An investment broker reports that yearly returns on common stocks are normally distributed with a mean of 12.4 percent and a standard deviation of 20.6 percent. (Round all k amounts to three decimal places. A negative sign should be used instead of the parentheses. For Z, use the inverse normal function and round to 3 decimal places.
(a)What percentage of yearly returns are at or below the 7th percentile of the distribution of yearly returns? What percentage are at or above the 7th percentile? Find the 7th percentile of the distribution of yearly returns.
Percentage of yearly returns are at or below the 7th percentile =
Percentage of yearly returns are at or above the 7th percentile =
k =
(b)If the mean changes to 12.7 percent and the standard deviation changes to 18.6 percent, find the first quartile, Q1 , and the third quartile, Q3, of the distribution of yearly returns.
To answer part (a) of the question, we need to find the percentage of yearly returns that are at or below the 7th percentile and the percentage that are at or above the 7th percentile.
First, let's find the 7th percentile of the distribution of yearly returns. The 7th percentile represents the value below which 7% of the data falls.
To find the 7th percentile (k), we can use the inverse normal function (also known as the Z-score formula), which helps us find the corresponding Z-score for a given cumulative probability.
The formula to find the Z-score is: Z = (x - μ) / σ
Where:
Z is the Z-score,
x is the value of interest,
μ (mu) is the mean of the distribution, and
σ (sigma) is the standard deviation of the distribution.
Given that the mean (μ) is 12.4 percent and the standard deviation (σ) is 20.6 percent, we can calculate the Z-score for the 7th percentile:
Z = (7 - 12.4) / 20.6
Z ≈ -0.259
Next, we can use the inverse normal function (e.g., a table lookup or a calculator with this function) to find the cumulative probability for this Z-score. Note that the inverse normal function gives us the percentage below a specific Z-score.
Using the inverse normal function, we find that the cumulative probability for Z ≈ -0.259 is approximately 0.397.
This means that approximately 39.7% of yearly returns are at or below the 7th percentile.
To find the percentage of yearly returns that are at or above the 7th percentile, subtract the cumulative probability from 1 (since the distribution is a standard normal distribution):
Percentage at or above 7th percentile = 1 - 0.397
Percentage at or above 7th percentile ≈ 0.603 (or 60.3%)
Finally, the 7th percentile (k) is 7 percent.
Therefore, to summarize:
Percentage of yearly returns at or below the 7th percentile = 39.7%
Percentage of yearly returns at or above the 7th percentile = 60.3%
Value of the 7th percentile (k) = 7
Moving on to part (b) of the question, we need to find the first quartile (Q1) and the third quartile (Q3) of the distribution of yearly returns under the new mean and standard deviation values.
Using the same procedure as before, but with the new mean (12.7 percent) and standard deviation (18.6 percent), we can find the Z-scores for Q1 and Q3, respectively. The first quartile corresponds to the 25th percentile (k = 25) and the third quartile corresponds to the 75th percentile (k = 75).
With the new mean and standard deviation, calculate Z1 (the Z-score for Q1) and Z3 (the Z-score for Q3) using the formula Z = (x - μ) / σ, where x is the value of interest.
Once you find Z1 and Z3, you can use the inverse normal function to find the respective cumulative probabilities, and then convert those probabilities to percentages.
By following these steps, you will find the first quartile (Q1) and the third quartile (Q3) of the distribution of yearly returns.