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PPD303 Statistics for Policy, Planning and Development Megan Besecker

Fall 2023

Problem Set 4

(50 points)

Sampling Distributions and Confidence Intervals

1. (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.

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% Returns (Bin)

Distribution of Annual S&P Returns, 1969-2022

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2. (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.

3. (8 points) The records of the 100 postal employees show that the average time these employees have worked for the postal service is 𝑥𝑥 = 9 𝑦𝑦𝑦𝑦𝑦𝑦𝑦𝑦𝑦𝑦. Assume we know that the time the population of U.S. postal service employees have spent with the postal service is approximately Normal, with standard deviation 𝜎𝜎 = 5 𝑦𝑦𝑦𝑦𝑦𝑦𝑦𝑦𝑦𝑦.

a. (2 points) Which value is a parameter, and which value is a statistic?

b. (2 points) Calculate a 95% confidence interval for the mean time 𝜇𝜇 the population of U.S. postal service employees have spent with the postal service. Show your work. Round to the nearest 0.1 year and report your answer as (*,*).

c. (2 points) Interpret the 95% confidence interval in the context of this data.

d. (4 points) You learn the records solely came from a postal station in a large city, and that we do not know how the 100 records were obtained. Based on the data collection method, you determine the 95% confidence interval for the mean time 𝜇𝜇 the population of U.S. postal service employees is not reliable.

(i) (2 points) What three conditions are needed to estimate a confidence interval?

(ii) (2 points) Identify two reasons why the confidence interval is not trustworthy.

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Hypothesis Testing

4. (24 points) In the absence of special preparation, SAT Mathematics (SATM) scores in 2019 varied Normally with mean 𝜇𝜇 = 528 and 𝜎𝜎 = 117.

a. (8 points) For a simple random sample of 50 students: Show your work for any questions involving calculations. Round answers to the nearest 0.1.

(i) (2 points) What is the sample mean, 𝑥𝑥?

(ii) (2 points) What is the standard deviation of the average SATM score 𝑥𝑥?

(iii) (2 points) What is the probability that a randomly selected individual from the 50 students had a SATM score greater than 650?

(iv) (2 points) What is the probability that a randomly selected individual from the population had a SATM score greater than 650?

b. (8 points) Fifty students go through a rigorous training program designed to raise their SATM scores by improving their mathematics skills. The students' average score is 𝑥𝑥 = 555.

(i) (2 points) What is the null hypothesis H0 and the alternative hypothesis Ha?

(ii) (2 points) Identify whether the alternative hypothesis is one-sided or two-sided.

(iii) (2 points) Calculate the test statistic, 𝑧𝑧. Show your work and round to the nearest 0.01.

(iv) (2 points) State the p-value and interpret the p-value in the context of this data.

c. (4 points) Consider your answers part (b). Do the scores provide good evidence that the students’ SATM scores improved?

(i) (2 points) Is this result statistically significant at the 5% level?

(ii) (2 points) Do you reject or fail to reject your null hypothesis? Explain why.

d. (4 points) The program continues to collect data on students’ SATM scores and finds improvements in scores are statistically significant at the 1% level. The program advertises these results to interested students and their families.

(i) (2 points) Can you conclude the program causes improvements in students’ SATM scores? Explain why or why not.

(ii) (2 points) Suppose that all students who participated in the program come from high-income families. What type of variable is family-income in this situation, and how does it impact your interpretation of the results?

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