Question

Technology. In Exercises 9–12, test the given claim by using the display provided from technology. Use a 0.05 significance level. Identify the null and alternative hypotheses, test statistic, P-value (or range of P-values), or critical value(s), and state the final conclusion that addresses the original claim.

Body Temperatures Data Set 3 “Body Temperatures” in Appendix B includes 93 body temperatures measured at 12 ²³ on day 1 of a study, and the accompanying XLSTAT display results from using those data to test the claim that the mean body temperature is equal to 98.6°F. Conduct the hypothesis test using these results

Short answer

The hypotheses are as follows.

\[\begin{array}{l}{H_0}:\mu = 98.6^\circ \,{\rm{F}}\\{H_1}:\mu \ne 98.6^\circ \,{\rm{F}}\end{array}\]

The test statistic is -7.102.

The p-value is <0.0001.

The null hypothesis is rejected, and it is concluded that there is not sufficient evidence to support the claim that the population mean of the body temperatures is equal to .

Step-by-step solution

Given information

A sample is taken from body temperatures with a sample size of 93 with the claim that the population mean of the body temperature is equal to \[98.6^\circ \,{\rm{F}}\].

State the hypotheses

The null hypothesis\[{H_0}\]represents the mean body temperature equal to. Also, the alternate hypothesis\[{H_1}\]represents the mean body temperature, which is not equal to.

Let\[\mu \]be the population mean of the body temperatures.

State the null and alternate hypotheses.

\[\begin{array}{l}{H_0}:\mu = 98.6^\circ \,{\rm{F}}\\{H_1}:\mu \ne 98.6^\circ \,{\rm{F}}\end{array}\]

State the test statistic and the p-value from the summary given

State the test statistic and the p-value obtained from the second row and the fourth row of the given output, respectively. The critical value can also be observed from the third row.

\[\begin{array}{c}t\left( {{\rm{observed}}} \right) \approx - 7.102\\{\rm{p - value}}\left( {Two - Tailed} \right) < 0.0001\\t\left( {{\rm{critical}}} \right) \approx 1.986\end{array}\]

State the decision

Reject the null hypothesis when the absolute value of the observed test statistics is greater than the critical value. Otherwise, fail to reject the null hypothesis.

In this case,

\[\begin{array}{c}\left| { - 7.102} \right| = 7.102\\ > 1.986\\t\left( {{\rm{observed}}} \right) > t\left( {{\rm{critical}}} \right)\end{array}\].

The absolute value of the observed test statistic is significantly larger than the critical value. This implies that the null hypothesis is rejected.

Conclusion

As the null hypothesis is rejected, it can be concluded that there is insufficient evidence to support the claim that the population mean of the body temperature is equal to .