Page 72 - Intermediate Statistics for Dummies

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Chapter 3: Building Confidence and Testing Models
The general formula for a confidence interval is the following:
Confidence interval = Sample statistic ± Margin of error
The confidence interval has a certain level of precision (measured by the
margin of error). Precision calculates how close you expect your results
to be to the truth.
For example, you want to know the average amount of time a student at Ohio
State University spends listening to music per day, using an MP3 player. The
average time for the entire population of OSU students that are MP3-player
users is the parameter you’re looking for. Certain that you can’t ask every
student who uses an MP3 player at OSU this question, you take a random
sample of students and find the average from there.
Suppose the average time a student uses an MP3 player per day to listen to
music based on a random sample of 1,000 OSU students is 2.5 hours, and the
standard deviation is 0.5 hours. Is it right to say that the population of all 51
OSU-student MP3-player owners use their players an average of 2.5 hours
per day for music listening? No. You hope and may assume that the average
for the whole population is close to 2.5, but it probably isn’t exact. After all,
you’re only sampling a tiny fraction of the 60,000 member population of all
OSU students. The fact is that sample results vary from sample to sample.
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. You want to cover your bases, so to speak
(at least most of the time). The number that you use to represent this level of
precision in your results is called the margin of error. You take your sample
average and add and subtract the margin of error (to get that plus-or-minus
factor going), which gives you a confidence interval for the average time all
OSU students use their MP3 players.
Finding the confidence interval
for a population mean
The sample statistic part of the confidence-interval formula is fairly straight-
forward. If you want to estimate the population mean, you use the sample
mean. If you want to estimate the population proportion, use the sample
proportion. If you want to find the difference of two population means,
take two samples, find their sample means, and subtract them.