Study Notes on Hypothesis Testing for Two Proportions

Hypothesis Testing for Difference Between Two Proportions

Introduction

• Overview of hypothesis testing for comparing two proportions, similar to that of a single proportion.
• Emphasis on consistent steps of hypothesis testing, unique formulas per test type.

General Steps in Hypothesis Testing

1. Check Conditions
   • Ensure data meets necessary criteria.
2. Calculate Sample Statistic
   • Focus on the difference between two sample proportions.
3. Compute Z-Score
   • Assess whether to reject the null hypothesis or not.
4. Contextualize the Conclusion
   • Explain findings in relation to the research question.

Research Scenario: Prisoner's Dilemma

• Description: Game format involving independent choices between splitting or stealing money.
• Decision Outcomes:
  • If both choose to split: they share the money.
  • If both choose to steal: neither gets anything.
  • If one chooses steal and the other split: stealer gets everything.
• Analogy: Related to the original prisoner's dilemma involving incarceration and incentive structures.

Sample Data Collection

• Participants: 600 individuals (approximately equal gender representation).
• Gender Groups:
  • Male: 269 participants
  • Chose to split: 140
  • Chose to steal: 129
  • Female: 305 participants
  • Chose to split: 163
  • Chose to steal: 142

Research Question

• Main Query: Does gender influence the decision to split or steal in the prisoner's dilemma?
• Specific Hypothesis Test: Comparison of proportions of males and females choosing to split.

Hypothesis Testing Conditions

• Conditions for Test:
  • Each category must contain at least 10 entries.
  • Groups must be independent (participants cannot belong to both categories).
  • Example validity checks include:
  • Proportion of males who chose split and steal > 10
  • Proportion of females who chose split and steal > 10
• Conclusion: Sample meets criteria with hundreds in each sub-group.

Hypothesis Statements

Null and Alternative Hypotheses

• Null Hypothesis ($H_0$):
  • Form: Proportion of males ($pM$) equals proportion of females ($pF$).
  • Notation: $pM = pF$ or equivalently $pM - pF = 0$.
• Alternative Hypothesis ($H_a$):
  • Non-directional: There is a difference between groups.
  • Form: $pM eq pF$.
  • Directional examples: $pM > pF$ or $pM < pF$ based on research expectations.
• Choice of Hypothesis:
  • Non-directional used here due to uncertainty in expected outcomes.

Sample Statistics

• Calculation of sample proportions:
  • For males: $p_M = \frac{140}{269} ext{ (approximately 0.520) }$
  • For females: $p_F = \frac{163}{305} ext{ (approximately 0.534) }$
  • Sample statistic difference: $p<em>M - p</em>F = 0.520 - 0.534 = -0.014$

Calculation of Z-Score

• Z-Score Formula:
  $Z = \frac{(p<em>M - p</em>F)}{ ext{Standard Error}}$
• Standard Error Calculation:
  • Pooled proportion: $p_{hat} = \frac{140 + 163}{269 + 305} = 0.528$
  • Standard error formula: $SE = ext{sqrt}\bigg(\frac{p<em>{hat} imes (1 - p</em>{hat})}{n<em>M} + \frac{p</em>{hat} imes (1 - p<em>{hat})}{n</em>F}\bigg)$
  • Substitute values and solve:
    $SE = ext{sqrt}\bigg(\frac{0.528 imes (1 - 0.528)}{269} + \frac{0.528 imes (1 - 0.528)}{305}\bigg)$
  • Resulting Z-Score approximately:
    $Z ext{ (final calculation)} ext{ approximates to } -0.335.$

P-Value Calculation

• Determine whether the hypothesis is one-tailed or two-tailed.
• Here, testing for difference implies a two-tailed hypothesis:
  • P-Value derived by checking the Z-Score against a Z-table yields a certain value which needs further confirmation.
  • Average P-Value if Z-Score is mid-range between two values.

Conclusion on Hypotheses

• Compare P-Value to significance level (e.g., 0.05).
• Findings:
  • P-Value indicates no sufficient evidence to reject $H_0$.
  • Outcome: "We fail to reject the null hypothesis, indicating no significant difference in proportions of gender choice in the prisoner's dilemma."
• Final Interpretation: No evidence that gender influences the decision to split versus steal within the sampled population.