Confidence Interval for a Proportion

This lesson describes how to create a confidence interval for a population proportion.

Checking Conditions

Before we can create a confidence interval for a population proportion, we first need to make sure the following conditions are met to ensure that our confidence interval will be accurate:

  • Random: A random sample or random experiment should be used to collect the data.
  • Normal: The sampling distribution of the sample proportion needs to be approximately normal – need at least 10 expected successes and 10 expected failures.
  • Independent: Individual observations are independent. If sampling without replacement, our sample size shouldn’t be more than 10% of the population.

If these three conditions are met, then we can construct a confidence interval for a population proportion.

The formula to find a confidence interval for a population proportion is:

p +/- (z critical value) * (√p(1-p) / n)

where p is the sample proportion and n is the sample size.

Let’s walk through an example of how to find a confidence interval for a population proportion.

Example: How to Find a Confidence Interval for a Population Proportion

There are 500 students at a certain school. The principal of the school wants to estimate what proportion of all students prefer chocolate milk over regular milk in the cafeteria. He takes a simple random sample of 50 students and finds that 20 of the students prefer chocolate milk.

Based on this sample, find a 99% confidence interval for the proportion of students at this school who prefer chocolate milk over regular milk.

Solution:

Step 1: Identify the sample proportion.

p = 20 / 50 = 0.4

Step 2: Find the z critical value.

Since we are using a 99% confidence level, we need to look up (1 – 0.99)/2 = .01/2 = .005 in the body of the z table and find the corresponding z value. It turns out to be -2.575

Step 3: Find the sample standard deviation.

p(1-p) / n = √.4(1-.4) / 50 = .0048

Step 4: Plug all of the numbers we just found into the confidence interval formula.

p +/- (z critical value) * (√p(1-p) / n)

.4 +- (-2.575)*(.0048)

.4 +/- (.01236) = (.3876, .4124)

The 99% confidence interval for the proportion of students at this school who prefer chocolate milk over regular milk is (.3876, .4124)

Leave a Reply

Your email address will not be published. Required fields are marked *