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: $N\left(\mu, \frac{\sigma^2}{n}\right) = N\left(5, \frac{1}{40}\right) = N\left(5, 0.025\right).$

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, 6.25) N(7, 2.5) N(7, 0.15625) N(7, 0.0625)

N(7, 0.15625)

4.2 Population distribution unknown and n≥30n \geq 30

0 of 1 correct

4.2 Population distribution unknown and $n \geq 30$