Page 223 - Statistics for Dummies

Chapter 13: Confidence Intervals: Making Your Best Guesstimate

The formula shown in the preceding example for a CI for p is used under the
condition that the sample size is large enough for the Central Limit Theorem
to kick in and allow us to use a z*-value (see Chapter 11), which happens in
cases when you are estimating proportions based on large scale surveys (see
Chapter 9). For small sample sizes, confidence intervals for the proportion are
typically beyond the scope of an intro statistics course.
For example, suppose you want to estimate the percentage of the time you’re 207
expected to get a red light at a certain intersection.
1. Because you want a 95% confidence interval, your z*-value is 1.96.
2. You take a random sample of 100 different trips through this intersec-
tion and find that you hit a red light 53 times, so .
3. Find .
4. Take the square root to get 0.0499.
The margin of error is, therefore, plus or minus 1.96 ∗ (0.0499) = 0.0978,
or 9.78%.
5. Your 95% confidence interval for the percentage of times you will ever
hit a red light at that particular intersection is 0.53 (or 53%), plus or
minus 0.0978 (rounded to 0.10 or 10%). (The lower end of the interval is
0.53 – 0.10 = 0.43 or 43%; the upper end is 0.53 + 0.10 = 0.63 or 63%.)
To interpret these results within the context of the problem, you can say
that with 95% confidence the percentage of the times you should expect
to hit a red light at this intersection is somewhere between 43% and
63%, based on your sample.

While performing any calculations involving sample percentages, use the
decimal form. After the calculations are finished, convert to percentages by
multiplying by 100. To avoid round-off error, keep at least 2 decimal places
throughout.

Creating a Confidence Interval for

the Difference of Two Means

The goal of many surveys and studies is to compare two populations, such
as men versus women, low versus high income families, and Republicans
versus Democrats. When the characteristic being compared is numerical (for
example, height, weight, or income), the object of interest is the amount of
difference in the means (averages) for the two populations.

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