild colt is greater than 66 kg. What would you use for the alternate hypothesis? μ > 66 kg μ < 66 kg μ = 66 kg μ ≠ 66 kg (d) Suppose you want to test the claim that the average weight of such a wild colt is different from 66 kg. What would you use for the alternate hypothesis? μ > 66 kg μ < 66 kg μ = 66 kg μ ≠ 66 kg (e) For each of the tests in parts (b), (c), and (d), respectively, would the area corresponding to the P-value be on the left, on the right, or on both sides of the mean? both; left; right left; both; right right; left; both left; right; both 5. –/1.29 pointsBBUnderStat12 8.1.016. How much customers buy is a direct result of how much time they spend in the store. A study of average shopping times in a large national houseware store gave the following information (Source: Why We Buy: The Science of Shopping by P. Underhill). Women with female companion: 8.3 min. Women with male companion: 4.5 min. Suppose you want to set up a statistical test to challenge the claim that a woman with a female friend spends an average of 8.3 minutes shopping in such a store. (a) What would you use for the null and alternate hypotheses if you believe the average shopping time is less than 8.3 minutes? Ho: μ = 8.3; H1: μ ≠ 8.3 Ho: μ = 8.3; H1: μ < 8.3 Ho: μ = 8.3; H1: μ > 8.3 Ho: μ < 8.3; H1: μ = 8.3 Is this a right-tailed, left-tailed, or two-tailed test? two-tailed left-tailed right-tailed (b) What would you use for the null and alternate hypotheses if you believe the average shopping time is different from 8.3 minutes? Ho: μ = 8.3; H1: μ < 8.3 Ho: μ ≠ 8.3; H1: μ = 8.3 Ho: μ = 8.3; H1: μ > 8.3 Ho: μ = 8.3; H1: μ ≠ 8.3 Is this a right-tailed, left-tailed, or two-tailed test? two-tailed left-tailed right-tailed Stores that sell mainly to women should figure out a way to engage the interest of men! Perhaps comfortable seats and a big TV with sports programs. Suppose such an entertainment center was installed and you now wish to challenge the claim that a woman with a male friend spends only 4.5 minutes shopping in a houseware store. (c) What would you use for the null and alternate hypotheses if you believe the average shopping time is more than 4.5 minutes? Ho: μ = 4.5; H1: μ ≠ 4.5 Ho: μ = 4.5; H1: μ > 4.5 Ho: μ = 4.5; H1: μ < 4.5 Ho: μ > 4.5; H1: μ = 4.5 Is this a right-tailed, left-tailed, or two-tailed test? right-tailed left-tailed two-tailed (d) What would you use for the null and alternate hypotheses if you believe the average shopping time is different from 4.5 minutes? Ho: μ ≠ 4.5; H1: μ = 4.5 Ho: μ = 4.5; H1: μ < 4.5 Ho: μ = 4.5; H1: μ ≠ 4.5 Ho: μ = 4.5; H1: μ > 4.5 Is this a right-tailed, left-tailed, or two-tailed test? two-tailed left-tailed right-tailed 6. –/1.29 pointsBBUnderStat12 8.1.020. Gentle Ben is a Morgan horse at a Colorado dude ranch. Over the past 8 weeks, a veterinarian took the following glucose readings from this horse (in mg/100 ml). 94 89 83 107 97 112 83 91 The sample mean is x ≈ 94.5. Let x be a random variable representing glucose readings taken from Gentle Ben. We may assume that x has a normal distribution, and we know from past experience that σ = 12.5. The mean glucose level for horses should be μ = 85 mg/100 ml.† Do these data indicate that Gentle Ben has an overall average glucose level higher than 85? Use α = 0.05. (a) What is the level of significance? State the null and alternate hypotheses. Will you use a left-tailed, right-tailed, or two-tailed test? H0: μ = 85; H1: μ > 85; right-tailed H0: μ > 85; H1: μ = 85; right-tailed H0: μ = 85; H1: μ < 85; left-tailed H0: μ = 85; H1: μ ≠ 85; two-tailed (b) What sampling distribution will you use? Explain the rationale for your choice of sampling distribution. The standard normal, since we assume that x has a normal distribution with known σ. The standard normal, since we assume that x has a normal distribution with unknown σ. The Student's t, since n is large with unknown σ. The Student's t, since we assume that x has a normal distribution with known σ. Compute the z value of the sample test statistic. (Round your answer to two decimal places.) (c) Find (or estimate) the P-value. (Round your answer to four decimal places.) Sketch the sampling distribution and show the area corresponding to the P-value. (d) Based on your answers in parts (a) to (c), will you reject or fail to reject the null hypothesis? Are the data statistically significant at level α? At the α = 0.05 level, we reject the null hypothesis and conclude the data are statistically significant. At the α = 0.05 level, we reject the null hypothesis and conclude the data are not statistically significant. At the α = 0.05 level, we fail to reject the null hypothesis and conclude the data are statistically significant. At the α = 0.05 level, we fail to reject the null hypothesis and conclude the data are not statistically significant. (e) State your conclusion in the context of the application. There is sufficient evidence at the 0.05 level to conclude that Gentle Ben's glucose is higher than 85 mg/100 ml. There is insufficient evidence at the 0.05 level to conclude that Gentle Ben's glucose is higher than 85 mg/100 ml. 7. –/1.29 pointsBBUnderStat12 8.1.024. Total blood volume (in ml) per body weight (in kg) is important in medical research. For healthy adults, the red blood cell volume mean is about μ = 28 ml/kg.† Red blood cell volume that is too low or too high can indicate a medical problem. Suppose that Roger has had seven blood tests, and the red blood cell volumes were as follows. 