1 2 pts A development economist is studying income growth in a rural area of a developing country. The last census of the population of this area, several years earlier, showed that mean household annual income was 425 dollars, and the variance of household income was 2500 (dollarssquared). A current random sample of 100 households yields a sample mean income of $433.75. Assume that household annual incomes are approximately normally distributed, and that the population variance is known still to be 2500. Test the null hypothesis that population mean income has not increased against the alternative hypothesis that it has increased, at a 1% level of significance. What rejection region should be used to conduct this test at a significance level of 1%? I.e., what rejection region controls the maximum probability of Type I error at 1%? Reject if Reject if Reject if Reject if or if Question 2 2 pts What is the critical value for question 1? 425 433.75 436.63 437.88 Question 3 2 pts What is the conclusion of the hypothesis test? Fail to reject the null hypothesis. Reject the null in favor of the alternative hypothesis. Question 4 0 pts If the population mean is $420, what is the probability of rejecting the null hypothesis? Please give your answer in the following format: 0.0450 (if it is 0.045) or 0.2345 (if it is for example 0.23452...). 1/8 Question 5 0 pts If the population mean is $430, what is the power of the test? Please give your answer in the following format: 0.0450 (if it is 0.045) or 0.2345 (if it is for example 0.23452...). Question 6 0 pts If the population mean is $435, what is the power of the test? Please give your answer in the following format: 0.0450 (if it is 0.045) or 0.2345 (if it is for example 0.23452...). Question 7 2 pts A firm manufactures metal wheels. Diameters of metal wheels produced by this process are approximately normally distributed. The variance of wheel diameters characteristic of the firm's old production process is .01 (inchessquared). The firm's engineers have proposed a new process. They claim that the variance of wheel diameters characteristic of the new process is less. An evaluation team wants to test the null hypothesis that the variance of the new process is no less than the variance of the old process, against the alternative that it is less. In a random sample of size 51, the variance is 0.0042 (inchessquared). What rejection region controls the probability of Type I error at 1%? What is the form of the rejection region? Reject H0 if s2 = cv Reject H0 if s2 < cv1 or if s2 > cv2 Reject H0 if s2 < cv Reject H0 if s2 > cv Question 8 2 pts When calculating the rejection region that controls the probability of Type I error at 1%, the table value you need is the _____ percentile of the _____ random variable. 99th percentile 1st percentile 1st percentile 99th percentile 2/8 Question 9 2 pts Without looking it up, what is the table value? (You can look it up, but you should not need to.) 76.15 50 29.71 Question 10 2 pts What is the conclusion of the test? Fail to reject the null. Reject H0 in favor of the alternative hypothesis. Question 11 2 pts If the population variance of the new process is 0.0025, what is the power of the test? Question 12 2 pts As part of a study of informal financial markets in developing countries, you are investigating whether moneylenders charge \"usurious\" interest rates. You take a random sample of 61 loans made by moneylenders to farmers, for seeds and fertilizer. The sample mean interest rate is 37.4 (percentage points), and the sample variance is 16.8 (squared percentage points). Farmers who are able to obtain bank loans for seeds and fertilizer pay 35.9 percentage points. Preliminary analysis shows it is reasonable to assume that interest rates charged by moneylenders on loans for seeds and fertilizer are approximately normally distributed. Test the null hypothesis that the mean interest rate charged by moneylenders is no greater than 35.9 percent, against the alternative that it is greater. Give the test statistic and rejection region, as well as the conclusion of the test. Use a 5% level of significance. Assume there is no difference between the creditworthiness of farmers who get bank loans for seeds and fertilizer and those who borrow from moneylenders. Otherwise, this hypothesis test would not make sense, because any observed difference in interest rates then could be due to a difference in the riskiness of the loans. What are the test statistic and its sampling distribution? by the CLT since n=61 is large. since the population is normal and is unknown. by the CLT since n=61 is large, and is unknown. 3/8 since the population is normal. Question 13 0 pts (Contd.) Before conducting this hypothesis test using the rejection region approach, let's conduct it using the pvalue approach. What is the pvalue? Please give your answer in the following format: 0.0450 (if it is 0.045) or 0.2345 (if it is for example 0.23452...). Question 14 0 pts Now taking the rejection region approach, what is the critical value for the test in Question 9? Please give your answer in the following format: 0.04 (if it is 0.042) or 234.23 (if it is for example 234.228...). Question 15 2 pts What is the conclusion of the test? Reject the null hypothesis in favor of the alternative hypothesis. The test fails to reject the null. Question 16 2 pts Historically, a real estate agency has found houses for 0.7 of its clients. The agency wants to test whether its new web site has increased the proportion of clients who find houses (this ratio is denoted by p). The agency plans to test the null hypothesis against the alternative on the basis of its sales record in a random sample of 15 clients. Suppose the rejection region is X>12, where X is the number of clients in the sample who find a house. What is the maximum probability of Type I error? 0.1221 0.17004 0.127 0.0353 4/8 Question 17 2 pts If p = 0.8, what is the probability of Type II error (using the rejection region X > 12)? 0.6372 0.8329 0.873 0.602 Question 18 2 pts What rejection region controls the maximum probability of Type I error at 5%? x>14 X>13 None X>11 Question 19 2 pts A medical study showed that girls under sixteen who are treated for Hodgkin's disease face an exceptionally high risk of developing a second type of cancer later in life (New York Times, 3/21/96). The study followed 1,380 girls after their original treatment, and 88 of them, or 0.0638, developed second cancers. Only four of them, or 0.0029, would have been expected based on data for the population at large. Test the null hypothesis that girls treated for Hodgkin's disease are not at increased risk for a second cancer (H0: p 0.0029) against the alternative hypothesis that they are at increased risk (H0: p > 0.0029). Use a 10% level of significance. What is the test statistic and its sampling distribution? Question 20 0 pts What is your critical value? Question 21 0 pts What is the result of your test? 5/8 The test fails to reject the null. Reject the null. Question 22 2 pts A random sample of 1562 undergraduates enrolled in a marketing course was asked to respond on a scale from one (strongly disagree) to seven (strongly agree) with the proposition: \"Advertising helps raise our standard of living.\" The sample mean response was 4.27, and the sample standard deviation was 1.32. Test the null hypothesis that the population mean is 4, against the alternative hypothesis that it is not 4. Use a 1% level of significance. The test statistic and its sampling distribution are because the population is normal and is unknown. by the CLT since n=1562 is large and is unknown. by the CLT since n=1562 is large by the CLT, since n=1562 is large, and is unknown. Question 23 The 99% CI interval for is 2 pts . Therefore the test fails to reject the null hypothesis. the test rejects the null in favor of the alternative. Question 24 2 pts What is the shape of the rejection probability curve? 6/8 Question 25 2 pts An engineer is designing a new production process to reduce energy consumption per unit produced. The old process had a mean level of energy consumption of 50 per unit, and a standard deviation of 6. The relevant hypothesis is the same for the new process as it was for the old ( . Assume the variance of energy consumption per unit is ). Also, assume energy consumption per unit is approximately normally distributed. The engineer decides to use a significance level of 10%. The engineer also decides the probability should be 90% of detecting that H0 is false in the event that equals 48. What is the form of the rejection region? Reject if Reject if Reject if Reject if or Question 26 2 pts What sample size should be utilized? (Collecting a sample is costly.) n=55 n=50 n=60 n=65 Question 27 2 pts What is the shape of the rejection probability curve? 7/8 8/8