A simple random sample of 31 men from a normally distributed population results in a standard deviation of 8.1 beats per minute. The normal range of pulse rates of adults is typically given as 60 to 100 beats per minute. If the range rule of thumb is applied to that normal range, the result is a standard deviation of 10 beats per minute. Use the sample results with a 0.10 significance level to test the claim that pulse rates of men have a standard deviation equal to 10 beats per minute. Complete parts (a) through (d) below.
\[
\chi^{-}=19.683
\]
(Round to three decimal places as needed.)
c. Find the P-value.
\[
\text { P-value }=0.1509
\]
(Round to four decimal places as needed.)
d. State the conclusion.
v $\mathrm{H}_{0}$, because the $\mathrm{P}$-value is the level of significance. There is evidence to warrant rejection of the claim that the standard deviation of men's pulse rates is equal to 10 beats per minute.

Step 1 ：Given a sample size (n) of 31 men, a sample standard deviation (s) of 8.1 beats per minute, a population standard deviation (sigma) of 10 beats per minute, and a significance level (alpha) of 0.10.

Step 2 ：The null hypothesis (H0) is that the population standard deviation is equal to 10 beats per minute. The alternative hypothesis (H1) is that the population standard deviation is not equal to 10 beats per minute.

Step 3 ：Calculate the chi-square test statistic (chi2) using the formula \(\chi^2 = (n - 1) * (s^2) / \sigma^2\). Substituting the given values, we get \(\chi^2 = (31 - 1) * (8.1^2) / 10^2 = 19.683\).

Step 4 ：Compare the test statistic to the critical value from the chi-square distribution with n - 1 degrees of freedom. If the test statistic is greater than the critical value, we reject the null hypothesis.

Step 5 ：The test statistic (19.683) is less than the critical value (43.772), so we do not reject the null hypothesis.

Step 6 ：Calculate the p-value. The p-value (1.849) is greater than the significance level (0.10), which also suggests that we do not reject the null hypothesis.

Step 7 ：Therefore, there is not enough evidence to reject the claim that the standard deviation of men's pulse rates is equal to 10 beats per minute.

Step 8 ：\(\boxed{\text{Final Answer: The standard deviation of men's pulse rates is equal to 10 beats per minute.}}\)