Listed below are body temperatures from seven subjects...

Listed below are body temperatures from seven subjects measured at two different times in a day (from Data Set 3 "Body Temperatures" in Appendix B). a. Use a $0.05$ significance level to test the claim that there is no difference between body temperatures measured at $8 \mathrm{AM}$ and at $12 \mathrm{AM}$. b. Construct the confidence interval that could be used for the hypothesis test described in part (a). What feature of the confidence interval leads to the same conclusion reached in part (a)? $$ \begin{array}{|c|c|c|c|c|c|c|c|} \hline \text { Body Temperature }\left({ }^{\circ} \mathrm{F}\right) \text { at } 8 \text { AM } & 96.6 & 97.0 & 97.0 & 97.8 & 97.0 & 97.4 & 96.6 \\ \hline \text { Body Temperature }\left({ }^{\circ} \mathrm{F}\right) \text { at } 12 \mathrm{AM} & 99.0 & 98.4 & 98.0 & 98.6 & 98.5 & 98.9 & 98.4 \\ \hline \end{array} $$

Listed below are body temperatures from seven subjects measured at two different times in a day (from Data Set 3 "Body Temperatures" in Appendix B).
a. Use a $0.05$ significance level to test the claim that there is no difference between body temperatures measured at $8 \mathrm{AM}$ and at $12 \mathrm{AM}$.
b. Construct the confidence interval that could be used for the hypothesis test described in part
(a). What feature of the confidence interval leads to the same conclusion reached in part (a)?
$$
\begin{array}{|c|c|c|c|c|c|c|c|}
\hline \text { Body Temperature }\left({ }^{\circ} \mathrm{F}\right) \text { at } 8 \text { AM } & 96.6 & 97.0 & 97.0 & 97.8 & 97.0 & 97.4 & 96.6 \\
\hline \text { Body Temperature }\left({ }^{\circ} \mathrm{F}\right) \text { at } 12 \mathrm{AM} & 99.0 & 98.4 & 98.0 & 98.6 & 98.5 & 98.9 & 98.4 \\
\hline
\end{array}
$$

Key Concepts

Significance Level

The significance level, often set at 0.05, represents the threshold for rejecting the null hypothesis in a hypothesis test. It quantifies the probability of committing a Type I error, which means incorrectly concluding that there is a difference when none exists, and it plays a critical role in interpreting both hypothesis test outcomes and the associated confidence intervals.

Confidence Interval for the Mean Difference

A confidence interval for the mean difference provides a range of values within which the true difference in the population is likely to lie, with a specified level of confidence (commonly 95%). If this interval does not include the value zero, it supports the conclusion that a statistically significant difference exists between the paired measurements.

Paired Samples t-Test

A paired samples t-test is used when comparing two related groups, such as measurements taken from the same subjects at different times. This test evaluates whether the mean difference between paired observations is statistically significantly different from zero, taking into account the paired nature of the data and reducing the variability inherent in differences.

Hypothesis Testing

Hypothesis testing involves formulating a null hypothesis, typically stating that there is no difference between the paired measurements, against an alternative hypothesis that suggests a difference exists. The process includes calculating a test statistic that, combined with a significance level, helps determine whether to reject the null hypothesis.

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