31 23 43 36 29 39 31 The sample mean is x ≈ 33.1 ml/kg. Let x be a random variable that represents Roger's red blood cell volume. Assume that x has a normal distribution and σ = 4.75. Do the data indicate that Roger's red blood cell volume is different (either way) from μ = 28 ml/kg? Use a 0.01 level of significance. (a) What is the level of significance? State the null and alternate hypotheses. Will you use a left-tailed, right-tailed, or two-tailed test? H0: μ = 28 ml/kg; H1: μ < 28 ml/kg; left-tailed H0: μ = 28 ml/kg; H1: μ > 28 ml/kg; right-tailed H0: μ = 28 ml/kg; H1: μ ≠ 28 ml/kg; two-tailed H0: μ ≠ 28 ml/kg; H1: μ = 28 ml/kg; two-tailed (b) What sampling distribution will you use? Explain the rationale for your choice of sampling distribution. The standard normal, since we assume that x has a normal distribution with unknown σ. The Student's t, since n is large with unknown σ. The Student's t, since we assume that x has a normal distribution with known σ. The standard normal, since we assume that x has a normal distribution with known σ. Compute the z value of the sample test statistic. (Round your answer to two decimal places.) (c) Find (or estimate) the P-value. (Round your answer to four decimal places.) Sketch the sampling distribution and show the area corresponding to the P-value. (d) Based on your answers in parts (a) to (c), will you reject or fail to reject the null hypothesis? Are the data statistically significant at level α? At the α = 0.01 level, we reject the null hypothesis and conclude the data are statistically significant. At the α = 0.01 level, we reject the null hypothesis and conclude the data are not statistically significant. At the α = 0.01 level, we fail to reject the null hypothesis and conclude the data are statistically significant. At the α = 0.01 level, we fail to reject the null hypothesis and conclude the data are not statistically significant. (e) State your conclusion in the context of the application. There is sufficient evidence at the 0.01 level to conclude that Roger's average red cell volume differs from the average for healthy adults. There is insufficient evidence at the 0.01 level to conclude that Roger's average red cell volume differs from the average for healthy adults. 8. –/1.29 pointsBBUnderStat12 8.2.001. For the same sample data and null hypothesis, how does the P-value for a two-tailed test of μ compare to that for a one-tailed test? The P-value for a two-tailed test is three times the P-value for a one-tailed test. The P-value for a two-tailed test is half the P-value for a one-tailed test. The P-value for a two-tailed test is twice the P-value for a one-tailed test. The P-value for a two-tailed test is the same as the P-value for a one-tailed test. 9. –/1.29 pointsBBUnderStat12 8.2.002. To test μ for an x distribution that is mound-shaped using sample size n ≥ 30, how do you decide whether to use the normal or Student's t distribution? If σ is known, use the standard normal distribution. If σ is unknown, use the Student's t distribution with n degrees of freedom. If σ is known, use the standard normal distribution. If σ is unknown, use the Student's t distribution with n – 1 degrees of freedom. If σ is unknown, use the standard normal distribution. If σ is known, use the Student's t distribution with n – 1 degrees of freedom. For large samples we always the standard normal distribution. 10.–/1.29 pointsBBUnderStat12 8.2.009. A random sample of 36 values is drawn from a mound-shaped and symmetric distribution. The sample mean is 13 and the sample standard deviation is 2. Use a level of significance of 0.05 to conduct a two-tailed test of the claim that the population mean is 12.5. (a) Is it appropriate to use a Student's t distribution? Explain. Yes, because the x distribution is mound-shaped and symmetric and σ is unknown. No, the x distribution is skewed left. No, the x distribution is skewed right. No, the x distribution is not symmetric. No, σ is known. How many degrees of freedom do we use? (b) What are the hypotheses? H0: μ > 12.5; H1: μ = 12.5 H0: μ = 12.5; H1: μ ≠ 12.5 H0: μ < 12.5; H1: μ = 12.5 H0: μ = 12.5; H1: μ < 12.5 H0: μ = 12.5; H1: μ > 12.5 (c) Compute the t value of the sample test statistic. (Round your answer to three decimal places.) t= (d) Estimate the P-value for the test. P-value > 0.250 0.100 < P-value < 0.250 0.050 < P-value < 0.100 0.010 < P-value < 0.050 P-value < 0.010 (e) Do we reject or fail to reject H0? At the α = 0.05 level, we reject the null hypothesis and conclude the data are statistically significant. At the α = 0.05 level, we reject the null hypothesis and conclude the data are not statistically significant. At the α = 0.05 level, we fail to reject the null hypothesis and conclude the data are statistically significant. At the α = 0.05 level, we fail to reject the null hypothesis and conclude the data are not statistically significant. (f) Interpret the results. There is sufficient evidence at the 0.05 level to reject the null hypothesis. There is insufficient evidence at the 0.05 level to reject the null hypothesis. 11.–/1.29 pointsBBUnderStat12 8.2.011. Weatherwise is a magazine published by the American Meteorological Society. One issue gives a rating system used to classify Nor'easter storms that frequently hit New England and can cause much damage near the ocean. A severe storm has an average peak wave height of μ = 16.4 feet for waves hitting the shore. Suppose that a Nor'easter is in progress at the severe storm class rating. Peak wave heights are usually measured from land (using binoculars) off fixed cement piers. Suppose that a reading of 33 waves showed an average wave height of x = 17.3 feet. Previous studies of severe storms indicate that σ = 3.5 feet. Does this information suggest that the storm is (perhaps temporarily) increasing above the severe rating? Use α = 0.01. (a) What is the level of significance? State the null and alternate hypotheses. H0: μ > 16.4 ft; H1: μ = 16.4 ft H0: μ = 16.4 ft; H1: μ < 16.4 ft H0: μ = 16.4 ft; H1: μ > 16.4 ft H0: μ = 16.4 ft; H1: μ ≠ 16.4 ft H0: μ < 16.4 ft; H1: μ = 16.4 ft (b) What sampling distribution will you use? Explain the rationale for your choice of sampling distribution. The standard normal, since the sample size is large and σ is unknown. The standard normal, since the sample size is large and σ is known. The Student's t, since the sample size is large and σ is unknown. The Student's t, since the sample size is large and σ is known. What is the value of the sample test statistic? (Round your answer to two decimal places.) (c) Estimate the P-value. P-value > 0.250 0.100 < P-value < 0.250 0.050 < P-value < 0.100 0.010 < P-value < 0.050 P-value < 0.010 Sketch the sampling distribution and show the area corresponding to the P-value. (d) Based on your answers in parts (a) to (c), will you reject or fail to reject the null hypothesis? Are the data statistically significant at level α? At the α = 0.01 level, we reject the null hypothesis and conclude the data are statistically significant. At the α = 0.01 level, we reject the null hypothesis and conclude the data are not statistically significant. At the α = 0.01 level, we fail to reject the null hypothesis and conclude the data are statistically significant. At the α = 0.01 level, we fail to reject the null hypothesis and conclude the data are not statistically significant. (e) Interpret your conclusion in the context of the application. There is sufficient evidence at the 0.01 level to conclude that the storm is increasing above the severe rating. There is insufficient evidence at the 0.01 level to conclude that the storm is increasing above the severe rating. 