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Chapter 8, continued...
III. Interpretation of Confidence
Intervals
Remember, we don’t know the population mean. We
take a sample to estimate µ, then construct a
confidence interval (CI) to provide some measure
of accuracy for that estimate.
An accurate interpretation for a 95% CI:
“Before sampling, there is a 95% chance that the
 will include µ.
interval:
x 196
.
n
More interpretation.
In other words, if 100 samples are taken, each of size
n, on average 95 of these intervals will contain µ.
Important: this statement can only be made before
we sample, when x-bar is still an undetermined
random variable. After we sample, x-bar is no
longer a random variable, thus there is no
probability.
An example of interpretation.
Suppose that the CJW company samples 100
customers and finds this month’s customer service
mean is 82, with a population standard deviation
of 20. We wish to construct a 95% confidence
interval. Thus, =.05 and z.025=1.96.
Before vs. After sampling
• Before we sample, there is a 95% chance that µ

will be in the interval:
x 196
.
n
• After sampling we create an interval:
82 ± 3.92, or (78.08 to 85.92).
We can only say that under repeated sampling, 95%
of similarly constructed intervals would contain
the true µ . This one particular interval may or
may not contain µ .
IV. Interval Estimate of µ: Small
Sample
A small sample is one in which n<30. If the
population has a normal probability distribution,
we can use the following methods. However, if
you can’t assume the normal population, you must
increase n30 so the Central Limit Theorem can
be invoked.
A. The t-distribution
William Sealy Gosset (“student”) founded the tdistribution. An Oxford graduate in math and
chemistry, he worked for Guinness Brewing in
Dublin and developed a new small-sample theory
of statistics while working on small-scale
materials and temperature experiments. “The
probable error of a mean” was published in 1908,
but it wasn’t until 1925 when Sir Ronald A. Fisher
called attention to it and its many applications.
The idea behind the t.
Specific t-distributions are associated with a different
degree of freedom.
Degree of freedom: the # of observations allowed to
vary in calculating a statistic = n-1.
As the degrees of freedom increase (n), the closer
the t-distribution gets to the standard normal
distribution.
B. An Example.
Suppose n=20 and you are constructing a 99%
(=.01) confidence interval.
x  t /2
s
n
First we need to be able to read a t-table to find t.005.
See Table 8.3 in the text.
The t-table.
I see where
you’re going!
/2
0
t/2
We need to find t.005 with 19 degrees of freedom in a ttable like Table 8.3.
Our Example
D.F.
.
.
.
.
18
.10
.05
.01
.025
.005
How do I get back to
the brewery?
19 1.328 1.729 2.093 2.539 2.861