STRAIGHT STATISTICS

Hypothesis Test: Difference Between Proportions

This lesson explains how to conduct a hypothesis test to determine whether the difference between two proportions is significant. The test procedure, called the two-proportion z-test, is appropriate when the following conditions are met:

• The sampling method for each population is simple random sampling.
• The samples are independent.
• Each sample includes at least 10 successes and 10 failures. (Some texts say that 5 successes and 5 failures are enough.)
• Each population is at least 10 times as big as its sample.

This approach consists of four steps: (1) state the hypotheses, (2) formulate an analysis plan, (3) analyze sample data, and (4) interpret results.

State the Hypotheses

Every hypothesis test requires the analyst to state a null hypothesis and an alternative hypothesis. The table below shows three sets of hypotheses. Each makes a statement about the difference d between two population proportions, P₁ and P₂. (In the table, the symbol ≠ means " not equal to ".)

Set	Null hypothesis	Alternative hypothesis	Number of tails
1	P1 - P2 = 0	P1 - P2 ≠ 0	2
2	P1 - P2 > 0	P1 - P2 < 0	1
3	P1 - P2 < 0	P1 - P2 > 0	1

The first set of hypotheses (Set 1) is an example of a two-tailed test, since an extreme value on either side of the sampling distribution would cause a researcher to reject the null hypothesis. The other two sets of hypotheses (Sets 2 and 3) are one-tailed tests, since an extreme value on only one side of the sampling distribution would cause a researcher to reject the null hypothesis.
When the null hypothesis states that there is no difference between the two population proportions (i.e., d = 0), the null and alternative hypothesis for a two-tailed test are often stated in the following form.

H₀: P₁ = P₂
H_a: P₁ ≠ P₂

Formulate an Analysis Plan

The analysis plan describes how to use sample data to accept or reject the null hypothesis. It should specify the following elements.

• Significance level. Often, researchers choose significance levels equal to 0.01, 0.05, or 0.10; but any value between 0 and 1 can be used.
  
• Test method. Use the two-proportion z-test (described in the next section) to determine whether the hypothesized difference between population proportions differs significantly from the observed sample difference.

Analyze Sample Data

Using sample data, complete the following computations to find the test statistic and its associated P-Value.

• Pooled sample proportion. Since the null hypothesis states that P₁=P₂, we use a pooled sample proportion (p) to compute the standard error of the sampling distribution.
  p = (p₁ * n₁ + p₂ * n₂) / (n₁ + n₂)
  where p₁ is the sample proportion from population 1, p₂ is the sample proportion from population 2, n₁ is the size of sample 1, and n₂ is the size of sample 2.
  
• Standard error. Compute the standard error (SE) of the sampling distribution difference between two proportions.
  SE = sqrt{ p * ( 1 - p ) * [ (1/n₁) + (1/n₂) ] }
  where p is the pooled sample proportion, n₁ is the size of sample 1, and n₂ is the size of sample 2.
  
• Test statistic. The test statistic is a z-score (z) defined by the following equation.
  z = (p₁ - p₂) / SE
  where p₁ is the proportion from sample 1, p₂ is the proportion from sample 2, and SE is the standard error of the sampling distribution.
  
• P-value. The P-value is the probability of observing a sample statistic as extreme as the test statistic. Since the test statistic is a z-score, use the Normal Distribution Calculator to assess the probability associated with the z-score. (See sample problems at the end of this lesson for examples of how this is done.)

The analysis described above is a two-proportion z-test.

by: J-Lynn B. Ramos