Advanced High-School Mathematics

384 CHAPTER 6 Inferential Statistics



x−zα/ 2 √σ

n

,x+zα/ 2

σ

√

n



.

Furthermore, we expect that (1−α)×100 percent of the intervals so
constructed will contain the true population mean.

Note, finally, that as the confidence level rises, the width of the
confidence interval also increases. This is obvious for the wider the
interval, the more confident we should be that it will “capture” the
true population mean!

Exercises

1. Suppose that we are going to sample the random variable X =
   4 ×rand. Is this a normal random variable? What is the mean
   and variance of X? Suppose that we instead sample X, where
   X= 4×randandXis computed by taking 50 independent samples
   and forming the average. IsXclose to being normally distributed?
   To help in answering this question, write the simple TI code into
   your calculator

PROGRAM: NORMCHECK

:{ 0 }→L 1

:For(I,1,100)

:4*rand(50)→L 2

:mean(L 2 )→L 1 (I)

:END

A moment’s thought reveals that this program collects 100 samples

of X, where each mean is computed from 50 samples each and

putting the result into list variableL 1. Finally draw a histograms

of these 100 samples of the mean; does it look normal? This

little experiment is tantamount to sending out 100 statisticians

and having each collecting 50 independent samples and computing

the mean. The statisticians all return to combine their results into

a single histogram.