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Chapter 3: Building Confidence and Testing Models
What changes the margin of error?
What do you need to know in order to come up with a margin of error?
Margin of error, in general, depends on three elements:
The standard deviation of the population, σ (or an estimate of it,
denoted by s, the sample standard deviation)
The sample size, n
The level of confidence you need
You can see these elements in action in the following formula for margin of
s
error of the sample mean: t n 1- *
. Here I assume that σ isn’t known; t n – 1
n
represents the value on the t-distribution (Table A-1 in the Appendix) with
n – 1 degrees of freedom.
Each of these three elements has a major role in determining how large the 53
margin of error will be when you estimate the mean of a population. At times
it may seem that different elements work against each other (and they do!),
but you can find ways around that. In the following sections, I show how each
of the elements of the margin of error formula work separately and together
to affect the size of the margin of error.
The population standard deviation’s affect on margin of error
The standard deviation of the population is typically combined with the
sample size in the margin of error formula, with the population standard
deviation on top of the fraction, and n in the bottom. (In this case, the
standard error of the population, σ, is estimated by the standard deviation
of the sample, s, because σ is typically unknown.)
This combination of standard deviation of the population and sample size is
known as the standard error of your statistic. It measures how much the sample
statistic deviates from its mean in the long term.
How does the standard deviation of the population (σ) affect margin of error?
As the standard deviation of the population (or its estimate, s) gets larger, the
margin of error increases, so your range of likely values is wider. That’s why
you typically see the population standard deviation in the numerator of
margin of error formulas. The formula for the margin of error for one popula-
tion is an example of this.
Suppose you have two gas stations, one on a busy corner (gas station #1)
and one farther off the main drag (gas station #2). You want to estimate the
average time between customers at each station. At the busy gas station #1,