7.3 Using the central limit theorem (modified r. bloom)

It is important to understand when to use the CLT . Use the CLT for means or averages when you are asked to find the probability for a sample average or mean, or when working with percentiles for sample averages. (If youare being asked to find the probability or percentile of a sum or total, use the CLT for sums.)

Law of large numbers

The Law of Large Numbers says that if you take samples of larger and larger size from any population, then the mean $\overline{x}$ of the sample gets closer and closer to $\mu$ . From the Central Limit Theorem, we know that as $n$ gets larger and larger, the sample averages follow a normal distribution. The larger n gets, the smaller thestandard deviation gets. (Remember that the standard deviation for $\overline{X}$ is $\frac{\sigma }{\sqrt{n}}$ .) This means that the sample mean $\overline{x}$ must be close to the population mean $\mu$ . We can say that $\mu$ is the value that the sample averages approach as $n$ gets larger. The Central Limit Theorem illustrates the Law of Large Numbers.

Suppose that a market research analyst for a cell phone company conducts a study of their customers who exceed the time allowance included on their basic cell phone contract; the analyst finds that for those people who exceed the time included in their basic contract, the excess time used follows an exponential distribution with a mean of 22 minutes.

Consider a random sample of 80 customers who exceed the time allowance included in their basic cell phone contract.

Let $X$ = the excess time used by one INDIVIDUAL cell phone customer who exceeds his contracted time allowance.

$X$ ~ $\mathrm{Exp}\left(\frac{1}{22}\right)$ From Chapter 5, we know that $\mu =22$ and $\sigma =22$ .

Let $\overline{X}$ = the AVERAGE excess time used by a sample of $n=80$ customers who exceed their contracted time allowance.

$\overline{X}$ ~ $N(22,\frac{22}{\sqrt{80}})$ by the CLT for Sample Means or Averages