Page 59 - Statistics II for Dummies

Chapter 3: Reviewing Confidence Intervals and Hypothesis Tests
your value of n. If you sample 500 homes, the margin of error decreases to 43
, which brings you down to $1,314.81.

You can use a formula to find the sample size you need to meet a desired

margin of error. That formula is , where MOE is the desired margin
of error (as a proportion), s is the sample standard deviation, and t is the
value on the t-distribution that corresponds with the confidence level you
want. (For large sample sizes, the t-distribution is approximately equal to the
Z-distribution; you can use the last line of Table A-1 in the appendix for the
appropriate t-values, or use a Z-table from your Stats I textbook.)

Interpreting a confidence interval

Interpreting a confidence interval involves a couple of subtle but important
issues. The big idea is that a confidence interval presents a range of likely
values for the population parameter, based on your sample. However, you
interpret it not in terms of your own sample, but in terms of an infinite
number of other samples out there that could have been selected, yours just
being one of them. For example, suppose 1,000 people each took a sample
and they each formed a 95 percent confidence interval for the mean. The
“95 percent confidence” part means that of those 1,000 confidence intervals,
about 950 of them can be expected to be correct on average. (Correct means
the confidence interval actually contains the true value of the parameter.)

A 95 percent confidence interval doesn’t mean that your particular confidence
interval has a 95 percent chance of capturing the actual value of the parameter;
after the sample has been taken, the parameter is either in the interval or it isn’t.
A confidence interval represents the chances of capturing the actual value of
the population parameter over many different samples.

Suppose a polling organization wants to estimate the percentage of people in the
United States who drive a car with more than 100,000 miles on it, and it wants to
be 95 percent confident in its results. The organization takes a random sample
of 1,200 people and finds that 420 of them (35 percent) drive a car with that mini-
mum mileage; the margin of error turns out to be plus or minus 3 percent. (See
your Stats I text for determining margin of error for percentages.)

The meaty part of the interpretation lies in the confidence level — in this
case, the 95 percent. Because the organization took a sample of 1,200 people
in the U.S., asked each of them whether his or her car has more than 100,000
miles on it, and made a confidence interval out of the results, the polling
organization is, in essence, accounting for all the other samples out there
that it could have gotten by building in the margin of error (± 3 percent). The
organization wants to cover its bases on 95 percent of those other situations,
and ± 3 percent satisfies that.

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