Page 57 - Statistics II for Dummies

Chapter 3: Reviewing Confidence Intervals and Hypothesis Tests
This combination of standard deviation of the population and sample size 41
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 it gets larger, the margin of error increases, so your range of likely values is
wider.
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, customers are constantly using the gas pumps, so you basically have no
downtime between customers. At gas station #2, customers sometimes come
all at once, and sometimes you don’t see a single person for an hour or more.
So the time between customers varies quite a bit.
For which gas station would it be easier to estimate the overall average time
between customers as a whole? Gas station #1 has much more consistency,
which represents a smaller standard deviation of time between customers.
Gas station #2 has much more variability in time between customers. That
means σ for gas station #1 is smaller than σ for gas station #2. So the average
time between customers is easier to estimate at gas station #1.

Sample size
Sample size affects margin of error in a very intuitive way. Suppose you’re
trying to estimate the average number of pets per household in your city.
Which sample size would give you better information: 10 homes or 100
homes? I hope you’d agree that 100 homes would give more precise informa-
tion (as long as the data on those 100 homes was collected properly).

If you have more data to base your conclusions on and that data is collected
properly, your results will be more precise. Precision is measured by margin
of error, so as the sample size increases, the margin of error of your estimate
goes down.
Bigger is only better in terms of sample size if the data is collected properly —
that is, with minimal bias. If the quality of the data can’t be maintained with a
larger sample size, it does no good to have it.

Confidence level
For each problem at hand, you have to address how confident you need to
be in your results over the long term, and, of course, more confidence comes
with a price in the margin of error formula. This level of confidence in your
results over the long term is reflected in a number called the confidence level,
which you report as a percentage. In general, more confidence requires a
wider range of likely values. So, as the confidence level increases, so does the
margin of error.

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