Q 7.66. - Elementary Statistics

Question

According to the central limit theorem, for a relatively large sample size, the variable $\tilde{x}$ is approximately normally distributed.

a. What rule of thumb is used for deciding whether the sample size is relatively large?

b. Roughly speaking, what property of the distribution of the variable under consideration determines how large the sample size must be for a normal distribution to provide an adequate approximation to the distribution of $\tilde{x}$ ?

Step-by-Step Solution

Answer

Part (a)$\text{ the sample size is greater than }30\text{. }$

Part (b) For a normal distribution to provide an adequate approximation to the distribution of $\overline{x}$ the sample size must be large.

Part (a) Step 1: Given information

$\text{ the variable }\tilde{x}\text{ is approximately normally distributed. }$

Part (a) Step 2: Concept

Formula used: $\text{ population mean and standard deviation: }{\mu }_{\overline{x}}=\mu \text{ and }{\sigma }_{\overline{x}}=\sigma /\sqrt{n}\text{. }$

Part (a) Step 3: Explanation

$\text{ We consider a sample as a large sample if the sample size is greater than }30\text{. }$

Part (b) Step 1: Explanation

The probability density function's symmetry and bell-shapedness determine how large sample size needed to be for a normal distribution to provide a good approximation to the distribution of $\overline{x}$