Page 217 - Statistics for Dummies

Chapter 13: Confidence Intervals: Making Your Best Guesstimate
Counting On Population Variability
One of the factors influencing variability in sample results is the fact that the
population itself contains variability. For example, in a population of houses
in a fairly large city like Columbus, Ohio, you see a great deal of variety in not
only the types of houses, but also the sizes and the prices. And the variability
in prices of houses in Columbus should be more than the variability in prices 201
of houses in a selected housing development in Columbus.
That means if you take a sample of houses from the entire city of Columbus
and find the average price, the margin of error should be larger than if you
take a sample from that single housing development in Columbus, even if you
have the same confidence level and the same sample size.
Why? Because the houses in the entire city have more variability in price, and
your sample average would change more from sample to sample than it would
if you took the sample only from that single housing development, where
the prices tend to be very similar because houses tend to be comparable in
a single housing development. So you need to sample more houses if you’re
sampling from the entire city of Columbus in order to have the same amount
of accuracy that you would get from that single housing development.

The standard deviation of the population is denoted . Notice that appears
in the numerator of the standard error in the formula for margin of error for
the sample mean: .

Therefore, as the standard deviation (the numerator) increases, the standard
error (the entire fraction) also increases. This results in a larger margin of
error and a wider confidence interval. (Refer to Chapter 11 for more info on
the standard error.)

More variability in the original population increases the margin of error,
making the confidence interval wider. This increase can be offset by increas-
ing the sample size.

Calculating a Confidence Interval

for a Population Mean

When the characteristic that’s being measured (such as income, IQ, price,
height, quantity, or weight) is numerical, most people want to estimate the
mean (average) value for the population. You estimate the population mean,
, by using a sample mean, , plus or minus a margin of error. The result is

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