Question

Test the given claim. Identify the null hypothesis, alternative hypothesis, test statistic, P-value, or critical value(s), then the conclusion about the null hypothesis, as well as the final conclusion that address the original claim. Assume that a simple random sample is selected from a normally distributed population.

Body Temperature: Example 5 in Section 8-3 involved a test of the claim that humans have body temperatures with a mean equal to 98.6°F. The sample of 106 body temperatures has a standard deviation of 0.62°F. The conclusion in that example would change if the sample standard deviation swere 2.08°F or greater. Use a 0.01 significance level to test the claim that the sample of 106 body temperatures is from a population with a standard deviation less than 2.08°F. What does the result tell us about the validity of the hypothesis test in Example 5 in Section 8-3?

Short answer

The hypotheses are as follows.

$$\begin{gathered} H_0:\sigma =2.08°F \\ {H}_1:\sigma <2.08°F \end{gathered}$$

The test statistic ${\chi }^2=9.329$, and the critical value ${{\chi }_{\alpha }}^2=70.065$. Reject ${\text{H}}_{\text{0}}$.

There is sufficient evidence to support the claim that the body temperatures have a standard deviation of less than $2.08°$F.

The results of Example 5 from sections 8 to 3 are not likely to change because of the difference in the standard deviation value from a different sample.

Step-by-step solution

Given information 

Refer to Example 5 in sections 8 to 3 for the test of mean human body temperatures of 98.6F. Assume that the standard deviation measure for a sample of 106 temperatures changes to $2.08F$.

The level of significance is 0.01 to test the claim that the standard deviation of the population is less than 2.08F.

State the hypothesis

To test that the sample is taken from the population of body temperatures with a standard deviation less than 2.08 F, the null and alternative hypotheses are formulated as follows.

$$\begin{gathered} H_0:\sigma =2.08 \\ {H}_1:\sigma <2.08 \end{gathered}$$

Here, $\sigma$is the actual standard deviation for the population.

Compute the test statistic 

The test statistic ${\chi }^2$with $\left(n-1\right)$degrees of freedom is given as follows.

$$\begin{gathered} {\chi }^2=\frac{\left(n-1\right)s^2}{{\sigma }^2} \\ =\frac{\left(106-1\right){\left(0.62\right)}^2}{{\left(2.08\right)}^2} \\ =9.329 \end{gathered}$$

The degree of freedom for the test is computed as follows.

$$\begin{gathered} df=n-1 \\ =106-1 \\ =105 \end{gathered}$$

State the critical value 

The test is left-tailed.

The critical value of $P\left({\chi }^2<{{\chi }^2}_{0.01}\right)=0.01$.

From the chi-square table, the critical value is obtained corresponding to the degree of freedom 105 as 70.065.

The decision rule states that if the critical value is greater than the test statistic, reject the null hypothesis; otherwise, fail to reject the null hypothesis.

Conclusion of the test

As the test statistic is less than the critical value, the null hypothesis is rejected.

There is sufficient evidence to support the claim that the body temperatures have a standard deviation of less than .

In Example 5, from sections 8 to 3, the change in the sample standard deviation is unlikely to affect the results in the hypothesis test as the sample changes. This is because the test statistic observed in the example is extreme and will not change significantly to alter the test results.