Respuesta :
Answer:
a) [tex]\mu = -4.8, s = \frac{25}{\sqrt{7}} = 9.45[/tex]
b) 31.92% probability that a randomly chosen stock showed a return of at least 7% in 1987.
c) 20.61% probability that a randomly chosen portfolio of 3 stocks showed a return of at least 7% in 1987.
d) 62.93% of 3-stock portfolios lost money in 1987.
Step-by-step explanation:
The Central Limit Theorem estabilishes that, for a random variable X, with mean [tex]\mu[/tex] and standard deviation [tex]\sigma[/tex], a large sample size can be approximated to a normal distribution with mean [tex]\mu[/tex] and standard deviation [tex]\frac{\sigma}{\sqrt{n}}[/tex].
Normal probability distribution:
Problems of normally distributed samples can be solved using the z-score formula.
In a set with mean [tex]\mu[/tex] and standard deviation [tex]\sigma[/tex], the zscore of a measure X is given by:
[tex]Z = \frac{X - \mu}{\sigma}[/tex]
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 pvalue, we get the probability that the value of the measure is greater than X.
In this problem, we have that:
[tex]\mu = -4.8, \sigma = 25[/tex]
(a) What are the mean and the standard deviation of the distribution of 7-stock portfolios in 1987.
By the Central Limit Theorem, we have that:
[tex]\mu = -4.8, s = \frac{25}{\sqrt{7}} = 9.45[/tex]
(b) Assuming that the population distribution of returns on individual common stocks is normal, what is the probability that a randomly chosen stock showed a return of at least 7% in 1987?
This is 1 subtracted by the pvalue of Z when X = 7. So
[tex]Z = \frac{X - \mu}{\sigma}[/tex]
[tex]Z = \frac{7 - (-4.8)}{25}[/tex]
[tex]Z = 0.47[/tex]
[tex]Z = 0.47[/tex] has a pvalue of 0.6808.
So 1-0.6808 = 0.3192 = 31.92% probability that a randomly chosen stock showed a return of at least 7% in 1987.
(c) Assuming that the population distribution of returns on individual common stocks is normal, what is the probability that a randomly chosen portfolio of 3 stocks showed a return of at least 7% in 1987?
By the central Limit theorem, we use [tex]\mu = -4.8, \sigma = \frac{25}{\sqrt{3}} = 14.43[/tex]
This is 1 subtracted by the pvalue of Z when X = 7. So
[tex]Z = \frac{X - \mu}{\sigma}[/tex]
[tex]Z = \frac{7 - (-4.8)}{14.43}[/tex]
[tex]Z = 0.82[/tex]
[tex]Z = 0.82[/tex] has a pvalue of 0.7939.
So 1-0.7939 = 0.2061 = 20.61% probability that a randomly chosen portfolio of 3 stocks showed a return of at least 7% in 1987.
(d) What percentage of 3-stock portfolios lost money in 1987?
This is the pvalue of Z when X = 0. So:
[tex]Z = \frac{X - \mu}{\sigma}[/tex]
[tex]Z = \frac{0 - (-4.8)}{14.43}[/tex]
[tex]Z = 0.33[/tex]
[tex]Z = 0.33[/tex] has a pvalue of 0.6293.
So 62.93% of 3-stock portfolios lost money in 1987.
