Velma Vasquez,fund manager of the Vasquez Value Fund,manages a portfolio of 250 common stocks. Velma relies on various statistics,such as variance,to assess the overall risk of stocks in an economic sector.Her staff reported that for a sample 14 utility stocks the mean annualized return was 14% and that the variance was 3%. Assume that annualized returns are normally distributed. The 90% confidence interval for the population variance of annualized returns is _______.
A) 0.018 to 0.064
B) 0.016 to 0.078
C) 0.017 to 0.066
D) 0.016 to 0.075
E) 0.020 to 0.080

Question

Quizplus · Accepted Answer

First, we need to find the chi-squared values for the lower and upper limits of the confidence interval. This can be done using the chi-squared distribution table with n-1 degrees of freedom, where n is the sample size. In this case, n = 14, so we have 13 degrees of freedom.

The lower chi-squared value is given by the 5% tail probability in the left-hand tail of the chi-squared distribution with 13 degrees of freedom, which is 5.229.

The upper chi-squared value is given by the 5% tail probability in the right-hand tail of the chi-squared distribution with 13 degrees of freedom, which is 22.362.

Next, we can use the formula for the confidence interval for the population variance:

[(n-1)*s^2]/X^2 <= sigma^2 <= [(n-1)*s^2]/Y^2

where s^2 is the sample variance, X^2 and Y^2 are the lower and upper chi-squared values, and sigma^2 is the population variance.

Plugging in the values from the problem, we get:

[(14-1)*0.03]/22.362 <= sigma^2 <= [(14-1)*0.03]/5.229

0.017 <= sigma^2 <= 0.066

Therefore, the 90% confidence interval for the population variance of annualized returns is 0.017 to 0.066, which corresponds to choice (C).

Velma Vasquez,fund Manager of the Vasquez Value Fund,manages a Portfolio