PPD 303 Statistics for Policy, Planning and Development- Problem Set #4

Please provide unique step by step answers for the following 4 questions below, as well as the the excel spreadsheet and relevant formulas needed to solve these problems. Thank you.

Problem Set 4

(50 points)

Sampling Distributionsand Confidence Intervals

1.(8points)Nikplans toretire in 40years.Theyplan to invest partof theirretirement funds instocks, sotheyseekout information on past returns.Theylearn that from1928to 2022, theannual returns on S&P 500had mean7.7% and standard deviation19.0%. They assume themeanreturn overeven a moderate number of years is close to Normaland assume that the past pattern of variationcontinues.

a.(2points)Most returns (95%) fall betweenwhat range?Usethe 68-95-99.7 rule.Round your answerto the nearesttenth of a percentage and report youranswer as follows: (*,*).

b.(4points)What is the probability,????1, (assuming that the past patternof variationcontinues) that the mean annual return oncommonstocks over the next 40 years will:

(i)(2points)Exceed 10%?Give your answer totwo decimal places.

(ii)(2points)Lessthan 5%? Give your answer totwo decimal places.

c.(2points)Before investing, they lookthe distributionof S&P returns from 1968to 2022. Based onthe distribution, canthey trust their estimatesin part (b)? Explain why or whynot.

051015202530Frequency% Returns(Bin)Distribution ofAnnualS&P Returns, 1969-2022

This contentis protected and may not be shared, uploaded,ordistributed.

2 This content is protected and may not be shared, uploaded, or distributed.

2. (10 points) To estimate the mean score ???? 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 ???? to estimate ???? with an error of no more than 1 point in either direction. (i) (2 points) What standard deviation must ???? have so that 99.7% of all samples give an ???? within 1 point of ????? 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 ???? 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 ????=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 ????? 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 ????=9 ????????????????????. 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 ????=5 ????????????????????. 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 ???? 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 ???? 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.

3 This content is protected and may not be shared, uploaded, or distributed.

Hypothesis Testing

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

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

(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 ????=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, ????. 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?