## 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 $$\stackrel{\tiny \text{approx.}}\sim$$.

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 $$\overline{X}$$? $$\overline{X} \sim$$...