12.–/1.29 pointsBBUnderStat12 8.2.014. Pyramid Lake is on the Paiute Indian Reservation in Nevada. The lake is famous for cutthroat trout. Suppose a friend tells you that the average length of trout caught in Pyramid Lake is μ = 19 inches. However, a survey reported that of a random sample of 51 fish caught, the mean length was x = 18.6 inches, with estimated standard deviation s = 3.0 inches. Do these data indicate that the average length of a trout caught in Pyramid Lake is less than μ = 19 inches? Use α = 0.05. (a) What is the level of significance? State the null and alternate hypotheses. H0: μ = 19 in; H1: μ > 19 in H0: μ = 19 in; H1: μ ≠ 19 in H0: μ = 19 in; H1: μ < 19 in H0: μ > 19 in; H1: μ = 19 in H0: μ < 19 in; H1: μ = 19 in (b) What sampling distribution will you use? Explain the rationale for your choice of sampling distribution. The standard normal, since the sample size is large and σ is unknown. The standard normal, since the sample size is large and σ is known. The Student's t, since the sample size is large and σ is known. The Student's t, since the sample size is large and σ is unknown. What is the value of the sample test statistic? (Round your answer to three decimal places.) (c) Estimate the P-value. P-value > 0.250 0.100 < P-value < 0.250 0.050 < P-value < 0.100 0.010 < P-value < 0.050 P-value < 0.010 Sketch the sampling distribution and show the area corresponding to the P-value. (d) Based on your answers in parts (a) to (c), will you reject or fail to reject the null hypothesis? Are the data statistically significant at level α? At the α = 0.05 level, we reject the null hypothesis and conclude the data are statistically significant. At the α = 0.05 level, we reject the null hypothesis and conclude the data are not statistically significant. At the α = 0.05 level, we fail to reject the null hypothesis and conclude the data are statistically significant. At the α = 0.05 level, we fail to reject the null hypothesis and conclude the data are not statistically significant. (e) Interpret your conclusion in the context of the application. There is sufficient evidence at the 0.05 level to conclude that the average fish length is less than 19 inches. There is insufficient evidence at the 0.05 level to conclude that the average fish length is less than 19 inches. 13.–/1.29 pointsBBUnderStat12 8.2.015. Socially conscious investors screen out stocks of alcohol and tobacco makers, firms with poor environmental records, and companies with poor labor practices. Some examples of "good," socially conscious companies are Johnson and Johnson, Dell Computers, Bank of America, and Home Depot. The question is, are such stocks overpriced? One measure of value is the P/E, or price-to-earnings ratio. High P/E ratios may indicate a stock is overpriced. For the S&P Stock Index of all major stocks, the mean P/E ratio is μ = 19.4. A random sample of 36 "socially conscious" stocks gave a P/E ratio sample mean of x = 17.9, with sample standard deviation s = 5.6. Does this indicate that the mean P/E ratio of all socially conscious stocks is different (either way) from the mean P/E ratio of the S&P Stock Index? Use α = 0.05. (a) What is the level of significance? State the null and alternate hypotheses. H0: μ = 19.4; H1: μ > 19.4 H0: μ = 19.4; H1: μ < 19.4 H0: μ ≠ 19.4; H1: μ = 19.4 H0: μ = 19.4; H1: μ ≠ 19.4 H0: μ > 19.4; H1: μ = 19.4 (b) What sampling distribution will you use? Explain the rationale for your choice of sampling distribution. The Student's t, since the sample size is large and σ is known. The standard normal, since the sample size is large and σ is unknown. The Student's t, since the sample size is large and σ is unknown. The standard normal, since the sample size is large and σ is known. What is the value of the sample test statistic? (Round your answer to three decimal places.) (c) Estimate the P-value. P-value > 0.250 0.100 < P-value < 0.250 0.050 < P-value < 0.100 0.010 < P-value < 0.050 P-value < 0.010 Sketch the sampling distribution and show the area corresponding to the P-value. (d) Based on your answers in parts (a) to (c), will you reject or fail to reject the null hypothesis? Are the data statistically significant at level α? At the α = 0.05 level, we reject the null hypothesis and conclude the data are statistically significant. At the α = 0.05 level, we reject the null hypothesis and conclude the data are not statistically significant. At the α = 0.05 level, we fail to reject the null hypothesis and conclude the data are statistically significant. At the α = 0.05 level, we fail to reject the null hypothesis and conclude the data are not statistically significant. (e) Interpret your conclusion in the context of the application. There is sufficient evidence at the 0.05 level to conclude that the mean P/E ratio of all socially conscious stocks differs from the mean P/E ratio of the S&P Stock Index. There is insufficient evidence at the 0.05 level to conclude that the mean P/E ratio of all socially conscious stocks differs from the mean P/E ratio of the S&P Stock Index. 14.