Chapter 8: Inference for Proportions

Section 8.2: Comparing Two Proportions

In this section, we discuss how to

1. compare two population proportions
2. responses to two treatments of two independent SRS samples

Here are the parameters and statistics:

Population	Population proportion	Sample Size	Sample proportion
1	p1	n1	1
2	p2	n2	2

We use ₁ – ₂ to estimate p₁ – p₂.

The sampling distribution of ₁ – ₂

Here is the information about this sampling distribution:

• when n is large, the distribution of ₁ – ₂ is approximately normal
• the mean of the sampling distribution is p₁ – p₂ (i.e., ₁ – ₂ is an unbaised estimator of p₁ – p₂)
• the standard deviation is

Confidence intervals for ₁ – ₂

Since p is NOT known, we need to use the standard error as an estimate of the s.d.: SE = . This is valid because for large n, is close to p.

A level C CI for p₁ – p₂ is ₁ – ₂ ± z^*×SE. Luckily, the TI83 can calculate the two–sample proportion z CI!

Assumptions for the two–sample proportion CI:

1. the population is at least 10 times larger than the sample
2. there are at least 5 successes and 5 failures in each sample

Significance tests for ₁ – ₂

The one–sample z–test for a population proportion:

1. State the hypotheses:
   • one-tailed tests:
     1. 
     2. 
   • two-tailed test:
     
2. Calculate the P–value
3. Make your conclusion

If H₀ is true, then all observations in both samples come from the same population. Thus, the two samples can be pooled together. The pooled sample proportion is:

.

Thus, the z–statistic is: .

Recall: H₀ and H_a always refer to the population and NOT to a particular outcome. It is often easier (and more appropriate) to state H₀ and H_a before looking at the data.