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

Chapter 16Confidence Intervals for a Single Proportion

Idea

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.

Problem

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

Solution

We take a single, properly selected, random sample of sufficient size (there must be at least 10 “Y” and at least 10 “N” in the sample).

We calculate for this sample.

We use the standard error of the distribution of to estimate the standard deviation.

Confidence Intervals

A confidence interval for a population parameter X (this could be , , 2, p…) is a range of possible values that has some measure of confidence as to whether the actual value of X is actually in the interval.

If the 95% confidence interval for X is (min, max), this means that we are 95% confident that the true value of X satisfies min < X < max.

Margin of Error

The margin of error (ME) of a confidence interval is half its width.

The higher the confidence, the larger the ME.

The larger the sample size, the smaller the ME.

Confidence Level

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.

Confidence Interval for p

Assumptions/Requirements

We have a properly collected, random sample of size n

We have a large enough sample size (at least 10 “Y” and at least 10 “N” are in the sample)

The sample size is not more than 10% of the total population.

Confidence Interval for p

Formula

p is in the interval:

• z* = 1.645 for 90% confidence

• z* = 1.96 for 95% confidence

• z* = 2.33 for 98% confidence

Example 1

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.

Example 1via technology

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

Example 2

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.

Choosing n

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:

Choosing n

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.

Choosing n

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

Example 3

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 .

Example 4

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.

ME and n Relationship

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 k2n.

Example 5

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?