–/1.29 pointsBBUnderStat12 8.2.016. Unfortunately, arsenic occurs naturally in some ground water†. A mean arsenic level of μ = 8.0 parts per billion (ppb) is considered safe for agricultural use. A well in Texas is used to water cotton crops. This well is tested on a regular basis for arsenic. A random sample of 36 tests gave a sample mean of x = 7.1 ppb arsenic, with s = 2.1 ppb. Does this information indicate that the mean level of arsenic in this well is less than 8 ppb? Use α = 0.01. (a) What is the level of significance? State the null and alternate hypotheses. H0: μ < 8 ppb; H1: μ = 8 ppb H0: μ = 8 ppb; H1: μ ≠ 8 ppb H0: μ > 8 ppb; H1: μ = 8 ppb H0: μ = 8 ppb; H1: μ < 8 ppb H0: μ = 8 ppb; H1: μ > 8 ppb (b) What sampling distribution will you use? Explain the rationale for your choice of sampling distribution. The Student's t, since the sample size is large and σ is unknown. The Student's t, since the sample size is large and σ is known. The standard normal, since the sample size is large and σ is known. The standard normal, since the sample size is large and σ is unknown. What is the value of the sample test statistic? (Round your answer to three decimal places.) (c) Estimate the P-value. P-value > 0.250 0.100 < P-value < 0.250 0.050 < P-value < 0.100 0.010 < P-value < 0.050 P-value < 0.010 Sketch the sampling distribution and show the area corresponding to the P-value. (d) Based on your answers in parts (a) to (c), will you reject or fail to reject the null hypothesis? Are the data statistically significant at level α? At the α = 0.01 level, we reject the null hypothesis and conclude the data are statistically significant. At the α = 0.01 level, we reject the null hypothesis and conclude the data are not statistically significant. At the α = 0.01 level, we fail to reject the null hypothesis and conclude the data are statistically significant. At the α = 0.01 level, we fail to reject the null hypothesis and conclude the data are not statistically significant. (e) Interpret your conclusion in the context of the application. There is sufficient evidence at the 0.01 level to conclude that the mean level of arsenic in the well is less than 8 ppb. There is insufficient evidence at the 0.01 level to conclude that the mean level of arsenic in the well is less than 8 ppb. 15.–/1.29 pointsBBUnderStat12 8.2.017. Let x be a random variable that represents red blood cell count (RBC) in millions of cells per cubic millimeter of whole blood. Then x has a distribution that is approximately normal. For the population of healthy female adults, suppose the mean of the x distribution is about 4.68. Suppose that a female patient has taken six laboratory blood tests over the past several months and that the RBC count data sent to the patient's doctor are as follows. 4.9 4.2 4.5 4.1 4.4 4.3 (i) Use a calculator with sample mean and standard deviation keys to find x and s. (Round your answers to two decimal places.) x= s= (ii) Do the given data indicate that the population mean RBC count for this patient is lower than 4.68? Use α = 0.05. (a) What is the level of significance? State the null and alternate hypotheses. H0: μ = 4.68; H1: μ > 4.68 H0: μ = 4.68; H1: μ < 4.68 H0: μ > 4.68; H1: μ = 4.68 H0: μ < 4.68; H1: μ = 4.68 H0: μ = 4.68; H1: μ ≠ 4.68 (b) What sampling distribution will you use? Explain the rationale for your choice of sampling distribution. The standard normal, since we assume that x has a normal distribution and σ is known. The Student's t, since we assume that x has a normal distribution and σ is unknown. The standard normal, since we assume that x has a normal distribution and σ is unknown. The Student's t, since we assume that x has a normal distribution and σ is known. What is the value of the sample test statistic? (Round your answer to three decimal places.) (c) Estimate the P-value. P-value > 0.250 0.100 < P-value < 0.250 0.050 < P-value < 0.100 0.010 < P-value < 0.050 P-value < 0.010 Sketch the sampling distribution and show the area corresponding to the P-value. (d) Based on your answers in parts (a) to (c), will you reject or fail to reject the null hypothesis? Are the data statistically significant at level α? At the α = 0.05 level, we reject the null hypothesis and conclude the data are statistically significant. At the α = 0.05 level, we reject the null hypothesis and conclude the data are not statistically significant. At the α = 0.05 level, we fail to reject the null hypothesis and conclude the data are statistically significant. At the α = 0.05 level, we fail to reject the null hypothesis and conclude the data are not statistically significant. (e) Interpret your conclusion in the context of the application. There is sufficient evidence at the 0.05 level to conclude that the population mean RBC count for the patient is lower than 4.68. There is insufficient evidence at the 0.05 level to conclude that the population mean RBC count for the patient is lower than 4.68. 16.–/1.29 pointsBBUnderStat12 8.2.018. Let x be a random variable that represents hemoglobin count (HC) in grams per 100 milliliters of whole blood. Then x has a distribution that is approximately normal, with population mean of about 14 for healthy adult women. Suppose that a female patient has taken 10 laboratory blood tests during the past year. The HC data sent to the patient's doctor are as follows. 16 17 16 19 13 12 15 16 16 11 (i) Use a calculator with sample mean and standard deviation keys to find x and s. (Round your answers to two decimal places.) x= s= (ii) Does this information indicate that the population average HC for this patient is higher than 14? Use α = 0.01. (a) What is the level of significance? State the null and alternate hypotheses. H0: μ = 14; H1: μ < 14 H0: μ < 14; H1: μ = 14 H0: μ = 14; H1: μ > 14 H0: μ > 14; H1: μ = 14 H0: μ = 14; H1: μ ≠ 14 (b) What sampling distribution will you use? Explain the rationale for your choice of sampling distribution. The standard normal, since we assume that x has a normal distribution and σ is known. The standard normal, since we assume that x has a normal distribution and σ is unknown. The Student's t, since we assume that x has a normal distribution and σ is known. The Student's t, since we assume that x has a normal distribution and σ is unknown. What is the value of the sample test statistic? (Round your answer to three decimal places.) (c) Estimate the P-value. P-value > 0.250 0.100 < P-value < 0.250 0.050 < P-value < 0.100 0.010 < P-value < 0.050 P-value < 0.010 Sketch the sampling distribution and show the area corresponding to the P-value. (d) Based on your answers in parts (a) to (c), will you reject or fail to reject the null hypothesis? Are the data statistically significant at level α? At the α = 0.01 level, we reject the null hypothesis and conclude the data are statistically significant. At the α = 0.01 level, we reject the null hypothesis and conclude the data are not statistically significant. At the α = 0.01 level, we fail to reject the null hypothesis and conclude the data are statistically significant. At the α = 0.01 level, we fail to reject the null hypothesis and conclude the data are not statistically significant. (e) Interpret your conclusion in the context of the application. There is sufficient evidence at the 0.01 level to conclude that the population average HC for this patient is higher than 14. There is insufficient evidence at the 0.01 level to conclude that the population average HC for this patient is higher than 14. 