Statistics Essentials For Dummies (22 page)

in the appendix.) The general conversion formula is
.

Substituting the appropriate values of the mean and standard

error of
the conversion formula becomes
.

Suppose
X
is the time it takes a worker to type and send 5 letters of recommendation. Suppose
X
(the times for all the workers) has a normal distribution and the reported mean is 10 minutes and the standard deviation 2 minutes. You take a random sample of 50 workers and measure their times. What is the chance that their average time is less than 9.5 minutes?

This question translates to finding P(
< 9.5). As
X
has a normal distribution to start with, we know
also has a normal

distribution. Converting to
z
-value. we get
.

So we want P(
Z
< -1.77), which equals 0.0384 from the Z-table (Table A-1 in the appendix). So the chance that these 50 randomly selected workers average less than 9.5 minutes to complete this task is 3.84%.

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.

How do you find probabilities for
if
X
is not normal, or is 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 probabilities using the
Z
-table (Table A-1, appendix).)

When you do have to use the CLT to find a probability for
you need to say that your answer is an approximation and that you've got a large enough
n
to proceed because of the CLT. (If
n
is not large enough for the CLT, you use the
t
-distribution in many cases — see Chapter 9.)

The Sampling Distribution of the Sample Proportion