Example:
Find the 90% CI for the population proportion of people who prefer chocolate over vanilla. Assume that your sample proportion is .67 for people who prefer chocolate. You have a sample size of 150 people.

Solution

What you are given from the problem:
sample size = 150
sample proportion = .67 (this is called p)
q = 1 – p = 1 – .67 = .33

NOTICE: In many textbooks, there are special symbols for the sample proportion, such as

For this example and for simplicity, I use “p” for the sample proportion and q=1 – p

The critical value or “zc” for 90% is 1.645

Why is the critical z value for a 90% Confidence Interval equal to 1.645?

Next, using the formula for the error E we have:
E = zc * sqrt [ p*q / n]
E = 1.645 * sqrt [.67 *.33/150]
E = 1.645 * sqrt(.0015)
E = 1.645 * .04
E = .07

To get the 90% Confidence Interval, we need to subtract and add E to the sample proportion.

sample prop – E < population prop < sample prop + E

.67 – .07 < population proportion < .67 + .07
.60 < population proportion < .74

The 90% Confidence Interval can also be written as:
{.60, .74}