17.–/1.29 pointsBBUnderStat12 8.2.020. A national newspaper reported that the state with the longest mean life span is Hawaii, where the population mean life span is 78 years. A random sample of 20 obituary notices in the Honolulu Advertizer gave the following information about life span (in years) of Honolulu residents. 72 68 81 93 56 19 78 94 83 84 77 69 85 97 75 71 86 47 66 27 (i) Use a calculator with sample mean and standard deviation keys to find x and s. (Round your answers to two decimal places.) x= yr s= yr (ii) Assuming that life span in Honolulu is approximately normally distributed, does this information indicate that the population mean life span for Honolulu residents is less than 78 years? Use a 5% level of significance. (a) What is the level of significance? State the null and alternate hypotheses. H0: μ = 78 yr; H1: μ > 78 yr H0: μ = 78 yr; H1: μ ≠ 78 yr H0: μ = 78 yr; H1: μ < 78 yr H0: μ > 78 yr; H1: μ = 78 yr H0: μ < 78 yr; H1: μ = 78 yr (b) What sampling distribution will you use? Explain the rationale for your choice of sampling distribution. The standard normal, since we assume that x has a normal distribution and σ is unknown. The Student's t, since we assume that x has a normal distribution and σ is known. The standard normal, since we assume that x has a normal distribution and σ is known. The Student's t, since we assume that x has a normal distribution and σ is unknown. What is the value of the sample test statistic? (Round your answer to three decimal places.) (c) Estimate the P-value. P-value > 0.250 0.100 < P-value < 0.250 0.050 < P-value < 0.100 0.010 < P-value < 0.050 P-value < 0.010 Sketch the sampling distribution and show the area corresponding to the P-value. (d) Based on your answers in parts (a) to (c), will you reject or fail to reject the null hypothesis? Are the data statistically significant at level α? At the α = 0.05 level, we reject the null hypothesis and conclude the data are statistically significant. At the α = 0.05 level, we reject the null hypothesis and conclude the data are not statistically significant. At the α = 0.05 level, we fail to reject the null hypothesis and conclude the data are statistically significant. At the α = 0.05 level, we fail to reject the null hypothesis and conclude the data are not statistically significant. (e) Interpret your conclusion in the context of the application. There is sufficient evidence at the 0.05 level to conclude that the population mean life span of Honolulu residents is less than 78 years. There is insufficient evidence at the 0.05 level to conclude that the population mean life span of Honolulu residents is less than 78 years. 18.–/1.29 pointsBBUnderStat12 8.3.001. To use the normal distribution to test a proportion p, the conditions np > 5 and nq > 5 must be satisfied. Does the value of p come from H0, or is it estimated by using ̂p from the sample? Neither, the value of p is guessed by the person in charge of the study. The value of p is estimated using ̂p from the sample. The value of p comes from both H0 and ̂p. The value of p comes from H0. 19.–/1.29 pointsBBUnderStat12 8.3.005.MI. A random sample of 40 binomial trials resulted in 16 successes. Test the claim that the population proportion of successes does not equal 0.50. Use a level of significance of 0.05. (a) Can a normal distribution be used for the ̂p distribution? Explain. No, np is greater than 5, but nq is less than 5. No, nq is greater than 5, but np is less than 5. Yes, np and nq are both greater than 5. Yes, np and nq are both less than 5. No, np and nq are both less than 5. (b) State the hypotheses. H0: p = 0.5; H1: p < 0.5 H0: p = 0.5; H1: p > 0.5 H0: p < 0.5; H1: p = 0.5 H0: p = 0.5; H1: p ≠ 0.5 (c) Compute ̂p. Compute the corresponding standardized sample test statistic. (Round your answer to two decimal places.) (d) Find the P-value of the test statistic. (Round your answer to four decimal places.) (e) Do you reject or fail to reject H0? Explain. At the α = 0.05 level, we reject the null hypothesis and conclude the data are statistically significant. At the α = 0.05 level, we reject the null hypothesis and conclude the data are not statistically significant. At the α = 0.05 level, we fail to reject the null hypothesis and conclude the data are statistically significant. At the α = 0.05 level, we fail to reject the null hypothesis and conclude the data are not statistically significant. (f) What do the results tell you? The sample ̂p value based on 40 trials is not sufficiently different from 0.50 to justify rejecting H0 for α = 0.05. The sample ̂p value based on 40 trials is sufficiently different from 0.50 to not reject H0 for α = 0.05. The sample ̂p value based on 40 trials is sufficiently different from 0.50 to justify rejecting H0 for α = 0.05. The sample ̂p value based on 40 trials is not sufficiently different from 0.50 to not reject H0 for α = 0.05. 20.–/1.29 pointsBBUnderStat12 8.3.008. Recall that Benford's Law claims that numbers chosen from very large data files tend to have "1" as the first nonzero digit disproportionately often. In fact, research has shown that if you randomly draw a number from a very large data file, the probability of getting a number with "1" as the leading digit is about 0.301. Now suppose you are the auditor for a very large corporation. The revenue file contains millions of numbers in a large computer data bank. You draw a random sample of n = 224 numbers from this file and r = 86 have a first nonzero digit of 1. Let p represent the population proportion of all numbers in the computer file that have a leading digit of 1. (i) Test the claim that p is more than 0.301. Use α = 0.10. (a) What is the level of significance? State the null and alternate hypotheses. H0: p = 0.301; H1: p < 0.301 H0: p = 0.301; H1: p ≠ 0.301 H0: p > 0.301; H1: p = 0.301 H0: p = 0.301; H1: p > 0.301 (b) What sampling distribution will you use? The standard normal, since np > 5 and nq > 5. The standard normal, since np < 5 and nq < 5. The Student's t, since np < 5 and nq < 5. The Student's t, since np > 5 and nq > 5. What is the value of the sample test statistic? (Round your answer to two decimal places.) (c) Find the P-value of the test statistic. (Round your answer to four decimal places.) Sketch the sampling distribution and show the area corresponding to the P-value. (d) Based on your answers in parts (a) to (c), will you reject or fail to reject the null hypothesis? Are the data statistically significant at level α? At the α = 0.10 level, we reject the null hypothesis and conclude the data are statistically significant. At the α = 0.10 level, we reject the null hypothesis and conclude the data are not statistically significant. At the α = 0.10 level, we fail to reject the null hypothesis and conclude the data are statistically significant. At the α = 0.10 level, we fail to reject the null hypothesis and conclude the data are not statistically significant. (e) Interpret your conclusion in the context of the application. There is sufficient evidence at the 0.10 level to conclude that the true proportion of numbers with a leading 1 in the revenue file is greater than 0.301. There is insufficient evidence at the 0.10 level to conclude that the true proportion of numbers with a leading 1 in the revenue file is greater than 0.301. (ii) If p is in fact larger than 0.301, it would seem there are too many numbers in the file with leading 1's. Could this indicate that the books have been "cooked" by artificially lowering numbers in the file? Comment from the point of view of the Internal Revenue Service. Comment from the perspective of the Federal Bureau of Investigation as it looks for "profit skimming" by unscrupulous employees. No. There does not seem to be too many entries with a leading digit 1. Yes. There seems to be too many entries with a leading digit 1. No. There seems to be too many entries with a leading digit 1. Yes. There does not seem to be too many entries with a leading digit 1. (iii) Comment on the following statement: If we reject the null hypothesis at level of significance α , we have not proved H0 to be false. We can say that the probability is α that we made a mistake in rejecting Ho. Based on the outcome of the test, would you recommend further investigation before accusing the company of fraud? We have not proved H0 to be false. Because our data lead us to accept the null hypothesis, more investigation is not merited. We have not proved H0 to be false. Because our data lead us to reject the null hypothesis, more investigation is merited. We have proved H0 to be false. Because our data lead us to reject the null hypothesis, more investigation is not merited. We have not proved H0 to be false. Because our data lead us to reject the null hypothesis, more investigation is not merited. 21.–/1.29 pointsBBUnderStat12 8.3.009. Is the national crime rate really going down? Some sociologists say yes! They say that the reason for the decline in crime rates in the 1980s and 1990s is demographics. It seems that the population is aging, and older people commit fewer crimes. According to the FBI and the Justice Department, 70% of all arrests are of males aged 15 to 34 years†. Suppose you are a sociologist in Rock Springs, Wyoming, and a random sample of police files showed that of 35 arrests last month, 27 were of males aged 15 to 34 years. Use a 1% level of significance to test the claim that the population proportion of such arrests in Rock Springs is different from 70%. (a) What is the level of significance? State the null and alternate hypotheses. H0: p = 0 .7; H1: p < 0.7 H0: p ≠ 0.7; H1: p = 0.7 H0: p < 0 .7; H1: p = 0.7 H0: p = 0.7; H1: p > 0.7 H0: p = 0.7; H1: p ≠ 0.7 (b) What sampling distribution will you use? The standard normal, since np < 5 and nq < 5. The standard normal, since np > 5 and nq > 5. The Student's t, since np < 5 and nq < 5. The Student's t, since np > 5 and nq > 5. What is the value of the sample test statistic? (Round your answer to two decimal places.) (c) Find the P-value of the test statistic. (Round your answer to four decimal places.) Sketch the sampling distribution and show the area corresponding to the P-value. (d) Based on your answers in parts (a) to (c), will you reject or fail to reject the null hypothesis? Are the data statistically significant at level α? At the α = 0.01 level, we reject the null hypothesis and conclude the data are statistically significant. At the α = 0.01 level, we reject the null hypothesis and conclude the data are not statistically significant. At the α = 0.01 level, we fail to reject the null hypothesis and conclude the data are statistically significant. At the α = 0.01 level, we fail to reject the null hypothesis and conclude the data are not statistically significant. (e) Interpret your conclusion in the context of the application. There is sufficient evidence at the 0.01 level to conclude that the true proportion of arrests of males aged 15 to 34 in Rock Springs differs from 70%. There is insufficient evidence at the 0.01 level to conclude that the true proportion of arrests of males aged 15 to 34 in Rock Springs differs from 70%. 22.–/1.29 pointsBBUnderStat12 8.3.012. What is your favorite color? A large survey of countries, including the United States, China, Russia, France, Turkey, Kenya, and others, indicated that most people prefer the color blue. In fact, about 24% of the population claim blue as their favorite color.