Page 213 - Statistics for Dummies

Chapter 13: Confidence Intervals: Making Your Best Guesstimate
You know that this process will result in intervals that capture the popula-
tion mean 95% of the time. The other 5% of the time, the data collected in the
sample just by random chance has abnormally high or low values in it and
doesn’t represent the population. This 5% measures errors due to random
chance only and doesn’t include bias.
The margin of error is meaningless if the data that went into the study were
biased and/or unreliable. However, you can’t tell that by looking at anyone’s 197
statistical results. My best advice is to look at how the data were collected
before accepting a reported margin of error as the truth (see Chapters 16 and
17 for details on data collection issues). That means asking questions before
you believe a study.
Zooming In on Width
The width of your confidence interval is two times the margin of error. For
example, suppose the margin of error is ± 5%. A confidence interval of 7%, plus
or minus 5%, goes from 7% – 5% = 2%, all the way up to 7% + 5% = 12%. So the
confidence interval has a width of 12% – 2% = 10%. A simpler way to calculate
this is to say that the width of the confidence interval is two times the margin
of error. In this case, the width of the confidence interval is 2 ∗ 5% = 10%.
The width of a confidence interval is the distance from the lower end of the
interval (statistic minus margin of error) to the upper end of the interval (sta-
tistic plus margin of error). You can always calculate the width of a confidence
interval quickly by taking two times the margin of error.
The ultimate goal when making an estimate using a confidence interval is
to have a narrow width, because that means you’re zooming in on what the
parameter is. Having to add and subtract a large margin of error only makes
your result much less accurate.

So, if a small margin of error is good, is smaller even better? Not always. A
narrow confidence interval is a good thing — to a point. To get an extremely
narrow confidence interval, you have to conduct a much larger — and
expensive — study, so a point comes where the increase in price doesn’t jus-
tify the marginal difference in accuracy. Most people are pretty comfortable
with a margin of error of 2% to 3% when the estimate itself is a percentage
(like the percentage of women, Republicans, or smokers).

How do you go about ensuring that your confidence interval will be narrow
enough? You certainly want to think about this issue before collecting your
data; after the data are collected, the width of the confidence interval is set.

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