Page 190 - Statistics for Dummies

174 Part III: Distributions and the Central Limit Theorem

Don’t forget to divide by the square root of n in the denominator of z. Always
divide by square root of n when the question refers to the average of the
x- values.

Revisiting the clerical worker example from the previous section “Sample size
and standard error,” suppose X is the time it takes a randomly chosen cleri-
cal worker to type and send a standard letter of recommendation. Suppose X
has a normal distribution, and assume the mean is 10.5 minutes and the stan-
dard deviation 3 minutes. You take a random sample of 50 clerical workers
and measure their times. What is the chance that their average time is less
than 9.5 minutes?
This question translates to finding . As X has a normal distribution
to start with, you know also has an exact (not approximate) normal distri-
bution. Converting to z, you get:

So you want P(Z < –2.36), which equals 0.0091 (from the Z-table in the appen-
dix). So the chance that a random sample of 50 clerical workers average less
than 9.5 minutes to complete this task is 0.91% (very small).

How do you find probabilities for if X is not normal, or unknown? As a
result of the CLT , the distribution of X can be non-normal or even unknown
and as long as n is large enough, you can still find approximate probabilities
for using the standard normal (Z-)distribution and the process described
earlier. That is, convert to a z-value and find approximate probabilities using
the Z-table (in the appendix).

When you use the CLT to find a probability for (that is, when the distribu-
tion of X is not normal or is unknown), be sure to say that your answer is an
approximation. You also want to say the approximate answer should be close
because you’ve got a large enough n to use the CLT. (If n is not large enough
for the CLT, you can use the t-distribution in many cases — see Chapter 10.)
Beyond actual calculations, probabilities about can help you decide
whether an assumption or a claim about a population mean is on target, based
on your data. In the clerical workers example, it was assumed that the average
time for all workers to type up a recommendation letter was 10.5 minutes.
Your sample averaged 9.5 minutes. Because the probability that they would
average less than 9.5 minutes was found to be tiny (0.0091), you either got an
unusually high number of fast workers in your sample just by chance, or the
assumption that the average time for all workers is 10.5 minutes was simply
too high. (I’m betting on the latter.) The process of checking assumptions or
challenging claims about a population is called hypothesis testing; details are
in Chapter 14.

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