† Suppose a random sample of n = 57 college students were surveyed and r = 8 of them said that blue is their favorite color. Does this information imply that the color preference of all college students is different (either way) from that of the general population? Use α = 0.05. (a) What is the level of significance? State the null and alternate hypotheses. H0: p = 0.24; H1: p < 0.24 H0: p = 0.24; H1: p ≠ 0.24 H0: p ≠ 0.24; H1: p = 0.24 H0: p = 0.24; H1: p > 0.24 (b) What sampling distribution will you use? The Student's t, since np > 5 and nq > 5. The standard normal, since np < 5 and nq < 5. The standard normal, since np > 5 and nq > 5. The Student's t, since np < 5 and nq < 5. What is the value of the sample test statistic? (Round your answer to two decimal places.) (c) Find the P-value of the test statistic. (Round your answer to four decimal places.) Sketch the sampling distribution and show the area corresponding to the P-value. (d) Based on your answers in parts (a) to (c), will you reject or fail to reject the null hypothesis? Are the data statistically significant at level α? At the α = 0.05 level, we reject the null hypothesis and conclude the data are statistically significant. At the α = 0.05 level, we reject the null hypothesis and conclude the data are not statistically significant. At the α = 0.05 level, we fail to reject the null hypothesis and conclude the data are statistically significant. At the α = 0.05 level, we fail to reject the null hypothesis and conclude the data are not statistically significant. (e) Interpret your conclusion in the context of the application. There is sufficient evidence at the 0.05 level to conclude that the true proportion of college students favoring the color blue differs from 0.24. There is insufficient evidence at the 0.05 level to conclude that the true proportion of college students favoring the color blue differs from 0.24. 23.–/1.29 pointsBBUnderStat12 8.3.013. The following is based on information from The Wolf in the Southwest: The Making of an Endangered Species, by David E. Brown (University of Arizona Press). Before 1918, the proportion of female wolves in the general population of all southwestern wolves was about 50%. However, after 1918, southwestern cattle ranchers began a widespread effort to destroy wolves. In a recent sample of 37 wolves, there were only 11 females. One theory is that male wolves tend to return sooner than females to their old territories, where their predecessors were exterminated. Do these data indicate that the population proportion of female wolves is now less than 50% in the region? Use α = 0.01. (a) What is the level of significance? State the null and alternate hypotheses. H0: p < 0.5; H1: p = 0.5 H0: p = 0.5; H1: p > 0.5 H0: p = 0.5; H1: p < 0.5 H0: p = 0.5; H1: p ≠ 0.5 (b) What sampling distribution will you use? The standard normal, since np > 5 and nq > 5. The Student's t, since np > 5 and nq > 5. The standard normal, since np < 5 and nq < 5. The Student's t, since np < 5 and nq < 5. What is the value of the sample test statistic? (Round your answer to two decimal places.) (c) Find the P-value of the test statistic. (Round your answer to four decimal places.) Sketch the sampling distribution and show the area corresponding to the P-value. (d) Based on your answers in parts (a) to (c), will you reject or fail to reject the null hypothesis? Are the data statistically significant at level α? At the α = 0.01 level, we reject the null hypothesis and conclude the data are statistically significant. At the α = 0.01 level, we reject the null hypothesis and conclude the data are not statistically significant. At the α = 0.01 level, we fail to reject the null hypothesis and conclude the data are statistically significant. At the α = 0.01 level, we fail to reject the null hypothesis and conclude the data are not statistically significant. (e) Interpret your conclusion in the context of the application. There is sufficient evidence at the 1% level to conclude that the true proportion of female wolves in the region is less than 0.5. There is insufficient evidence at the 1% level to conclude that the true proportion of female wolves in the region is less than 0.5. 24.–/1.29 pointsBBUnderStat12 8.3.014. In a fishing lodge brochure, the lodge advertises that 75% of its guests catch northern pike over 20 pounds. Suppose that last summer 65 out of a random sample of 87 guests did, in fact, catch northern pike weighing over 20 pounds. Does this indicate that the population proportion of guests who catch pike over 20 pounds is different from 75% (either higher or lower)? Use α = 0.05. (a) What is the level of significance? State the null and alternate hypotheses. H0: p < 0.75; H1: p = 0.75 H0: p ≠ 0.75; H1: p = 0.75 H0: p = 0.75; H1: p ≠ 0.75 H0: p = 0.75; H1: p > 0.75 H0: p = 0.75; H1: p < 0.75 (b) What sampling distribution will you use? The Student's t, since np < 5 and nq < 5. The standard normal, since np < 5 and nq < 5. The standard normal, since np > 5 and nq > 5. The Student's t, since np > 5 and nq > 5. What is the value of the sample test statistic? (Round your answer to two decimal places.) (c) Find the P-value of the test statistic. (Round your answer to four decimal places.) Sketch the sampling distribution and show the area corresponding to the P-value. (d) Based on your answers in parts (a) to (c), will you reject or fail to reject the null hypothesis? Are the data statistically significant at level α? At the α = 0.05 level, we reject the null hypothesis and conclude the data are statistically significant. At the α = 0.05 level, we reject the null hypothesis and conclude the data are not statistically significant. At the α = 0.05 level, we fail to reject the null hypothesis and conclude the data are statistically significant. At the α = 0.05 level, we fail to reject the null hypothesis and conclude the data are not statistically significant. (e) Interpret your conclusion in the context of the application. There is sufficient evidence at the 0.05 level to conclude that the true proportion of guests who catch pike over 20 pounds differs from 75%. There is insufficient evidence at the 0.05 level to conclude that the true proportion of guests who catch pike over 20 pounds differs from 75%. 25.–/1.29 pointsBBUnderStat12 8.3.015. Prose rhythm is characterized as the occurrence of five-syllable sequences in long passages of text. This characterization may be used to assess the similarity among passages of text and sometimes the identity of authors. The following information is based on an article by D. Wishart and S. V. Leach appearing in Computer Studies of the Humanities and Verbal Behavior (Vol. 3, pp. 90-99). Syllables were categorized as long or short. On analyzing Plato's Republic, Wishart and Leach found that about 26.1% of the five-syllable sequences are of the type in which two are short and three are long. Suppose that Greek archaeologists have found an ancient manuscript dating back to Plato's time (about 427 - 347 B.C.). A random sample of 317 five-syllable sequences from the newly discovered manuscript showed that 61 are of the type two short and three long. Do the data indicate that the population proportion of this type of five syllable sequence is different (either way) from the text of Plato's