Answer to Question #258159 in Statistics and Probability for @1111mae

Question

Random sample of n=65 measurements is obtained from a population with µ=20 and σ²=400. Describe the sampling distribution for the sample means by computing µ˜x and σ2˜x. Consider a sampling without replacement.

Answer 1

The Central Limit Theorem

Let $X_1, X_2, . . . , X_n$ be a random sample from a distribution with mean $\mu$ and variance $\sigma^2.$ Then if $n$ is sufficiently large, $\bar{X}$ has approximately a normal distribution with $\mu_{\bar{X}}=\mu$ and $\sigma_{\bar{X}}^2=\sigma^2/n.$ The larger the value of $n,$ the better the approximation.

If $n > 30,$ the Central Limit Theorem can be used.

We have $n=65>30.$ Then the Central Limit Theorem can be used

$$\mu_{\bar{X}}=\mu=20$$ $$\sigma^2_{\bar{X}}=\sigma^2/n=400/65$$

The sample mean $\bar{X}\sim N(20, 400/65)$