Suppose a mutual fund qualifies as having moderate risk if the standard deviation of its monthly rate of return is less than $3\mathrm{\%}$. A mutual-fund rating agency randomly selects 21 months and determines the rate of return for a certain fund. The standard deviation of the rate of return is computed to be $2.04\mathrm{\%}$. 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.
What are the correct hypotheses for this test?
The null hypothesis is ${\mathrm{H}}_0$ :
Calculate the value of the test statistic.
${\chi }_0^2=◻$ (Round to two decimal places as needed.)
Use technology to determine the P-value for the test statistic.
The P-value is
(Round to three decimal places as needed.)
What is the correct conclusion at the $\alpha =0.05$ level of significance?
Since the P-value is less than the level of significance, reject $$ the null hypothesis. There $$ is sufficient evidence to conclude that the fund has moderate risk at the 0.05 level of significance.

Step 1 ：The null hypothesis is that the standard deviation is equal to 3%, and the alternative hypothesis is that the standard deviation is less than 3%.

Step 2 ：We are given a sample standard deviation of 2.04% from a sample of 21 months.

Step 3 ：We can use a chi-square test to test the hypotheses. The test statistic is calculated as $(n-1)\times (\text{sample standard deviation}{)}^2/(\text{population standard deviation}{)}^2$, where n is the sample size.

Step 4 ：After calculating, the test statistic is approximately 9.25.

Step 5 ：We can use a chi-square distribution with n-1 degrees of freedom to find the p-value.

Step 6 ：The p-value is approximately 0.98.

Step 7 ：Since the p-value is greater than the significance level of 0.05, we fail to reject the null hypothesis.

Step 8 ：This means that there is not sufficient evidence to conclude that the fund has moderate risk at the 0.05 level of significance.

Step 9 ：Final Answer: The test statistic is $\boxed{{\displaystyle 9.25}}$ and the p-value is $\boxed{{\displaystyle 0.98}}$. Since the p-value is greater than the level of significance, we fail to reject the null hypothesis. There is not sufficient evidence to conclude that the fund has moderate risk at the 0.05 level of significance.