4.2 Population distribution unknown and \(n \geq 30\)

If the underlying distribution is unknown, but the sample size is large (i.e. \(n \geq 30\)), then, from the Central Limit Theorem, we can assume:

\[ \overline{X}\stackrel{\tiny \text{approx.}}\sim N\left(\mu, \frac{\sigma^2}{n}\right).\]

For example, suppose a random sample of \(n = 40\) is taken from from a population with unknown distribution that has 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 approximately: \[N\left(\mu, \frac{\sigma^2}{n}\right) = N\left(5, \frac{1^2}{40}\right) = N\left(5, 0.025\right).\]

Although, from the CLT, it is known that \(\overline{X}\) approximately follows the above distribution, for ease of notation and without loss of generality, from this point onwards we will use \(\sim\) in place of \(\stackrel{\tiny \text{approx.}}\sim\).

Your turn:

Suppose a random sample of \(n = 40\) is taken from a population with an unknown distribution that has \(\mu = 7\) and \(\sigma^2 = 2.5^2\). What is the distribution of \(\overline{X}\)? \(\overline{X} \sim\)...

N(7, 0.15625)