Estimating the Population Proportion Using Intervals

Consider estimating the proportion of WMU undergraduates whose
permanent residence is in Southwest Michigan. We may take a random sample
of, say, *n*=25
students and compute the sample proportion .
If 9 out of the 25 students come from Southwest Michigan, then
.
Because of the ``luck of the draw'',
chances are the true proportion *p* is not exactly .36, but some number close
to .36. Can we find
an interval which we know contains *p* with say, at least
95% confidence?

Recall that the sample proportion
over different groups of 25
has a histogram that would look approximately normal with mean
*p* and SD
.
This means that there is
95% likelihood that
falls within
of *p*. Of course, this also means there is
95% likelihood that *p* falls within
of .
In statistical shorthand,

Recall the question posed earlier: ``Can we find an interval which we know contains

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

Example:

1. Suppose that 9 out of 25 randomly selected students live in Southwest Michigan.

a. Construct a 99% confidence interval for the true proportion of WMU students who live in Southwest Michigan .2. Suppose that 36 out of 100 randomly selected students live in Southwest Michigan.

b. Construct a 75% confidence interval for the true proportion of WMU students who live in Southwest Michigan .

a. Construct a 99% confidence interval for the true proportion of WMU students who live in Southwest Michigan .

b. Construct a 75% confidence interval for the true proportion of WMU students who live in Southwest Michigan .