4.1 Population distribution is normal
If it is known that the underlying population distribution is normal, then, from the Central Limit Theorem, it is straightforward to ascertain the distribution of the sample mean. In this instance, 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. From the Central Limit Theorem, the distribution of the sample mean would be: \[\overline{X}\sim N\left(\mu, \frac{\sigma^2}{n}\right) = N\left(5, \frac{1^2}{20}\right) = N\left(5, 0.05\right).\]
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)