Suppose a mutual fund qualifies as having moderate risk if the standard deviation of its monthly rate of return is less than $5 \%$. 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 $4.36 \%$. Is there sufficient evidence to conclude that the fund has moderate risk at the $\alpha=0.10$ 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.) Use technology to determine the P-value for the test statistic.

Step 1 ：We are given a problem where we need to determine if a mutual fund qualifies as having moderate risk. The criteria for moderate risk is a standard deviation of its monthly rate of return less than 5%. We have a sample of 26 months and the computed standard deviation of the rate of return is 4.36%. We are asked to determine if there is sufficient evidence to conclude that the fund has moderate risk at the 0.10 level of significance.

Step 2 ：We are performing a chi-square test for the standard deviation. The null hypothesis is that the standard deviation is less than or equal to 5%, and the alternative hypothesis is that the standard deviation is greater than 5%.

Step 3 ：The test statistic for a chi-square test for the standard deviation is calculated using the formula: \[\chi^{2} = \frac{(n - 1)s^{2}}{\sigma^{2}}\] where n is the sample size, s is the sample standard deviation, and σ is the population standard deviation under the null hypothesis.

Step 4 ：Substituting the given values into the formula, we get: n = 26, s = 4.36%, and σ = 5%. Plugging these values into the formula, we calculate the test statistic.

Step 5 ：The calculated value of the test statistic is approximately 19.01.

Step 6 ：We then use technology to determine the P-value for the test statistic. The calculated P-value is approximately 0.797.

Step 7 ：Since the P-value is greater than the significance level of 0.10, we do not reject the null hypothesis. This means that there is not sufficient evidence to conclude that the fund has a risk level that is not moderate.

Step 8 ：Final Answer: The value of the test statistic is \(\boxed{19.01}\) and the P-value is \(\boxed{0.797}\).