4.1 Population distribution is normal

If it is known that the underlying population distribution is normal, then we can assume:

$\text{If } X \sim N\left(\mu, \sigma^2\right), \text{then } \overline{X}\sim N\left(\mu, \frac{\sigma^2}{n}\right).$

This is true regardless of the sample size.

For example, suppose a random sample of $$n = 20$$ is taken from from an underlying normal population with $$\mu = 5$$ and $$\sigma^2 = 1$$, and a sample mean is calculated. The distribution of the sample mean would therefore be: $\overline{X}\sim N\left(\mu, \frac{\sigma^2}{n}\right) = N\left(5, \frac{1}{20}\right) = N\left(5, 0.05\right).$

1. Suppose a random sample of $$n = 20$$ is taken from a normally distributed population that has $$\mu = 7$$ and $$\sigma^2 = 2.5^2$$. What is the distribution of $$\overline{X}$$? $$\overline{X} \sim$$...
2. Suppose a random sample of $$n = 40$$ is taken from a normally distributed population that has $$\mu = 7$$ and $$\sigma^2 = 2.5^2$$. What is the distribution of $$\overline{X}$$? $$\overline{X} \sim$$...