Key Takeaways — Two-Sample Z Test for a Difference in Proportions

• The four-step process for a hypothesis test is: State, Plan, Do, Conclude.

• Null hypothesis assumes no difference: H₀: p_G − p_PL = 0

• Alternative hypothesis depends on the research question (e.g., Hₐ: p_G − p_PL < 0 for a one-sided test).

• Use the combined (pooled) proportion p̂_c to calculate the standard error when the null assumes no difference.

• The z statistic standardizes the difference in sample proportions:
  z = (p̂₁ − p̂₂ − 0) / standard error

• P-values are compared to the significance level to decide whether to reject the null.

• Interpretation: P-value represents the probability of observing a difference as extreme as the sample, assuming the null is true.

Two-Sample Z Test for a Difference in Proportions (Gem Study Example)

Hi everyone, Mr. Antonucci here. In this video, we’ll put everything together and actually calculate a two-sample Z test for a difference in population proportions.

Study Setup

The Helsinki Heart Study recruited middle-aged men with high cholesterol but no history of serious medical problems to investigate whether a cholesterol-reducing drug (Gem) could lower the risk of heart attacks.

• 251 men took Gem (treatment group)

• 2030 men took a placebo (control group)

During the next 5 years:

• 56 men in the Gem group had heart attacks

• 84 men in the placebo group had heart attacks

Research question: Does this study provide convincing evidence at α = 0.01 that Gem is effective in preventing heart attacks?

Step 1: State

Define parameters and hypotheses:

• p_G = true heart attack rate for middle-aged men taking Gem

• p_PL = true heart attack rate for middle-aged men taking placebo

Null hypothesis:
H₀: p_G − p_PL = 0

Alternative hypothesis:
Hₐ: p_G − p_PL < 0

Significance level: α = 0.01

Note: When stating parameters, always provide context for clarity.

Step 2: Plan

• Test: Two-sample Z test for a difference in proportions

• Random condition: Subjects were randomly assigned to Gem or placebo, so satisfied.

• 10% condition: Not needed for experiments with randomized assignment rather than sampling without replacement.

• Large counts condition: Calculate the combined (pooled) proportion:

p̂_c = (number of successes in both groups) / (sum of both sample sizes)

Then check expected successes and failures for each group using p̂_c. All expected counts must be ≥ 10.

Step 3: Do

1. Calculate sample proportions:
   p̂_G = 56 / 251 ≈ 0.237
   p̂_PL = 84 / 2030 ≈ 0.414

2. Calculate z statistic:
   z = (p̂_G − p̂_PL − 0) / standard error ≈ −2.47

3. Find P-value:

• Since Hₐ is one-sided (< 0), we calculate the area below z = −2.47.

• Using a calculator: normal CDF → P ≈ 0.00675

• Using Table A: P ≈ 0.0066

• No need to multiply by 2 because this is a one-sided test.

Step 4: Conclude

• Compare P-value to α: 0.00675 < 0.01 → reject H₀

• Conclusion in context: There is convincing evidence that the heart attack rate is lower for middle-aged men taking Gem compared to those taking placebo.

Interpreting the P-Value

Assuming the null hypothesis is true (Gem is not more effective than placebo):

There is a 0.00675 probability of observing a difference in heart attack rates between the Gem and placebo groups of 0.141 or less by chance alone.

This explains what the P-value means in context.

That’s it! This demonstrates the full four-step process for a two-sample Z test for a difference in proportions.

Hope that was helpful. Take care.