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For a sample of 11 body temperatures of men, the mean temperature is equal to97.69 degrees Fahrenheit,and the standard deviation of temperatures is equal to 0.89 degrees Fahrenheit.For another sample of 59 body temperatures of women, the mean temperature is equal to 97.45 degrees Fahrenheitand the standard deviation of temperatures is equal to 0.66 degrees Fahrenheit.

It is claimed that the variation in men鈥檚 body temperature is more than the variation in women鈥檚 body temperature.

Let\({\sigma _1}\)and\({\sigma _2}\)be the population standard deviationsof the men鈥檚 body temperaturesand the women鈥檚 body temperature,respectively.

Null Hypothesis: The population standard deviation of men鈥檚 body temperature is equal to the population standard deviation of women鈥檚 body temperature.

Symbolically,

\({H_0}:{\sigma _1} = {\sigma _2}\)

Alternate Hypothesis: The population standard deviation of men鈥檚 body temperature is greater than the population standard deviation of women鈥檚 body temperature.

Symbolically,

\({H_1}:{\sigma _1} > {\sigma _2}\)

Since two independent samples involve a claim about the population standard deviation, apply an F-test.

Consider the larger sample variance to be\(s_1^2\)and the corresponding sample size to be\({n_1}\).

The following values are obtained:

\({\left( {0.89} \right)^2} = 0.7921\)

\({\left( {0.66} \right)^2} = 0.4356\)

Here\(s_1^2\)is the sample variance corresponding to men鈥檚 body temperature and has a value equal to 0.7921\({\left( {^ \circ F} \right)^2}\).

\(s_2^2\)is the sample variance corresponding to women鈥檚 body temperature and has a value equal to 0.4356\({\left( {^ \circ F} \right)^2}\).

Substitute the respective values to calculate the F statistic:

\(\begin{array}{c}F = \frac{{s_1^2}}{{s_2^2}}\\ = \frac{{{{\left( {0.89} \right)}^2}}}{{{{\left( {0.66} \right)}^2}}}\\ = 1.818\end{array}\)

The value of the numerator degrees of freedom is equal to:

\(\begin{array}{c}{n_1} - 1 = 11 - 1\\ = 10\end{array}\)

The value of the denominator degrees of freedom is equal to:

\(\begin{array}{c}{n_2} - 1 = 59 - 1\\ = 58\end{array}\)

For the F test, the critical value corresponding to the right-tail is considered.

The critical value can be obtained using the F-distribution table with numerator degrees of freedom equal to 10and denominator degrees of freedom equal to 58 for a right-tailed test.

The level of significance is equal to 0.05.

Thus, the critical value is equal to 1.9983.

The right-tailed p-value for F equal to 1.818 is equal to 0.0775.

Since the test statistic value is less than the critical value and the p-value is greater than 0.05, the null hypothesis is failed to reject.

Thus, there is not enough evidence to supportthe claimthat men have body temperatures that vary more than the body temperatures of women.