The purpose of this question is to compare the variability of $\overline{x}_1$ and $\overline{x}_2$ with the variability of $(\overline{x}_1 - \overline{x}_2)$.

(a) $\ $ Suppose the first sample of $100$ observations is selected from a population with mean $\mu_1 = 160$ and variance $\sigma_1^2 = 960$. Construct an interval extending 2 standard deviations of $\overline{x}_1$ on each side of $\mu_1$.

$\leq \mu_1 \leq$

(b) $\ $ Suppose the second sample of $100$ observations is selected from a population with mean $\mu_2 = 160$ and variance $\sigma_2^2 = 1260$. Construct an interval extending 2 standard deviations of $\overline{x}_2$ on each side of $\mu_2$.

$\leq \mu_2 \leq$

(c) $\ $ Consider the difference between the two sample means $(\overline{x}_1 - \overline{x}_2)$. Compute the mean and the standard deviation of the sampling distribution of $(\overline{x}_1 - \overline{x}_2)$.
mean =
standard deviation =

(d) $\ $ Based on $100$ observations, construct an interval extending 2 standard deviations of $(\overline{x}_1 - \overline{x}_2)$ on each side of $(\mu_1 - \mu_2)$

$\leq (\mu_1 - \mu_2) \leq$