Problem
Suppose a mutual fund qualifies as having moderate risk if the standard deviation of its monthly rate of return is less than $6 \%$. A mutual-fund rating agency randomly selects 26 months and determines the rate of return for a certain fund. The standard deviation of the rate of return is computed to be $5.21 \%$. Is there sufficient evidence to conclude that the fund has moderate risk at the $\alpha=0.05$ level of significance? A normal probability plot indicates that the monthly rates of return are normally distributed. Calculate the value of the test statistic. $\chi^{2}=\square$ (Round to three decimal places as needed.)
Solution
Step 1 ：Given that the sample size (n) is 26, the sample standard deviation (s) is 5.21%, and the population standard deviation (σ) is 6%.
Step 2 ：We are testing the null hypothesis that the population standard deviation is 6%.
Step 3 ：The chi-square test statistic is calculated using the formula: \[\chi^{2} = \frac{(n-1)s^{2}}{\sigma^{2}}\]
Step 4 ：Substitute the given values into the formula: \[\chi^{2} = \frac{(26-1)(5.21)^{2}}{6^{2}}\]
Step 5 ：Calculate the value to get the test statistic.
Step 6 ：Final Answer: The value of the test statistic is \(\boxed{18.850}\).
