Confidence Interval for a Difference in Proportions

This lesson describes how to create a confidence interval for the difference between two population proportions.

Checking Conditions

Before we can create a confidence interval for the difference between two population proportions, 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 obtain the data for both samples.
  • Normal: Each sample includes at least 10 successes and 10 failures.
  • Independent: The two samples are independent.

The formula to find a confidence interval for the difference between two population proportions is:

(p1 – p2) +/- (z critical value) * (√p1(1-p1) / n1 + p2(1-p2) / n2)

where p1 is the proportion of sample 1, p is the proportion of sample 2, n1 is the sample size of sample 1, and n2 is the sample size of sample 2.

Let’s walk through an example of how to find a confidence interval for the difference between two population proportions.

Example: Confidence Interval for the Difference Between Two Population Proportions

A researcher wants to know what percentage of students at two different universities study for more than one hour per night. He takes a simple random sample of 100 students from each school and finds that 40% of students at school 1 study for more than one hour per night and 35% of students at school 2 study for more than one hour per night.

Construct a 95% confidence interval for the difference between the proportion of students who study for more than one hour per night at these two universities. 

Solution:

Step 1: Find the difference between the two sample proportions.

p1 – p= 0.4 – 0.35 = 0.05

Step 2: Find the z critical value.

Since we are using a 95% confidence level, we need to look up (1 – 0.95)/2 = .05/2 = .025 in the body of the z table and find the corresponding z value. It turns out to be -1.96.

Step 3: Find the standard deviation of the difference between the two proportions.

p1(1-p1) / n1 + p2(1-p2) / n2 = √.4(1-.4) / 100 + .35(1-.35) / 100 = .068

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

(p1 – p2) +/- (z critical value) * (√p1(1-p1) / n1 + p2(1-p2) / n2)

(.05) +/ (-1.96)*(.068)

.05 +/- (.13328) = (-.0833, .1833)

The 95% confidence interval for the difference between the proportion of students who study for more than one hour per night at these two universities is  (-.0833, .1833)

2 Replies to “Confidence Interval for a Difference in Proportions”

  1. Since it crosses zero (-.0833, .1833), does this mean it is not valid and meaningful? How do you verbalize this result to a general population that may not understand statistics?

    1. Since zero is within the confidence interval, it means there is no significant difference between the proportion of students who study for more than one hour per night at these two universities.

      In other words, it’s perfectly reasonable for the actual difference in proportions to be zero.

Leave a Reply

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