5.4.1 The Case of a Normal Population Distribution

The simulation experiment of Example 5.23 indicated that when the population distribution is normal, a histogram of $\bar{x}$ values for any sample size $n$ is well approximated by a normal curve.

sum of sample from normal distribution

Let $X_1,X_2,\dots ,X_n$ be a random sample from a normal distribution with

• mean $\mu$
• standard deviation $\sigma$.

Then for any $n$,

• $\bar{X}$ is normally distributed with
  • mean $\mu$
  • standard deviation $\sigma /\sqrt{n}$
• as is $T_o$ with
  • mean $n\mu$
  • standard deviation $\sqrt{n}\sigma$

We know everything there is to know about the $\bar{X}$ and $T_o$ distributions when the population distribution is normal.
In particular, probabilities such as $P\left(a\le \bar{X}\le b\right)$ and $P\left(c\le T_o\le d\right)$ can be obtained simply by standardizing. Figure 5.15 illustrates the $\bar{X}$ part of the proposition.

Figure 5.15 A normal population distribution and $\bar{X}$ sampling distributions

• A proof of the result for $T_o$ when $n=2$ is possible using the method in Example 5.22, but the details are messy. The general result is usually proved using a theoretical tool called a moment generating function. One of the chapter references can be consulted for more information.

EXAMPLE 5.26