Answer:
a) 0.0885 = 8.85% probability that the mean annual return on common stocks over the next 40 years will exceed 13%.
b) 0.4129 = 41.29% probability that the mean return will be less than 8%
Step-by-step explanation:
To solve this question, we need to understand the normal probability distribution and the central limit theorem.
Normal Probability Distribution:
Problems of normal distributions can be solved using the z-score formula.
In a set with mean
and standard deviation
, the z-score of a measure X is given by:
![Z = \frac{X - \mu}{\sigma}](https://tex.z-dn.net/?f=Z%20%3D%20%5Cfrac%7BX%20-%20%5Cmu%7D%7B%5Csigma%7D)
The Z-score measures how many standard deviations the measure is from the mean. After finding the Z-score, we look at the z-score table and find the p-value associated with this z-score. This p-value is the probability that the value of the measure is smaller than X, that is, the percentile of X. Subtracting 1 by the p-value, we get the probability that the value of the measure is greater than X.
Central Limit Theorem
The Central Limit Theorem estabilishes that, for a normally distributed random variable X, with mean
and standard deviation
, the sampling distribution of the sample means with size n can be approximated to a normal distribution with mean
and standard deviation
.
For a skewed variable, the Central Limit Theorem can also be applied, as long as n is at least 30.
Mean 8.7% and standard deviation 20.2%.
This means that ![\mu = 8.7, \sigma = 20.2](https://tex.z-dn.net/?f=%5Cmu%20%3D%208.7%2C%20%5Csigma%20%3D%2020.2)
40 years:
This means that ![n = 40, s = \frac{20.2}{\sqrt{40}}](https://tex.z-dn.net/?f=n%20%3D%2040%2C%20s%20%3D%20%5Cfrac%7B20.2%7D%7B%5Csqrt%7B40%7D%7D)
(a) What is the probability (assuming that the past pattern of variation continues) that the mean annual return on common stocks over the next 40 years will exceed 13%?
This is 1 subtracted by the pvalue of Z when X = 13. So
![Z = \frac{X - \mu}{\sigma}](https://tex.z-dn.net/?f=Z%20%3D%20%5Cfrac%7BX%20-%20%5Cmu%7D%7B%5Csigma%7D)
By the Central Limit Theorem
![Z = \frac{X - \mu}{s}](https://tex.z-dn.net/?f=Z%20%3D%20%5Cfrac%7BX%20-%20%5Cmu%7D%7Bs%7D)
![Z = \frac{13 - 8.7}{\frac{20.2}{\sqrt{40}}}](https://tex.z-dn.net/?f=Z%20%3D%20%5Cfrac%7B13%20-%208.7%7D%7B%5Cfrac%7B20.2%7D%7B%5Csqrt%7B40%7D%7D%7D)
![Z = 1.35](https://tex.z-dn.net/?f=Z%20%3D%201.35)
has a pvalue of 0.9115
1 - 0.9115 = 0.0885
0.0885 = 8.85% probability that the mean annual return on common stocks over the next 40 years will exceed 13%.
(b) What is the probability that the mean return will be less than 8%?
This is the pvalue of Z when X = 8. So
![Z = \frac{X - \mu}{s}](https://tex.z-dn.net/?f=Z%20%3D%20%5Cfrac%7BX%20-%20%5Cmu%7D%7Bs%7D)
![Z = \frac{8 - 8.7}{\frac{20.2}{\sqrt{40}}}](https://tex.z-dn.net/?f=Z%20%3D%20%5Cfrac%7B8%20-%208.7%7D%7B%5Cfrac%7B20.2%7D%7B%5Csqrt%7B40%7D%7D%7D)
![Z = -0.22](https://tex.z-dn.net/?f=Z%20%3D%20-0.22)
has a pvalue of 0.4129
0.4129 = 41.29% probability that the mean return will be less than 8%