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# Two-proportion z-interval

last edited by 5 years, 3 months ago

A two-proportion z-interval is a confidence interval for the true difference in proportions, p1 – p2, in two independent groups. This confidence interval is useful because it can provide an estimate for the difference between the two population proportions. The formula is

where z* is a critical value from the standard normal model corresponding to the specified confidence level.

The standard error is given by

Where is the sample proportion for the first set of data and is the sample proportion for the second set of data.

For example, a study done by the University of Texas Southwestern Medical Center examined 626 people with and without tattoos. This study showed that of those with tattoos, 25 of the 113 had Hepatitis C, and of those without tattoos, 18 of the 513 had Hepatitis C. By hand, we check the conditions for both groups separately which are that each sample is random, below 10% of the population, and that the successes and failures of each group are above 10 each. The study was indeed representative and random, 626 people is well below 10% of the population of people with and without tattoos, and of those with tattoos, 25 had Hepatitis C and 88 did not, while of those without tattoos, 18 had Hepatitis C and 495 did not. Then, we find the proportions for Hepatitis C successes for the two different groups.

25/113= 0.221 and 18/513= 0.035

Next, we find the standard error between the two groups using

and

We use $z_{*}&space;=&space;1.96$ if we want a 95% confidence interval. If we wanted a 90% confidence interval we would use $z_{*}$=1.645 and if we want 99% we would use $z_{*}$=2.576. Since we want a 95% confidence interval in this example, we use $z_{*}&space;=&space;1.96$.

By hand, we have determined that we are 95% confident that the proportion of people with tattoos that develop Hepatitis C is between 10.8% and 26.4% higher than the proportion of people without tattoos that develop Hepatitis C.

Instructions for generating a Two-proportion Z-Interval in SPSS

•  Make sure you are looking at two categorical variables where each takes on only two values. In this case "Tattoo" and "Hashepatitisc" taking the values "Yes" and "No")
• Go to Analyze -> Generalized Linear Models -> Generalized Linear Models (Be careful, there is an option called "General Linear Models" right above it, but that's not the correct one.)
• In the first tab called "Type of Model", select "Custom" . Change Distribution from "Normal" to "Binomial".
• In the next tab called "Response", drag your response variable (in this case "Hashepatitisc") to the "Dependent Variable" slot.
• In the next tab called "Predictors", drag the explanatory variable (in this case"Tattoo") to the "Factors" box.
• In the next tab called "Model", drag the explanatory variable to the"Model" box.
• In the "Statistics" tab, change the confidence level from 95% or whatever confidence level you desire.
• Click "Okay".
• In the output, scroll all the way to the bottom to the box called "Parameter Estimates".
• The desired confidence interval will be the second one listed