Page 56 - Statistics II for Dummies

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Part I: Tackling Data Analysis and Model-Building Basics
If you know the population standard deviation, you should certainly use it.
In that case, you use the corresponding number from the Z-distribution
(standard normal distribution) in the confidence interval formula. (The
Z-distribution from your Stats I textbook can give you the numbers you need.)
But I would be remiss in saying that while textbooks and teachers always
include problems where σ is known, rarely is σ known in the real world. Why
teach it this way? This issue is up for debate; for now just go with it, and I can
keep you posted.

For the MP3 player example from the preceding section, a random sample of
1,000 all OSU students spends an average of 2.5 hours using their MP3 players
to listen to music. The standard deviation is 0.5 hours. Plugging this informa-
tion into the formula for a confidence interval, you get . You

conclude that all OSU-student MP3-owners spend an average of between 2.47
and 2.53 hours listening to music on their players.

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 formula for margin of error of the
sample mean: . Here I assume that σ isn’t known; t represents the
n – 1
value on the t-distribution table (see 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
margin of error will be when you estimate the mean of a population. 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.

Population standard deviation
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 on 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.)

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