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:

1. 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 $\overline{X}$? $\overline{X} \sim$...
   N(7, 0.125) N(7, 0.3125) N(7, 6.25) N(7, 2.5)
2. 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, 0.0625) N(7, 6.25) N(7, 2.5) N(7, 0.15625)

1. N(7, 0.3125)
2. N(7, 0.15625)

0 of 2 correct

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These notes have been prepared by Amanda J. Shaker. The copyright for the material in these notes resides with the author named above, with the Department of Mathematics and Statistics and with La Trobe University. Copyright in this work is vested in La Trobe University including all La Trobe University branding and naming. Unless otherwise stated, material within this work is licensed under a Creative Commons Attribution-Non Commercial-Non Derivatives License CC BY-NC-ND