CONCORDIA UNIVERSITY Department of Economics ECON 221/2 SECTIONS E, G, and FF STATISTICAL METHODS I WINTER 2017 - ASSIGNMENT 2 Due: Monday, March 27, before 3:00 pm Name: I.D: Section: Points Total: 60 points 1. (14 points) James Chadwick manages a mutual fund for Sigma Wealth Management. In a particular year, it is known that the return on the fund follows a normal distribution with a mean of 14.8% and a standard deviation of 6.3%. As it is only March, James would like to draw a sample of nine stocks from the fund to gauge performance thus far (and to see if the fund is performing up to his expectations. a. (2 points) What is the distribution of the mean return? b. (2 points) How would your answer change if the parent distribution was not normally distributed? c. (2 points) What is the probability that the sample mean return is greater than 19%? . 1 d. (2 points) What is the probability that the mean rate of return is between 10.6% and 19%? e. (2 points) What is the minimum return you would need to have to yield a probability of 0.25? f. (2 points) What is the minimum sample standard deviation yielding a probability is 0.10? g. (2 points) Calculate the probability that the sample standard deviation is less than 9.33. 2 2. (6 points) Suppose that in the previous question, the sample yielded a sample mean of 12.5 and sample standard variance (s2) of 5.4. a. (2 points) Provide a 99% confidence interval for the mean rate of return. b. (2 points) How would your answer to part (a) change if we did not know the population variance? c. (2 points) what is the 95% confidence interval for the population variance? 3. (10 points) Do Smart Boards make a difference in the age at which a child learns to read? To study this question, researchers designed an experiment in which one group of children spent 2 hours a day (for 6 months) in a room learning to read with a Smart Board as a companion. Another group spent 2 hours a day in a non-Smart Board room (for 6 months) learning the \"old fashioned\" way. Six pairs of identical twins were randomly selected with each being placed in one of the two groups. The data presented in the table below represents the number of months at which the child began reading at the primary level. The results are as follows. 3 Twin pair 1 2 3 4 5 6 Smart Group Reading Age (x1) 58 61 53 60 71 62 Non-Smart Group Reading Age (x2) 60 64 52 65 75 63 a. (2 points) Calculate the sample mean of the differences. b. (2 points) Calculate the sample standard deviation of the differences. c. (2 points) Calculate the margin of error for a 95 percent confidence interval for the difference between the means of the two populations. d. (2 points) Construct a 95 percent confidence interval for the difference between the means of the two populations. 4 e. (2 points) Explain which group performed better and why? 4. (14 points) In a recent study of Quebec high schools, it was found that 40% of high school students own personal computer and use them for their high school studies. Based on a random sample of 120 high school students, answer the following questions. a. (2 points) Calculate the mean and standard deviation of the sampling distribution of p . b. (2 points) Describe the sampling distribution of p . c. (2 points) what is the probability that less than 33% of students are using computers for their high school studies. . 5 d. (2 points) What is the probability that the sample proportion is between 0.38 and 0.46? e. (2 points) Suppose the sample in question found 58 students out of the 120 were using computers for their studies. What is a 95% confidence interval for the proportion of high school students using computers for their studies? f. (2 points) Do you think 120 is a sufficient sample size for 95% confidence and a margin of error of no more than 5%? g. (2 points) Briefly interpret what 95% means in this context. 5. (6 points) A recent sample of 200 Canadians, 100 men and 100 women, found that 121 were in favour of legalizing recreational marijuana usage. Further, 75 men and 46 women supported legalization. 6 a. (2 points) Treating these as random samples and letting p1 be the proportion of men who support legalization and p2 be the proportion of women who support legalization, calculate the margin of error for a 99 percent confidence interval for p1 p 2 . b. (2 points) Construct a 99 percent confidence interval for p1 p 2 . c. (2 points) Explain whether one can infer that men and women have a differing opinion on legalization. 6. (10 points) Carla is comparing automobile insurance premiums in Quebec with those in Ontario. A random sample of 31 insurance companies in Quebec yielded a sample mean premium $740 per year with sample standard deviation $115. An independent random sample of 34 insurance companies in Ontario gave a sample mean premium $815 with sample standard deviation $150 for similar coverage. Assume that the random sample of observations are from normally distributed populations and that the population variances are assumed to be equal. 7 a. (2 points) Calculate the pooled sample variance. b. (2 points) Calculate the margin of error for a 98 percent confidence interval for the difference between the insurance premiums. c. (2 points) Construct a 98 percent confidence interval for the difference between the insurance premiums. d. (2 points) How would your answer change if the standard deviations given were the population standard deviations? 8 e. (2 points) Based on the confidence interval constructed in part (c) and (d), explain what can be inferred about the difference in insurance rates. 9