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Part IV: Guesstimating and Hypothesizing with Confidence

plus or minus the margin of error from Step 5 to obtain
6. Take
the CI.

minus the margin of error, and the
The lower end of the CI is
upper end of the CI is
plus the margin of error.
The formula shown here for a CI for p – p is used under the condition that
2
1
both of the sample sizes are large enough for the Central Limit Theorem to kick
in and allow us to use a z*-value (see Chapter 11); this is true when you are
estimating proportions using large scale surveys, for example. For small sample
sizes, confidence intervals are beyond the scope of an intro statistics course.
Suppose you work for the Las Vegas Chamber of Commerce, and you want
to estimate with 95% confidence the difference between the percentage of
females who have ever gone to see an Elvis impersonator and the percentage
of males who have ever gone to see an Elvis impersonator, in order to help
determine how you should market your entertainment offerings.
1. Because you want a 95% confidence interval, your z*-value is 1.96.
2. Suppose your random sample of 100 females includes 53 females who
have seen an Elvis impersonator, so is 53 ÷ 100 = 0.53. Suppose also
that your random sample of 110 males includes 37 males who have ever
seen an Elvis impersonator, so is 37 ÷ 110 = 0.34.
3. The difference between these sample proportions (females – males) is
0.53 – 0.34 = 0.19.
4. Take 0.53 ∗ (1 – 0.53) and divide that by 100 to get 0.2491 ÷ 100 = 0.0025.
Then take 0.34 ∗ (1 – 0.34) and divide that by 110 to get 0.2244 ÷ 110 =
0.0020. Add these two results to get 0.0025 + 0.0020 = 0.0045; the square
root is 0.0671.
5. 1.96 ∗ 0.0671 gives you 0.13, or 13%, which is the margin of error.
6. Your 95% confidence interval for the difference between the percentage
of females who have seen an Elvis impersonator and the percentage of
males who have seen an Elvis impersonator is 0.19 or 19% (which you
got in Step 3), plus or minus 13%. The lower end of the interval is 0.19 –
0.13 = 0.06 or 6%; the upper end is 0.19 + 0.13 = 0.32 or 32%.
To interpret these results within the context of the problem, you can
say with 95% confidence that a higher percentage of females than males
have seen an Elvis impersonator, and the difference in these percent-
ages is somewhere between 6% and 32%, based on your sample.

Now I’m thinking there are some guys out there that wouldn’t admit they’d
ever seen an Elvis impersonator (although they’ve probably pretended to be

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