1) Let x be per capita income in thousands of dollars. Let y be the number of medical doctors per 10,000 residents. Six small cities in Oregon gave the following information about x and y. x 9.0 9.4 10.3 8.0 8.3 8.7 y 10.0 18.8 21.6 10.2 11.4 13.1 Complete parts (a) through (e), given x = 53.7, y = 85.1, x2 = 484.03, y2 = 1325.61, xy = 779.39, and r 0.882. (a) Verify the given sums x, y, x2, y2, xy, and the value of the sample correlation coefficient r. (Round your value for r to three decimal places.) x = y = x2 = y2 = xy = r= ^ (b) Find x, and y. Then find the equation of the least-squares line y = a + bx. (Round your answers for x and y to two decimal places. Round your answers for a and b to three decimal places.) = x Y = ^= Y + x (c) Find the value of the coefficient of determination r2. What percentage of the variation in y can be explained by the corresponding variation in x and the least-squares line? What percentage is unexplained? (Round your answer for r2 to three decimal places. Round your answers for the percentages to one decimal place.) r2 = explained % unexplained % (d) Suppose a small city in Oregon has a per capita income of 9.7 thousand dollars. What is the predicted number of M.D.s per 10,000 residents? (Round your answer to two decimal places.) M.D.s per 10,000 residents 2) 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 86 82 107 97 109 84 91 The sample mean is x 93.6. 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? b) State the null and alternate hypotheses. Will you use a left-tailed, right-tailed, or two-tailed test? (Please highlight the correct answer). H0: = 85; H1: 85; two-tailed H0: > 85; H1: = 85; right-tailed H0: = 85; H1: > 85; right-tailed H0: = 85; H1: < 85; left-tailed (c) What sampling distribution will you use? Explain the rationale for your choice of sampling distribution. (Please highlight the correct answer). 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 . The standard normal, since we assume that x has a normal distribution with unknown . d) What is the value of the sample test statistic? (Round your answer to two decimal places.) (e) Find (or estimate) the P-value. (Round your answer to four decimal places.) _____________. (f) 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 ? (Please highlight the correct answer). 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. (g) State your conclusion in the context of the application. (Please highlight the correct answer). 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. 3) A random sample of 41 adult coyotes in a region of northern Minnesota showed the average age to be x = 2.05 years, with sample standard deviation s = 0.78 years. However, it is thought that the overall population mean age of coyotes is = 1.75. Do the sample data indicate that coyotes in this region of northern Minnesota tend to live longer than the average of 1.75 years? Use = 0.01. (a) What is the level of significance? b) State the null and alternate hypotheses. H0: = 1.75 yr; H1: > 1.75 yr H0: = 1.75 yr; H1: 1.75 yr H0: = 1.75 yr; H1: < 1.75 yr H0: < 1.75 yr; H1: = 1.75 yr H0: > 1.75 yr; H1: = 1.75 yr c) 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 is unknown. 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. d) What is the value of the sample test statistic? (Round your answer to three decimal places.) (e) Find the P-value. (Round your answer to four decimal places.) (f) 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. (g) Interpret your conclusion in the context of the application. There is sufficient evidence at the 0.01 level to conclude that coyotes in the specified region tend to live longer than 1.75 years. There is insufficient evidence at the 0.01 level to conclude that coyotes in the specified region tend to live longer than 1.75 years. 4) 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 = 19.4. A random sample of 36 "socially conscious" stocks gave a P/E ratio sample mean of x = 18.0, with sample standard deviation s = 5.2.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 = 0.01. (a) What is the level of significance? b) State the null and alternate hypotheses. (Please highlight the correct answer). H0: = 19.4; H1: > 19.4 H0: = 19.4; H1: 19.4 H0: > 19.4; H1: = 19.4 H0: = 19.4; H1: < 19.4 H0: 19.4; H1: = 19.4 c) 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 is known. The standard normal, since the sample size is large and is unknown. The Student's t, since the sample size is large and is known. The Student's t, since the sample size is large and is unknown. d) What is the value of the sample test statistic? (Round your answer to three decimal places.) e) Find the P-value. (Round your answer to four decimal places.) (f) 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 ? (Please highlight the correct answer). 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. (g) Interpret your conclusion in the context of the application. (Please highlight the correct answer). There is sufficient evidence at the 0.01 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 0.01 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