Republic? Use α = 0.01. (a) What is the level of significance? State the null and alternate hypotheses. H0: p ≠ 0.261; H1: p = 0.261 H0: p = 0.261; H1: p < 0.261 H0: p = 0.261; H1: p ≠ 0.261 H0: p = 0.261; H1: p > 0.261 (b) What sampling distribution will you use? The Student's t, since np > 5 and nq > 5. The standard normal, since np > 5 and nq > 5. The standard normal, since np < 5 and nq < 5. The Student's t, since np < 5 and nq < 5. What is the value of the sample test statistic? (Round your answer to two decimal places.) (c) Find the P-value of the test statistic. (Round your answer to four decimal places.) Sketch the sampling distribution and show the area corresponding to the P-value. (d) Based on your answers in parts (a) to (c), will you reject or fail to reject the null hypothesis? Are the data statistically significant at level α? At the α = 0.01 level, we reject the null hypothesis and conclude the data are statistically significant. At the α = 0.01 level, we reject the null hypothesis and conclude the data are not statistically significant. At the α = 0.01 level, we fail to reject the null hypothesis and conclude the data are statistically significant. At the α = 0.01 level, we fail to reject the null hypothesis and conclude the data are not statistically significant. (e) Interpret your conclusion in the context of the application. There is sufficient evidence at the 0.01 level to conclude that the true proportion of the five-syllable sequence differs from that of the text of Plato's Republic. There is insufficient evidence at the 0.01 level to conclude that the true proportion of the five-syllable sequence differs from that of the text of Plato's Republic. 26.–/1.29 pointsBBUnderStat12 8.3.017. USA Today reported that about 47% of the general consumer population in the United States is loyal to the automobile manufacturer of their choice. Suppose Chevrolet did a study of a random sample of 1007 Chevrolet owners and found that 488 said they would buy another Chevrolet. Does this indicate that the population proportion of consumers loyal to Chevrolet is more than 47%? Use α = 0.01. (a) What is the level of significance? State the null and alternate hypotheses. H0: p = 0.47; H1: p > 0.47 H0: p > 0.47; H1: p = 0.47 H0: p = 0.47; H1: p ≠ 0.47 H0: p = 0.47; H1: p < 0.47 (b) What sampling distribution will you use? The standard normal, since np > 5 and nq > 5. The standard normal, since np < 5 and nq < 5. The Student's t, since np < 5 and nq < 5. The Student's t, since np > 5 and nq > 5. What is the value of the sample test statistic? (Round your answer to two decimal places.) (c) Find the P-value of the test statistic. (Round your answer to four decimal places.) Sketch the sampling distribution and show the area corresponding to the P-value. (d) Based on your answers in parts (a) to (c), will you reject or fail to reject the null hypothesis? Are the data statistically significant at level α? At the α = 0.01 level, we reject the null hypothesis and conclude the data are statistically significant. At the α = 0.01 level, we reject the null hypothesis and conclude the data are not statistically significant. At the α = 0.01 level, we fail to reject the null hypothesis and conclude the data are statistically significant. At the α = 0.01 level, we fail to reject the null hypothesis and conclude the data are not statistically significant. (e) Interpret your conclusion in the context of the application. There is sufficient evidence at the 0.01 level to conclude that the true proportion of customers loyal to Chevrolet is more than 0.47. There is insufficient evidence at the 0.01 level to conclude that the true proportion of customers loyal to Chevrolet is more than 0.47. 27.–/1.46 pointsBBUnderStat12 8.3.019. This problem is based on information taken from The Merck Manual (a reference manual used in most medical and nursing schools). Hypertension is defined as a blood pressure reading over 140 mm Hg systolic and/or over 90 mm Hg diastolic. Hypertension, if not corrected, can cause long-term health problems. In the college-age population (18-24 years), about 9.2% have hypertension. Suppose that a blood donor program is taking place in a college dormitory this week (final exams week). Before each student gives blood, the nurse takes a blood pressure reading. Of 191 donors, it is found that 27 have hypertension. Do these data indicate that the population proportion of students with hypertension during final exams week is higher than 9.2%? Use a 5% level of significance. (a) What is the level of significance? State the null and alternate hypotheses. Will you use a left-tailed, right-tailed, or two-tailed test? H0: p > 0.092; H1: p = 0.092; right-tailed H0: p = 0.092; H1: p < 0.092; left-tailed H0: p = 0.092; H1: p > 0.092; right-tailed H0: p = 0.092; H1: p ≠ 0.092; two-tailed (b) What sampling distribution will you use? Do you think the sample size is sufficiently large? The standard normal, since np < 5 and nq < 5. The standard normal, since np > 5 and nq > 5. The Student's t, since np < 5 and nq < 5. The Student's t, since np > 5 and nq > 5. What is the value of the sample test statistic? (Round your answer to two decimal places.) (c) Find the P-value of the test statistic. (Round your answer to four decimal places.) Sketch the sampling distribution and show the area corresponding to the P-value. (d) Based on your answers in parts (a) to (c), will you reject or fail to reject the null hypothesis? Are the data statistically significant at level α? At the α = 0.05 level, we reject the null hypothesis and conclude the data are statistically significant. At the α = 0.05 level, we reject the null hypothesis and conclude the data are not statistically significant. At the α = 0.05 level, we fail to reject the null hypothesis and conclude the data are statistically significant. At the α = 0.05 level, we fail to reject the null hypothesis and conclude the data are not statistically significant. (e) State your conclusion in the context of the application. There is sufficient evidence at the 0.05 level to conclude that the true proportion of students with hypertension during final exams week is higher than 0.092. There is insufficient evidence at the 0.05 level to conclude that the true proportion of students with hypertension during final exams week is higher than 0.092.