Advanced High-School Mathematics

394 CHAPTER 6 Inferential Statistics

We can similarly determine sample sizes needed to a given bound on
the margin of error in the case of confidence intervals for proportions,
as follows. In this case the margin of error for a confidence interval

with confidence (1−α)×100% is zα/ 2

Ã

pˆ(1−pˆ)

n

, where, as usual, ˆp

is the sampled population proportion. A very useful approximation is
obtained by noting that since 0 ≤ pˆ ≤ 1, then 0 ≤ pˆ(1−pˆ) ≤^14.
Therefore, if we wish for the margin of error to be less than a given
boundB, all we need is a sample size of at least

n≥

Çz

α/ 2

2 B

å 2

,

because regardless of the sampled value ˆpwe see that

zα/ 2

2 B

≥

zα/ 2

»

pˆ(1−pˆ)

B

Exercises

1. Assume that we need to estimate the mean diameter of a very
   critical bolt being manufactured at a given plant. Previous studies
   show that the machining process results in a standard deviation
   of approximately 0.012 mm. Estimate the sample size necessary
   to compute a 99% confidence interval for the mean bolt diameter
   with a margin of error of no more than 0.003 mm.


2. Assume that a polling agency wishes to survey the voting public to
   estimate the percentage of voters which prefer candidate A. What
   they seek is a sampling error of no more than .02% at a confidence
   level of 98%. Find a minimum sample size which will guarantee
   this level of confidence and precision.

6.5 Hypothesis Testing of Means and Proportions

Suppose we encounter the claim by a manufacturer that the precision
bolts of Exercise 1 above have a mean of 8.1 mm and that we are