Body Temperatures Listed below are body temperatures from six different subjects measured at two different times in a day (from Data Set 5 “Body Temperatures” in Appendix B).


b. Identify the null and alternative hypotheses for using the sample data to test the claim that the differences between 8 AM temperatures and 12 AM temperatures are from a population with a mean equal to 0°F

Verified step by step guidance

1. Step 1: Understand the problem. The goal is to test the claim that the differences between body temperatures measured at 8 AM and 12 AM are from a population with a mean equal to 0°F. This involves setting up null and alternative hypotheses for a paired sample test.
2. Step 2: Define the null hypothesis (H₀). The null hypothesis states that the mean difference between the 8 AM and 12 AM temperatures is equal to 0°F. Mathematically, this can be expressed as: $H₀:{\mu }_d=0$, where ${\mu }_d$ represents the mean difference.
3. Step 3: Define the alternative hypothesis (H₁). The alternative hypothesis states that the mean difference between the 8 AM and 12 AM temperatures is not equal to 0°F. Mathematically, this can be expressed as: $H₁:{\mu }_d\ne 0$.
4. Step 4: Identify the type of test. Since the data involves paired measurements (temperatures at 8 AM and 12 AM for the same subjects), a paired t-test is appropriate for testing the hypothesis.
5. Step 5: Calculate the differences between the paired temperatures for each subject. For each pair, subtract the 12 AM temperature from the 8 AM temperature. Then, compute the mean and standard deviation of these differences to proceed with the paired t-test.

Key Concepts

Null Hypothesis (H0)

The null hypothesis is a statement that there is no effect or no difference, and it serves as a starting point for statistical testing. In this context, the null hypothesis would assert that the mean difference between the body temperatures at 8 AM and 12 AM is equal to 0°F, indicating no significant change in temperature over the time period.

Alternative Hypothesis (H1)

The alternative hypothesis is a statement that contradicts the null hypothesis, suggesting that there is an effect or a difference. For this scenario, the alternative hypothesis would propose that the mean difference between the body temperatures at 8 AM and 12 AM is not equal to 0°F, indicating a significant change in temperature.

Mean Difference

The mean difference refers to the average of the differences between paired observations—in this case, the body temperatures measured at 8 AM and 12 AM. Calculating this mean difference is essential for hypothesis testing, as it helps determine whether the observed changes in temperature are statistically significant or could have occurred by random chance.