Respuesta :
Using the normal distribution, it is found that:
- There is a 0.1292 = 12.92% probability that an individual large-cap domestic stock fund had a three-year return of at least 20%.
- There is a 0.1587 = 15.87% probability that an individual large-cap domestic stock fund had a three-year return of 10% or less.
- To be in the top 10%, the return has to be of at least 20.72%.
Normal Probability Distribution
The z-score of a measure X of a normally distributed variable that has mean represented by [tex]\mu[/tex] and standard deviation represented by [tex]\sigma[/tex] is given by the following rule:
[tex]Z = \frac{X - \mu}{\sigma}[/tex]
- The z-score measures how many standard deviations the measure X is above or below the mean, depending if the z-score is positive or negative.
- From the z-score table, the p-value associated with the z-score is found, and it represents the percentile of the measure X.
In the context of this problem, the mean and the standard deviation are given as follows:
[tex]\mu = 14.7, \sigma = 4.7[/tex]
For a return of at least 20%, the probability is one subtracted by the p-value of Z when X = 20, hence:
[tex]Z = \frac{X - \mu}{\sigma}[/tex]
Z = (20 - 14.7)/4.7
Z = 1.13
Z = 1.13 has a p-value of 0.8708
1 - 0.8708 = 0.1292.
For the return of 10% or less, the probability is the p-value of Z when X = 10, hence:
[tex]Z = \frac{X - \mu}{\sigma}[/tex]
Z = (10 - 14.7)/4.7
Z = -1
Z = -1 has a p-value of 0.1587.
To be in the top 10%, the return has to be of at least the 90th percentile, which is X when Z = 1.28, hence:
[tex]Z = \frac{X - \mu}{\sigma}[/tex]
1.28 = (X - 14.7)/4.7
X - 14.7 = 1.28 x 4.7
X = 20.72%.
More can be learned about the normal distribution at https://brainly.com/question/4079902
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