Advanced High-School Mathematics

SECTION 6.4 Confidence Intervals 393

margin of error also decreases the confidence level. A natural question
to ask is whether we can decrease the margin of errorwithout at the
same time sacrificing confidence? The answer is yes: byincreasing the
sample size. We flesh this out in the following example.

Example. Suppose that we are interested in the average costμ of a
new house in the United States in 1966, and that a random selection
of the cost of 50 homes revealed the 95% confidence interval

$20, 116 ≤μ ≤$30, 614 ,

along with the sample mean x ≈ $25,365, and estimate σ ≈ sx =
$18,469. If we use this as an estimate of the population standard
deviation σ, then we see that a (1−α)×100% confidence interval
becomes

x−zα/ 2

σ

√

n

≤μ≤x+zα/ 2

σ

√

n

.

We see also that the margin of error associated with the above estimate
is one-half the width of the above interval, viz., $5,249.

Question: Suppose that we wish to take a new sample of new houses
and obtain a confidence interval forμ with the same level of confidence
(95%) but with a margin of error of at most$3, 000?

Solution. This is easy, for we wish to choose nto make the margin
of error no more than $3,000:

z. 025

σ

√

n

≤ $3, 000.

Usingz. 025 = 1.96 andσ≈$18,469 we quickly arrive at

n ≥

( 1. 96 × 18 , 469

3 , 000

) 2

≈ 146.

That is to say, if we take a sample of at least 146 data, then we will have
narrowed to margin of error to no more than $3,000 without sacrificing
any confidence.