6. 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). 93 88 80 107 98 109 86 90 The sample mean is x 93.9. 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 = 12.5. The mean glucose level for horses should be = 85 mg/100 ml. Do these data indicate that Gentle Ben has an overall average glucose level higher than 85? Use = 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: > 85; H1: = 85; right-tailed H0: = 85; H1: < 85; left-tailed H0: = 85; H1: > 85; right-tailed H0: = 85; H1: 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 unknown . The Student's t, since n is large with unknown . The standard normal, since we assume that x has a normal distribution with known . The Student's t, since we assume that x has a normal distribution with known . Compute the z value of the sample test statistic. (Round your answer to two decimal places.) (d) Based on your answers will you reject or fail to reject the null hypothesis? Are the data statistically significant at level ? At the = 0.05 level, we reject the null hypothesis and conclude the data are statistically significant. At the = 0.05 level, we reject the null hypothesis and conclude the data are not statistically significant. At the = 0.05 level, we fail to reject the null hypothesis and conclude the data are statistically significant. At the = 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. 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 = 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 27 41 35 30 36 29 The sample mean is x 32.7 ml/kg. Let x be a random variable that represents Roger's red blood cell volume. Assume that x has a normal distribution and = 4.75. Do the data indicate that Roger's red blood cell volume is different (either way) from = 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: = 28 ml/kg; H1: > 28 ml/kg; right-tailed H0: = 28 ml/kg; H1: 28 ml/kg; two-tailed H0: 28 ml/kg; H1: = 28 ml/kg; two-tailed H0: = 28 ml/kg; H1: < 28 ml/kg; left-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 . The standard normal, since we assume that x has a normal distribution with unknown . The Student's t, since we assume that x has a normal distribution with known . The Student's t, since n is large with unknown . 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. (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 ? At the = 0.01 level, we reject the null hypothesis and conclude the data are statistically significant. At the = 0.01 level, we reject the null hypothesis and conclude the data are not statistically significant. At the = 0.01 level, we fail to reject the null hypothesis and conclude the data are statistically significant. At the = 0.01 level, we fail to reject the null hypothesis and conclude the data are not statistically significant. 10. A random sample of 25 values is drawn from a mound-shaped and symmetric distribution. The sample mean is 11 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 10.5. (a) Is it appropriate to use a Student's t distribution? Explain. Yes, because the x distribution is mound-shaped and symmetric and is unknown. No, the x distribution is skewed left. No, the x distribution is skewed right. No, the x distribution is not symmetric. No, is known. How many degrees of freedom do we use? (b) What are the hypotheses? H0: = 10.5; H1: < 10.5 H0: > 10.5; H1: = 10.5 H0: = 10.5; H1: > 10.5 H0: = 10.5; H1: 10.5 H0: < 10.5; H1: = 10.5 (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 = 0.05 level, we reject the null hypothesis and conclude the data are statistically significant. At the = 0.05 level, we reject the null hypothesis and conclude the data are not statistically significant. At the = 0.05 level, we fail to reject the null hypothesis and conclude the data are statistically significant. At the = 0.05 level, we fail to reject the null hypothesis and conclude the data are not statistically significant. 11. 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 = 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 31 waves showed an average wave height of x = 16.9 feet. Previous studies of severe storms indicate that = 3.5 feet. Does this information suggest that the storm is (perhaps temporarily) increasing above the severe rating? Use = 0.01. (a) What is the level of significance? State the null and alternate hypotheses. H0: > 16.4 ft; H1: = 16.4 ft H0: < 16.4 ft; H1: = 16.4 ft H0: = 16.4 ft; H1: 16.4 ft H0: = 16.4 ft; H1: < 16.4 ft H0: = 16.4 ft; H1: > 16.4 ft (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 is unknown. The standard normal, since the sample size is large and is known. The Student's t, since the sample size is large and is known. The standard normal, since the sample size is large and is unknown. 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