Estimating the Population Mean Using Intervals

Problem revisited: Estimate the average GPA of the population of approximately 23000 WMU undergraduates.

Consider estimating the
population average by the sample average of, say *n*=25
randomly selected students.
Suppose this sample average, which we denote by
,
equals 3.05.
Now chances are the true average
is not equal to 3.05.
For sure, the true WMU average GPA is between 1.00 and 4.00,
and with high confidence between (2.50, 3.50); but what
level of confidence do we have that it is between say,
(2.75, 3.25) or (2.95, 3.15)? Even better, can we find
an interval (*a*, *b*) which will contain
with , say, 95% certainty?

Recall that if is the standard deviation for individuals,
then
is the standard deviation for averages
(also called the SE of the sample average).
More precisely, *n*-member averages
form an approximate normal
histogram with mean
and standard error
.
This means that there is 68% likelihood that the sample average
falls within
of ,
and 95% likelihood that
the sample average falls within
of ;
see figure below.

Selected areas under histogram for

In statistical shorthand, we write

where is read as ``the probability that ".

Of course, the distance of
from
is also the
distance of
from
,
so it is equally true that

Recall the question posed earlier: ``Can we find an interval which we know contains with 95% confidence?'' Answer: .

**Example 1:**Suppose that the SD of student GPA is .30. If a random sample of*n*=25 students has average GPA equal to 3.05, then we are 95% confident that is contained within or (2.93, 3.17).

Replacing
by
in Equation ( 7.1)
results in a 68% confidence interval. In general, we can create an
interval with any desired confidence level by replacing the
multiplier with the appropriate standard normal percentile *z*
(also called the *z*-**critical value**).

where

The following table gives the
appropriate critical value *z* for typical values of .

**Example 1 (con't.):**

a. Construct a 99% confidence interval for WMU GPA.

b. Construct a 75% confidence interval for WMU GPA.

c. Construct a 95.4% confidence interval for WMU GPA.

d. Construct a 95.0% confidence interval for WMU GPA.

Theoretically speaking, has a confidence level of 95.4%. Applied statisticians usually just round off and say 95%. We can, of course, choose the more precise critical value of 1.96 to get an exact 95% confidence level. However, normality of is at best approximate anyway, so the precision seems spurious. However, the distinction is important enough to merit a reminder.