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# 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 p with 95% confidence?'' Answer: .  In applications, we replace p inside the squareroot sign by . For our example, the 95% confidence interval for p is or . Furthermore, we can create an interval with any desired confidence level by replacing the critical value 2 with the appropriate standard normal percentile z.

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 .
b. Construct a 75% 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.
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 .

Next: Sample Size for Estimating Up: Confidence Intervals Previous: Comparing Averages of Two

2003-09-08