Question

A research analyst is examining a stock for possible inclusion in his client's portfolio. Over a 19-year period, the sample mean and the sample standard deviation of annual returns on the stock were 19% and 15%, respectively. The client wants to know if the risk, as measured by the standard deviation, differs from 14%. (You may find it useful to reference the appropriate table: chi-square table or F table)

a. Construct the 95% confidence intervals for the population variance and the population standard deviation. (Round intermediate calculations to at least 4 decimal places and final answers to 2 decimal places.)

b. What assumption did you make in constructing the confidence intervals?

• Annual return is not necessarily normally distributed.

• Annual return is normally distributed

c. Based on the above confidence intervals, can we state that the risk differs from 14%?

• Yes, since the confidence interval does not include the hypothesized value.

• No, since the confidence interval includes the hypothesized value.

• No, since the confidence interval does not include the hypothesized value.

• Yes, since the confidence interval includes the hypothesized value.

Answer 1

a. s = 15, n = 19

df = n-1 = 18

Critical value, χ²α/2 = CHISQ.INV.RT(0.05/2, 18) = 31.5264

Critical value, χ²1-α/2 = CHISQ.INV.RT(1-0.05/2, 18) = 8.2307

95% Confidence interval for population variance :

Lower Bound = (n-1)s²/χ²α/2 = (19 - 1)15²/31.5264 = 128.46

Upper Bound = (n-1)s²/χ²1-α/2 = (19 - 1)15²/8.2307 = 492.06

95% Confidence interval for population standard deviation :

Lower Bound = √((n-1)s²/χ²α/2) = √((19 - 1)15/31.5264) = 11.33

Upper Bound = √((n-1)s²/χ²1-α/2) = √((19 - 1)15/8.2307) = 22.18

b. Assumption:

Annual return is normally distributed

c. No, since the confidence interval includes the hypothesized value.