## 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: \[\overline{X}\sim N\left(\mu, \frac{\sigma^2}{n}\right) = N\left(5, \frac{1}{20}\right) = N\left(5, 0.05\right).\]

**Your turn:**

- 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 \(\overline{X}\)? \(\overline{X} \sim\)...
- 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 \(\overline{X}\)? \(\overline{X} \sim\)...

- N(7, 0.3125)
- N(7, 0.15625)