## Two Proportion Z Interval

**Overview:**

A two-proportion z-interval gives a confidence interval for the true difference in proportions, p1-p2, in two independent groups.

**Conditions**

- Randomization Condition: The data in each group should be drawn independently and at random from a homogenous population or generated by a randomized comparative experiment.
- Success/Failure Condition: Both groups are big enough that at least 10 successes and at least 10 failures have been observed in each.
- The 10% Condition: If the data are sampled without replacement, the sample should not exceed 10% of the population.
- Independent Groups Assumption: The two groups we are comparing must also be independent of each other.

**A Two-Proportion z-Interval**

When the conditions are met, we are ready to find the confidence interval for the difference of two proportions, p1 – p2. The confidence interval is

where we find the standard error of the difference,

from the observed proportions.

The critical value z* depends on the particular confidence level, C, that we specify.

TI Tip Go to the

A Gallup Poll asked whether the attribute “intelligent” described men in general. The poll revealed that 28% of 506 men thought it did, but only 14% of 520 women agreed. We want to estimate the true size of the gender gap by creating a 95% confidence interval.

2-PropZInt

(.09101, .18948)

p1= .2806

p2= .1404

n1 = 506

n2 = 520

The critical value z* depends on the particular confidence level, C, that we specify.

TI Tip Go to the

**STAT TESTS**menu and scroll down the list and select**B: 2-PropZInt**. For the first population, enter the observed number for**x1**, the sample size**n1**, and with the second population enter the observed number in**x2**and the sample size**n2**. Specify the desired confidence level and then select**Calculate.****Example 1:**A Gallup Poll asked whether the attribute “intelligent” described men in general. The poll revealed that 28% of 506 men thought it did, but only 14% of 520 women agreed. We want to estimate the true size of the gender gap by creating a 95% confidence interval.

*Solution: Go to the***STAT TESTS**menu and scroll down the list and select**B: 2-PropZInt**. Enter the observed number of males: .**28 x 506**, the sample size**506**, and with the second population enter the observed number of women**.14 x 520**and the sample size**520**. Specify the desired confidence level to be 95% and then select**Calculate.**2-PropZInt

(.09101, .18948)

p1= .2806

p2= .1404

n1 = 506

n2 = 520

*With the results given on the calculator, we are 95% confident that the proportion of men who think the attribute “intelligent” describe males in general is between 9 and 19 percentage points higher than the proportion of women who think so.*