Notes on Hypothesis Tests for 2 Proportions

Comparing Two Proportions

• Estimating Difference Between Two Proportions:
  • Example: Comparing admission rates at ages 17 and 18.
  • Statistic used: Difference in sample proportions
  • Population proportions:
    • Population 1: p1, size n1, sample proportion ar{p}_1
    • Population 2: p2, size n2, sample proportion ar{p}_2

Sampling Distribution of Proportions

• For independent samples of sizes n1 and n2 from populations with parameters p1 and p2:
  • Mean:
  • ext{Mean of the sampling distribution } ar{p} = p
  • ext{Mean of } (ar{p}1 - ar{p}2) = p1 - p2
  • Standard Deviation:
  • ext{Std. deviation of } ar{p} = rac{p(1 - p)}{n}
  • ext{Std. deviation of } (ar{p}1 - ar{p}2) = \sqrt{\frac{p1(1 - p1)}{n1} + \frac{p2(1 - p2)}{n2}}
  • Normality Conditions:
  • n p ext{ and } n(1 - p) ext{ should be } ext{≥} 10 for both samples

Assumptions and Conditions

• Independence Observations:
  • Randomization Condition:
  • Data drawn independently and randomly from a homogeneous population.
  • 10% Condition:
  • Sample should not exceed 10% of population when sampled without replacement.
• Independent Groups:
  • Two groups must be independent.
• Sample Size Condition:
  • Each group must be sufficiently large.
  • Success/Failure Condition:
  • n1 p1 ext{ and } n1(1 - p1) ext{ ≥ } 10
  • n2 p2 ext{ and } n2(1 - p2) ext{ ≥ } 10

Confidence Interval for 2 Population Proportions

• Formula:
  • ar{p}1 - ar{p}2 ext{ ± } z^* \sqrt{\frac{\bar{p}1(1 - \bar{p}1)}{n1} + \frac{\bar{p}2(1 - \bar{p}2)}{n2}}
• Example:
  • Smokers (n1=150): 95 with prominent wrinkles, ar{p}_1 = 0.63
  • Nonsmokers (n2=250): 105 with prominent wrinkles, ar{p}_2=0.42
  • 95% CI for smokers:
  • 0.63 ± 1.96 × 0.0394 = (0.55, 0.71)
  • 95% CI for nonsmokers:
  • 0.42 ± 1.96 × 0.0312 = (0.36, 0.48)
  • Check for overlap in intervals: indicates proportion differences.

Two-Proportion z Test

• Hypotheses:
  • Two-tailed: H0: p1 - p2 = p0 vs Ha: p1 - p2 eq p0
  • Upper-tailed: H0: p1 - p2 = p0 vs Ha: p1 - p2 > p0
  • Lower-tailed: H0: p1 - p2 = p0 vs Ha: p1 - p2 < p0
• Assumption & Conditions:
  • Random samples, independent observations, large sizes
  • Normal Conditions: n1 p1, n1(1 - p1), n2 p2, n2(1 - p2) ext{ ≥ } 10
• Test Statistic:
  • z0 = \frac{\bar{p}1 - \bar{p}2 - (p1 - p2)}{\sqrt{\frac{p1(1 - p1)}{n1} + \frac{p2(1 - p2)}{n_2}}}
  • If null is true, pool the proportions:
  • \bar{p}{pooled} = \frac{n1 \bar{p}1 + n2 \bar{p}2}{n1 + n_2}
• Modified test statistic:
  • z0 = \frac{\bar{p}1 - \bar{p}2}{\bar{p}{pooled}(1 - \bar{p}{pooled})(\frac{1}{n1} + \frac{1}{n_2})}

Decision Making

• p-value Criteria:
  • For two-tailed: p-value = 2 × P(Z > z_0)
  • For upper-tail: p-value = P(Z > z_0)
  • For lower-tail: p-value = P(Z < z_0)
• Decision rule:
  • If p-value ≤ \alpha, then reject H_0
  • If p-value > \alpha, do not reject H_0

Example Continuation

• Proportions of smokers and non-smokers with wrinkles at \alpha=0.05:
  • Test statistic:
  • z0 = \frac{\bar{p}1 - \bar{p}2}{\bar{p}{pooled}(1 - \bar{p}{pooled})(\frac{1}{n1} + \frac{1}{n_2})} = 4.2
  • Pooled proportion:
  • \bar{p}_{pooled} = 0.5
  • p-value calculation:
  • p-value = 0
• Conclusion:
  • Reject H_0, indicating a significant difference in proportions of smokers and non-smokers with wrinkles.