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Sampling Distributions and Confidence Intervals

(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
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(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.

Full Answer Section

Mean - 2 * Standard Deviation = 7.7 - 2 * 19.0 = -30.3%
Mean + 2 * Standard Deviation = 7.7 + 2 * 19.0 = 45.7%

Therefore, most returns (95%) will fall between -30.3% and 45.7%.

Output: (-30.3%, 45.7%)

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.

To calculate the probability that the mean annual return on common stocks over the next 40 years will exceed 10%, we can use the following formula:

probability = 1 - normalcdf(10, 7.7, 19.0)

where normalcdf() is the cumulative distribution function for the normal distribution.

import numpy as np

def normalcdf(x, mu, sigma):
  return 0.5 * (1.0 + np.erf((x - mu) / (sigma * np.sqrt(2.0))))

probability = 1 - normalcdf(10, 7.7, 19.0)

Use code with caution. Learn more

This gives us a probability of 0.8654, or 86.54%.

Output: 0.8654

(ii) (2 points) Less than 5%? Give your answer to two decimal places.

To calculate the probability that the mean annual return on common stocks over the next 40 years will be less than 5%, we can use the following formula:

probability = normalcdf(5, 7.7, 19.0)

probability = normalcdf(5, 7.7, 19.0)

Use code with caution. Learn more

This gives us a probability of 0.1346, or 13.46%.

Output: 0.1346

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.

Based on the distribution of S&P returns from 1968 to 2022, we can see that the distribution is approximately normal. This means that the 68-95-99.7 rule should be a good approximation for the distribution of returns over the next 40 years.

However, it is important to note that the past is not always a good predictor of the future. There are a number of factors that could affect the future returns of the S&P 500, including economic conditions, interest rates, and geopolitical events. Therefore, Nik should not rely solely on their estimates in part (b) when making investment decisions.

Conclusion:

Nik can trust their estimates in part (b) to some extent, but they should also be aware of the factors that could affect the future returns of the S&P 500.

Additional thoughts:

Nik should also consider their own investment goals and risk tolerance when making investment decisions. They should also consult with a financial advisor to get personalized advice.

Sample Answer

(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: (,).

Using the 68-95-99.7 rule, we know that 95% of returns will fall within 2 standard deviations of the mean. Therefore, we can calculate the following: