In this chapter we introduce the ideas of confidence intervals and look at how to construct one for a single population proportion.

We would now like to use a single, properly collected sample and its value of to try to predict the value of p for the population from which the sample is drawn. To do this, we would like to make use of the distribution of studied in the previous chapter.

To know the center and spread of the distribution of, we need to know p.

The confidence level is not a measurement of the probability that X is in the interval constructed. It is the percentage of samples of the given size that, if collected, would produce a confidence interval that contains the true value of X.

In a random sample of 100 XU students, 72% said that they “would like more dining choices on campus.” (a) Construct a 90% confidence interval for the proportion of all XU students that would say this. Interpret the results. (b) Construct a 98% confidence interval for the proportion of all XU students that would say this. Interpret the results.

This can be done in the TI by pressing, choosing “TESTS”, then choosing 1-PropZInt

In a random sample of 100 XU students, 72% said that they “would like more dining choices on campus.” Construct a 96.5% confidence interval for the proportion of all XU students that would say this.

Suppose we want to control the width of the confidence interval (likely make it narrower) while at the same time having a fairly high level of confidence. We can do this by selecting a sufficiently large sample. But how large? The margin of error for the confidence interval for a single proportion is:

If we solve this equation for n (using some basic algebra) we get: where z * is determined by the confidence level as earlier. If we have a value of from previous studies, then this formula works fine.

Otherwise, we can get a conservative estimate for n by allowing to get:

If we wanted to construct a 95% confidence interval for the proportion of all XU students that “want more dining choices on campus” with a margin of error no more than 3%, how many students must be polled. Use the 72% from example 1 as the value of.

Suppose that in a sample of 875 XU students, 633 said they “want more dining choices on campus.” Construct a 95% confidence interval for the proportion of all XU students that would feel the same way.

Suppose we have constructed a confidence interval based on a sample of size n. If we wish to construct another confidence interval (with the same confidence level) that has ME = 1 / k times the ME of the original, then we must choose a sample of size k 2 n.

Suppose we have a confidence interval constructed from a sample of size 80, and we want to construct another with ME ½ the size of the first. How large a sample must be randomly selected? What if we want the ME ¼ the size of the first?