# Thread: What is the mean in this problem?

1. ## What is the mean in this problem?

In a particular basketball league, the standard deviation in the distribution
of players' height is 2 inches. Twenty- five players are selected at random
and their heights are measured. Give an estimate for the probability that
the average height of the players in this sample of 25 is within 1 inch of the
league average height

This is a central limit theorem problem.Here variance of 25 players = (S.D.)^2/n
=4/25
and therefore S.D. of 25 players = 2/5

2. you dont need to know the population mean to answer this problem.

$\displaystyle \bar{X} \sim N(\mu, \frac{\sigma^2}{n})$

$\displaystyle \bar{X} - \mu \sim N(0, \frac{\sigma^2}{n})$

You want to find

$\displaystyle \mathbb{P} ( \mu -1 \leq \bar{X} \leq \mu + 1 )$

But this is just:
$\displaystyle \mathbb{P} ( -1 \leq \bar{X} - \mu \leq 1 )$

You know the distribution of $\displaystyle \bar{X} - \mu$. Can you finish from there?

$\displaystyle \mathbb{P} (-1 \leq Y \leq 1)$
where $\displaystyle Y \sim N(0,\frac{\sigma^2}{n})$