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

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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

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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.

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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 ????=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 *H**0 *and the alternative hypothesis *H**a*?

(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?

**Explanation & Answer**

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