To find the values of $\bar{d}$ (mean difference) and $s_d$ (standard deviation of the differences), we'll go through the following steps:

1. Calculate the difference $d$ between each pair of temperatures (Temperature at 12 AM - Temperature at 8 AM).
2. Calculate the mean of these differences, $\bar{d}$.
3. Calculate the standard deviation of these differences, $s_d$.

Step 1: Compute the differences $d$

Given dataTo find the values of $\bar{d}$ (mean difference) and $s_d$ (standard deviation of the differences), we'll go through the following steps:

1. Calculate the difference $d$ between each pair of temperatures (Temperature at 12 AM - Temperature at 8 AM).
2. Calculate the mean of these differences, $\bar{d}$.
3. Calculate the standard deviation of these differences, $s_d$.

Step 1: Compute the differences $d$

Given

• Temperature at 8 AM: $[98.3, 98.9, 97.6, 97.4, 97.5]$
• Temperature at 12 AM: $[98.9, 99.7, 98.1, 96.9, 97.9]$

Calculating each difference $d = \text{Temperature at 12 AM} - \text{Temperature at 8 AM}$:

$d_1 = 98.9 - 98.3 = 0.6$ $d_2 = 99.7 - 98.9 = 0.8$ $d_3 = 98.1 - 97.6 = 0.5$ $d_4 = 96.9 - 97.4 = -0.5$ $d_5 = 97.9 - 97.5 = 0.4$

So, the list of differences $d$ is: $[0.6, 0.8, 0.5, -0.5, 0.4]$.

Step 2: Calculate $\bar{d}$

$\bar{d} = \frac{0.6 + 0.8 + 0.5 - 0.5 + 0.4}{5} = \frac{1.8}{5} = 0.36$

Thus, $\bar{d} = 0.36$.

Step 3: Calculate $s_d$

To find $s_d$, we use the formula for the sample standard deviation of differences:

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

Calculating each squared deviation:

$(d_1 - \bar{d})^2 = (0.6 - 0.36)^2 = 0.0576$ $(d_2 - \bar{d})^2 = (0.8 - 0.36)^2 = 0.1936$ $(d_3 - \bar{d})^2 = (0.5 - 0.36)^2 = 0.0196$ $(d_4 - \bar{d})^2 = (-0.5 - 0.36)^2 = 0.7396$ $(d_5 - \bar{d})^2 = (0.4 - 0.36)^2 = 0.0016$

Sum of squared deviations:

$\sum (d_i - \bar{d})^2 = 0.0576 + 0.1936 + 0.0196 + 0.7396 + 0.0016 = 1.012$

Now, divide by $n - 1 = 4$:

$s_d = \sqrt{\frac{1.012}{4}} = \sqrt{0.253} \approx 0.503$

Final Answers

• $\bar{d} = 0.36$
• $s_d \approx 0.503$

Interpretation of $\mu_d$

The symbol $\mu_d$ represents the population mean difference, which is the average difference between the two sets of temperatures (8 AM and 12 AM) across the entire population, not just the sample provided.

Would you like further details on the calculation steps, or do you have any questions?

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Questions for Further Exploration:

1. What is the interpretation of a positive vs. a negative mean difference $\bar{d}$?
2. How would changing the sample size affect the accuracy of $\bar{d}$ and $s_d$?
3. How is the standard deviation of differences useful in hypothesis testing?
4. What assumptions are required for interpreting $\mu_d$ as the true mean difference?
5. How can $s_d$ inform confidence intervals for $\bar{d}$?

Tip: When interpreting sample statistics like $\bar{d}$ and $s_d$, remember that they estimate population parameters and reflect sample variability.