Let's compute the required values step by step.

Part (a): Calculate $d_i = X_i - Y_i$

The $d_i$ values for each observation are already provided: $d_1 = -3.5, \, d_2 = -1.6, \, d_3 = -3.3, \, d_4 = -4.7, \, d_5 = -1.4, \, d_6 = -0.1, \, d_7 = -1.4, \, d_8 = -2.1$

Part (b): Compute $\bar{d}$ (mean of $d_i$)

To calculate $\bar{d}$, use the formula: $\bar{d} = \frac{\sum d_i}{n}$ where $n = 8$ (number of observations).

$\sum d_i = (-3.5) + (-1.6) + (-3.3) + (-4.7) + (-1.4) + (-0.1) + (-1.4) + (-2.1)$

$\sum d_i = -18.1$

$\bar{d} = \frac{-18.1}{8} = -2.263$

Thus, $\bar{d} = -2.263$ (rounded to three decimal places).

Part (c): Compute $s_d$ (standard deviation of $d_i$)

To calculate $s_d$, use the formula: $s_d = \sqrt{\frac{\sum (d_i - \bar{d})^2}{n - 1}}$

Step 1: Compute $(d_i - \bar{d})^2$ for each $d_i$

$d_1 - \bar{d} = -3.5 - (-2.263) = -1.237, \quad (-1.237)^2 = 1.530$ $d_2 - \bar{d} = -1.6 - (-2.263) = 0.663, \quad (0.663)^2 = 0.440$ $d_3 - \bar{d} = -3.3 - (-2.263) = -1.037, \quad (-1.037)^2 = 1.076$ $d_4 - \bar{d} = -4.7 - (-2.263) = -2.437, \quad (-2.437)^2 = 5.939$ $d_5 - \bar{d} = -1.4 - (-2.263) = 0.863, \quad (0.863)^2 = 0.745$ $d_6 - \bar{d} = -0.1 - (-2.263) = 2.163, \quad (2.163)^2 = 4.678$ $d_7 - \bar{d} = -1.4 - (-2.263) = 0.863, \quad (0.863)^2 = 0.745$ $d_8 - \bar{d} = -2.1 - (-2.263) = 0.163, \quad (0.163)^2 = 0.027$

Step 2: Compute $\sum (d_i - \bar{d})^2$

$\sum (d_i - \bar{d})^2 = 1.530 + 0.440 + 1.076 + 5.939 + 0.745 + 4.678 + 0.745 + 0.027 = 15.180$

Step 3: Compute $s_d$

$s_d = \sqrt{\frac{15.180}{8 - 1}} = \sqrt{\frac{15.180}{7}} = \sqrt{2.16857} = 1.472$

Thus, $s_d = 1.472$ (rounded to three decimal places).

Final Results:

1. $\bar{d} = -2.263$
2. $s_d = 1.472$

Would you like further details or clarifications on these computations?

Expanding Questions:

1. How would the results change if one additional observation was included?
2. How does the assumption of normality affect the interpretation of these calculations?
3. Can you calculate a confidence interval for $\bar{d}$ given $s_d$ and the sample size?
4. How would you test whether $\bar{d}$ is significantly different from zero?
5. What is the practical interpretation of $\bar{d}$ and $s_d$ in a real-world context?

Tip:

Always check for outliers in your data, as they can heavily influence measures like the mean and standard deviation.