Hypothesis Testing for Two Proportions

Comparing Two Independent Proportions

• We compare the proportion of employees (%) who got bonuses in the Marketing and IT departments (two populations).
• We compare the survival rates (%) between natural remedies and medication (two treatments).

How to identify this type of situation

1. There are two samples: {Sample 1, Sample 2}
2. Each sample is randomly drawn from different, non-overlapping populations.

• Sample 1 is drawn from Population 1 with $x_1$ (number of successes in Sample 1)
• Sample 2 is drawn from Population 2 with $x_2$ (number of successes in Sample 2)

1. The two populations are independent (there is no matching of individuals or observations in the two samples)

• One group could be split into two randomly assigned treatments (randomized comparative experiment), in which case we can then compare the responses (comparing two proportions).

1. Let $i=1,2$. Each of the two samples will consist of:

• Sample proportion $\displaystyle{\hat p _i=\frac{x_i}{n_i}}$
• Sample size $n_i$
• It's okay if sample sizes differ as they do not need to be equal.
• Central Limit Theorem applies:
• If both population distributions are normal, then the sample sizes do not need to be large.
• If both population distributions are not normal, then the sample sizes need to be large.
• When in doubt, you should have sufficiently large sample sizes.

Hypothesis Test Steps: Comparing Two Proportions

• Step 1: State the hypotheses
• Is this a one-sided or two-sided test?
• To make inferences about the difference in the population proportions $\mu_1-\mu_2$ (two-sided test), the hypotheses are:

• You can also test if one population proportion is greater/less than the other population proportion at either direction (one-sided test), depending on the question:

• Step 2: Note the significance level $\alpha$
• You may find the critical value, depending on $\alpha$ and # of sides
• Step 3: Locate the relevant variables and run the appropriate test
• Find ${x}_1,\ {x}_2,\ n_1,\ n_2$
• Solve for $\hat p_1$and $\hat p_2$:

• Test statistic for comparing two proportions:

• where $\hat p$ is the pooled sample proportion (or combined sample proportion):

• $\hat q=1-\hat p$

• Step 4: Find the p-value
• The p-value is based on your test statistic and # of sides
• If p-value $<\alpha\ \rightarrow$ Reject $H_o$
• If p-value $>\alpha\ \rightarrow$ Fail to reject $H_o$
• You can also compare the critical value with the test statistic
• If $\left|z\right|>\left|CV\right|\rightarrow$ Reject $H_o$
• If $\left|z\right|<\left|CV\right|\rightarrow$ Fail to reject $H_o$
• Step 5: Draw your conclusion
• If you reject $H_o$, you conclude that there is evidence for $H_a$. Example: "There is evidence that the proportions differ."
• If you fail to reject $H_o$, you conclude that there is no evidence for $H_a$. Example: "There is no evidence that the proportions differ."