Advanced High-School Mathematics

SECTION 6.4 Confidence Intervals 385

2. Suppose that you go out and collect 50 samples of the random
   variable 4×rand and compute the mean x. Compute the 95%
   confidence interval so obtained. Does it contain the true meanμ?
   (See Exercise 1, above.)


3. We can build on Exercise 2, as follows. The following simple TI
   code can be used to count how many out of 100 95% confidence
   intervals for the meanμof the random variable 4*randwill actually
   contain the true mean (= 2):

PROGRAM: CONFINT

:0→C

:For(I,1,100)

:4*rand(50)→L 1

:mean(L 1 )→M

:M−. 32 →L

:M+. 32 →U

:C+ (L≤2)(2≤U)→C

:END

:Disp C

:Stop

(a) What is the number .32?

(b) What isC trying to compute?

(c) Run this a few times and explain what’s going on.

6.4.2 Confidence intervals for the mean; unknown variance

In this section we shall develop a method for finding confidence intervals
for the meanμof a population when we don’t already know the variance
σ^2 of the population. In the last section our method was based on the

fact that the statistic
X−μ
σ

was approximately normally distributed.

In the present section, since we don’t knowσ, we shall replaceσ^2 with