(8 points) Nik plans to retire in 40 years. They plan to invest part of their retirement funds in stocks, so they seek out information on past returns. They learn that from 1928 to 2022, the annual returns on S&P 500 had mean 7.7% and standard deviation 19.0%. They assume the mean return over even a moderate number of years is close to Normal and assume that the past pattern of variation continues. a. (2 points) Most returns (95%) fall between what range? Use the 68-95-99.7 rule. Round your answer to the nearest tenth of a percentage and report your answer as follows: (,). b. (4 points) What is the probability, 𝑝𝑝1, (assuming that the past pattern of variation continues) that the mean annual return on common stocks over the next 40 years will: (i) (2 points) Exceed 10%? Give your answer to two decimal places. (ii) (2 points) Less than 5%? Give your answer to two decimal places. c. (2 points) Before investing, they look the distribution of S&P returns from 1968 to 2022. Based on the distribution, can they trust their estimates in part (b)? Explain why or why not. 0 5 10 15 20 25 30 Frequency % Returns (Bin) Distribution of Annual S&P Returns, 1969-2022 2 This content is protected and may not be shared, uploaded, or distributed.
(10 points) To estimate the mean score 𝜇𝜇 of those who took the Medical College Admission Test on your campus, you will obtain the scores of a simple random sample (SRS) of students. From published information you know that the scores are approximately Normal with standard deviation about 6.3. a) (4 points) You want your sample mean 𝑥𝑥 to estimate 𝜇𝜇 with an error of no more than 1 point in either direction. (i) (2 points) What standard deviation must 𝑥𝑥 have so that 99.7% of all samples give an 𝑥𝑥 within 1 point of 𝜇𝜇? Use the 68-95-99.7 rule. Give your answer to two decimal places. (ii) (2 points) Determine the size of an SRS needed to reduce the standard deviation of 𝑥𝑥 to within the value you found in (i). b) (6 points) Suppose you decide to collect scores via a survey sent to the SRS size you determined in part (b) and 70% respond. You calculate a sample mean 𝑥𝑥 = 500. (i) (2 points) What happens to the margin of error when the sample size decreases? Explain. (ii) (2 points) How confident are you in using the sample mean to estimate the mean score 𝜇𝜇? Explain, and if present, identify any bias in the data.
Our customer support team is here to answer your questions. Ask us anything!
