To solve this problem, we need to find the values of d and sd, where:

• $d$ represents the mean of the differences between the paired data (the temperatures at 8 AM and 12 AM).
• $sd$ is the standard deviation of these differences.
• $\mu_d$ represents the population mean of the differences between the two time points.

Step 1: Find the Differences $d_i$

Let’s first calculate the difference between each subject's temperature at 8 AM and 12 AM. For each subject, we calculate:

$d_i = ( \text{Temperature at 12 AM} ) - ( \text{Temperature at 8 AM} )$

Here are the differences for each subject:

$d_1 = 98.6 - 98.2 = 0.4$ $d_2 = 99.5 - 99.2 = 0.3$ $d_3 = 97.7 - 97.3 = 0.4$ $d_4 = 97.6 - 97.7 = -0.1$ $d_5 = 97.3 - 97.2 = 0.1$

So, the list of differences $d$ is: $[0.4, 0.3, 0.4, -0.1, 0.1]$.

Step 2: Find the Mean of the Differences $d$

Now, we calculate the mean difference $d$. The formula for the mean is:

$\bar{d} = \frac{\sum d_i}{n}$

Where $n = 5$ is the number of subjects. The sum of the differences is:

$\sum d_i = 0.4 + 0.3 + 0.4 + (-0.1) + 0.1 = 1.1$

Thus, the mean of the differences is:

$\bar{d} = \frac{1.1}{5} = 0.22$

Step 3: Find the Standard Deviation of the Differences $sd$

The formula for the standard deviation of the differences is:

$s_d = \sqrt{\frac{\sum (d_i - \bar{d})^2}{n - 1}}$

First, calculate the squared differences from the mean for each $d_i$:

$(d_1 - \bar{d})^2 = (0.4 - 0.22)^2 = 0.0316$ $(d_2 - \bar{d})^2 = (0.3 - 0.22)^2 = 0.0064$ $(d_3 - \bar{d})^2 = (0.4 - 0.22)^2 = 0.0316$ $(d_4 - \bar{d})^2 = (-0.1 - 0.22)^2 = 0.1024$ $(d_5 - \bar{d})^2 = (0.1 - 0.22)^2 = 0.0144$

Now, sum these squared differences:

$\sum (d_i - \bar{d})^2 = 0.0316 + 0.0064 + 0.0316 + 0.1024 + 0.0144 = 0.1864$

Now calculate the standard deviation:

$s_d = \sqrt{\frac{0.1864}{5 - 1}} = \sqrt{\frac{0.1864}{4}} = \sqrt{0.0466} \approx 0.2159$

Conclusion:

• The mean difference $\bar{d}$ is 0.22.
• The standard deviation of the differences $s_d$ is approximately 0.2159.
• $\mu_d$ represents the population mean of the differences between the two time points (the average difference in body temperature between 8 AM and 12 AM for the entire population).

Would you like further details or have any questions?

Here are 5 related questions to explore further:

1. How would rounding the data affect the calculated values of $d$ and $sd$?
2. How do you interpret the positive mean difference $\bar{d}$ in this context?
3. Can the standard deviation ever be larger than the mean of the differences?
4. What additional insights could be gained by comparing temperatures at multiple times of the day?
5. How would the results change if more subjects were added to the study?

Tip: Always check whether your sample size affects the reliability of your standard deviation, especially in small samples.