Suppose a mutual fund qualifies as having moderate risk if the standard deviation of its monthly rate of return is less than $4 \%$. 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 retum is computed to be $3.78\%$. ls 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 retum are normally distributed.
What are the correct hypotheses for this test?
The null hypothesis is $H_{0} ; \quad \sigma=0.04$.
The alternative hypothesis is $\mathrm{H}_{4} ; \sigma< 0.04$.
Calculate the value of the test statistic.
$x^{2}=\square$ (Round to three decimal places as needed.)
Answer
\(\boxed{22.326}\) is the final answer.
Step 1 ：Define the null hypothesis as \(H_{0} ; \sigma=0.04\) and the alternative hypothesis as \(H_{4} ; \sigma<0.04\).
Step 2 ：Use the chi-square test statistic for a one-sample test of variance, which is given by the formula \(x^{2} = \frac{(n - 1)s^{2}}{\sigma^{2}}\), where n is the sample size, s is the sample standard deviation, and \(\sigma\) is the hypothesized population standard deviation.
Step 3 ：Substitute the given values into the formula: n = 26, s = 0.0378, and \(\sigma\) = 0.04.
Step 4 ：Calculate the test statistic to get \(x^{2} = \frac{(26 - 1) * 0.0378^{2}}{0.04^{2}}\).
Step 5 ：Compute the value to get \(x^{2} = 22.325625000000002\).
Step 6 ：Round the test statistic to three decimal places to get \(x^{2} = 22.326\).
Step 7 ：\(\boxed{22.326}\) is the final answer.
