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Confidence Interval for Population Proportion

Confidence Interval for Population Proportion


This lesson will explain finding a confidence interval for the population proportion.

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Introduction to Statistics

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What's Covered

In this tutorial you're going to learn about confidence intervals for a population proportion. Specifically you will focus on:

  1. Confidence Intervals for Population Proportion


Suppose you have this drug called Obecalp which is a popular prescription drug. It's thought to cause headaches as a side effect. To test they took a random sample of 206 patients who are taking Obecalp, and 23 got headaches. Construct a 95% confidence interval for the proportion of all Obecalp users that would experience headaches. So, if you gave this drug to all the people who are using it, what percent of all of them would be getting headaches? In our sample 23 of the 206 experienced headaches.

Term to Know

Confidence Interval for a Population Proportion

A confidence interval that gives a likely range for the value of a population proportion. It is the sample proportion, plus and minus the margin of error from the normal distribution.

Here is the process for constructing confidence intervals.

  1. First, verify the conditions necessary for inference. Stating the conditions isn't enough, and it's not just a formality, you have to verify.
  2. Second, calculate the confidence interval.
  3. Third, interpret what it actually means.

State what the conditions are.

The requirements are randomness, independence, and normality.

  • Randomness: the sample that you got of Obecalp users was a random sample, so that’s verified.
  • Independence: the sample of Obecalp users taken was a small fraction of the population of Obecalp users. There's no way to verify that empirically unless you had the whole list of people taking the drug. You're going to have to assume there at least 2060 people taking this drug.
  • Normality: this np is greater than or equal to 10 thing is a little harder to figure out. You don't know p. The true proportion of people who will get headaches, and you don't have a best guess for it either from a null hypothesis. There is no null hypothesis in this problem.

What you do have, as a point estimate for p, p hat. Verify normality by using p hat instead of p. Say n times p hat has to be at least 10. In this case 206 times p, 23 out of 206, is 23, which is bigger than 10. Times q hat is 183, which is also bigger than 10.

Again you need to use p hat to verify the normality condition because you don't know p.


Second, calculate the actual interval. Do the point estimate, p hat, plus or minus the z star critical value times the standard error of p hat, which is the square root of p hat, q hat over n. Again you're using the p hat and the q hat here, because you don't know what p and q are.

The population proportion is not known, so you’ll use p hat for the standard error. P hat is 23 out of 206. You knew that from the problem. The sample size is 206.

As you look at the graph below, you might be thinking this is the t distribution and you don't use the t distribution for proportions. Agreed. Where you have to look on this sheet is in the confidence levels. You need a confidence level for your confidence interval. You’ll find it down at the bottom.

* This table is saved as a pdf below this tutorial for your convenience.  View full screen or zoom in for clarity.

Z confidence level, critical values, are found in the last row of this t-table, under the infinity value. Essentially the normal distribution is the t distribution with infinite degrees of freedom. Look in the row highlighted to find that the z critical value that we should use is 1.96.

Take all of that and put it in the formula and obtain 0.112, which was your p hat, plus or minus 0.043. This is your margin of error, 4.3%. When you evaluate the integral it's going to be 0.069 all the way up 0.155.

Now you need to interpret this interval. You're 95% certain that if everyone who was taking Obecalp was in the study, the true proportion of all Obecalp users who would experience headaches is somewhere between 6.9% and 15.5%.

It's likely somewhere in that range. You don't know exactly where in that range, but the true proportion is probably somewhere in this range.


You can create point estimates for population proportions, which is your sample proportion, and then use that sample proportion to determine the margin of error for a confidence interval. First, verify the conditions for inference are met, then construct and interpret a confidence interval based on the data that you've gathered and the statistics that you've calculated.

Good luck.

Source: This work adapted from Sophia Author Jonathan Osters.

t-distribution of critical values

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  • Confidence Interval for a Population Proportion

    A confidence interval that gives a likely range for the value of a population proportion. It is the sample proportion, plus and minus the margin of error from the normal distribution.