To calculate $\overline{d}$ (the mean of the differences) and $s_d$ (the standard deviation of the differences), we proceed as follows:

1. Calculate the differences ($d$) for each pair of temperatures: $d = \text{Temperature at 12 AM} - \text{Temperature at 8 AM}$.

2. Calculate the mean difference ($\overline{d}$): $\overline{d} = \frac{\sum d}{n}$ where $n$ is the number of differences.

3. Calculate the standard deviation ($s_d$): $s_d = \sqrt{\frac{\sum (d - \overline{d})^2}{n - 1}}$

Step-by-step Calculation:

The data is:

The differences ($d$) are: $d = [0.6, 0.3, 0.2, -0.4, 0.3]$

1. Calculate $\overline{d}$:

$\overline{d} = \frac{\sum d}{n} = \frac{0.6 + 0.3 + 0.2 - 0.4 + 0.3}{5} = \frac{1.0}{5} = 0.2$

2. Calculate $s_d$:

First, compute $(d - \overline{d})^2$ for each difference:

[ d - \overline{d} = [0.6 - 0.2, 0.3 - 0.2, 0.2 - 0.2, -0.4 - 0.2, 0.3 - 0.2] = [0.4, 0.1, 0.0, -0.6, 0.1] ]

$(d - \overline{d})^2 = [0.16, 0.01, 0.00, 0.36, 0.01]$

Sum of squares: $\sum (d - \overline{d})^2 = 0.16 + 0.01 + 0.00 + 0.36 + 0.01 = 0.54$

Now calculate $s_d$: $s_d = \sqrt{\frac{\sum (d - \overline{d})^2}{n - 1}} = \sqrt{\frac{0.54}{5 - 1}} = \sqrt{\frac{0.54}{4}} = \sqrt{0.135} \approx 0.367$

Final Results:

• $\overline{d} = 0.2$
• $s_d \approx 0.367$

Interpretation of $\mu_d$:

The parameter $\mu_d$ represents the population mean of the differences between the temperatures at 12 AM and 8 AM. This is the average difference we would expect if we measured all such pairs in the population.

Let me know if you would like further clarification or if you have additional questions!

Expanding Questions:

1. How is the standard deviation $s_d$ used in hypothesis testing for paired samples?
2. Why is the denominator $n - 1$ used instead of $n$ in the standard deviation formula?
3. Can the sign of $d$ indicate a systematic temperature change between the times?
4. How would adding more measurements affect $\overline{d}$ and $s_d$?
5. How could we test if the difference between times is statistically significant?

Tip:

When calculating differences in paired data, always ensure to define the subtraction order clearly to maintain consistency.