Page 55 - Statistics II for Dummies

Chapter 3: Reviewing Confidence Intervals and Hypothesis Tests
MP3-player owners use their players an average of 2.5 hours per day for 39
music listening? You hope and may assume that the average for the whole
population is close to 2.5, but it probably isn’t exact.

What’s the solution to this problem? The solution is to not only report the
average from your sample but along with it report some measure of how
much you expect that sample average to vary from one sample to the next,
with a certain level of confidence. The number that you use to represent this
level of precision in your results is called the margin of error.

Finding the confidence interval
for a population mean

The sample statistic part of the confidence-interval formula is fairly
straightforward.

✓ To estimate the population mean, you use the sample mean plus or
minus a margin of error, which is based on standard error. The sample
mean has a standard error of . In this formula, you can see the popu-
lation standard deviation (σ) and the sample size (n).
✓ To estimate the population proportion, you use the sample proportion
plus or minus a margin of error.

In many cases, the standard deviation of the population, σ, is not known. To
estimate the population mean by using a confidence interval when σ

is unknown, you use the formula . This formula contains the

sample standard deviation (s), the sample size (n), and a t-value representing
how many standard errors you want to add and subtract to get the confi-
dence you need. To get the margin of error for the mean, you see the

standard error, , is being multiplied by a factor of t. Notice that t has n – 1

as a subscript to indicate which of the myriad t-distributions you use for your
confidence interval. The n – 1 is called degrees of freedom.

The value of t in this case represents the number of standard errors you add
and subtract to or from the sample mean to get the confidence you want. If
you want to be 95 percent confident, for example, you add and subtract 1.96
of those standard errors. If you want to be 99.7 percent confident, you add or
subtract about three of them. (See Table A-1 in the appendix to find t-values
for various confidence levels; use for the area to the
right and find the t-value that goes with it.)

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