Hypothesis Testing for Proportions

Hypothesis Tests for Proportions

Test Selection Guide

Test Type	When to Use	Key Conditions	Example
Single Proportion	Compare to hypothesized value	$n p_{0} \geq 10$ , $n (1 - p_{0}) \geq 10$	Testing if majority supports (p > 0.5)
Difference in Proportions	Compare two groups	All $n \hat{p} \geq 10$	Comparing treatment success rates

Single Proportion Test

Hypotheses Structure

Type	Symbolic Form	Example
Null	$H_{0} : p = p_{0}$	$H_{0} : p = 0.5$
Right-tailed	$H_{a} : p > p_{0}$	$H_{a} : p > 0.5$
Left-tailed	$H_{a} : p < p_{0}$	$H_{a} : p < 0.5$
Two-tailed	$H_{a} : p \neq p_{0}$	$H_{a} : p \neq 0.5$

Test Statistic Components

Component	Formula/Value	Description
Statistic	$Z = \frac{\hat{p} - p_{0}}{\sqrt{\frac{p_{0} (1 - p_{0})}{n}}}$	Test statistic
$\hat{p}$	$\frac{successes}{n}$	Sample proportion
$p_{0}$	Hypothesized value	Null value
$n$	Sample size	Number of observations

Conditions Checklist

Condition	Requirement	How to Verify
Random Sample	Independent observations	Study design
Success-Failure	$n p_{0} \geq 10$	Calculate $n p_{0}$
	$n (1 - p_{0}) \geq 10$	Calculate $n (1 - p_{0})$

Distribution Properties

Aspect	Details	Notes
Type	Normal approximation	Based on Central Limit Theorem
Center	$p_{0}$ under $H_{0}$	Null hypothesis value
Spread	$\sqrt{\frac{p_{0} (1 - p_{0})}{n}}$	Standard error

Difference of Proportions Test

Hypotheses Structure

Type	Symbolic Form	Interpretation
Null	$H_{0} : p_{1} - p_{2} = 0$	No difference
Alternative	$H_{a} : p_{1} - p_{2}$ [<, >, ≠] $0$	Difference exists

Test Statistic Components

Component	Formula	Purpose
Pooled Proportion	$\hat{p} = \frac{x_{1} + x_{2}}{n_{1} + n_{2}}$	Combined estimate under $H_{0}$
Test Statistic	$Z = \frac{({\hat{p}}_{1} - {\hat{p}}_{2})}{\sqrt{\hat{p} (1 - \hat{p}) (\frac{1}{n_{1}} + \frac{1}{n_{2}})}}$	Standardized difference

Success-Failure Conditions

Group	Conditions	Example Check
Group 1	$n_{1} \hat{p} \geq 10$	100(0.5) = 50 ✓
	$n_{1} (1 - \hat{p}) \geq 10$	100(0.5) = 50 ✓
Group 2	$n_{2} \hat{p} \geq 10$	100(0.5) = 50 ✓
	$n_{2} (1 - \hat{p}) \geq 10$	100(0.5) = 50 ✓

Common Applications

Application	Example	Key Considerations
A/B Testing	Website conversion	Independence
Clinical Trials	Treatment success	Randomization
Survey Analysis	Response differences	Sample selection
Quality Control	Defect rates	Time independence

Best Practices

Reporting Template

Element	Content	Example
Context	Research question	"Testing if new treatment improves success rate"
Method	Test type and conditions	"Two-proportion z-test, conditions met"
Results	Statistics and p-value	"z = 2.45, p = 0.014"
Conclusion	Interpretation	"Evidence suggests improvement of 12%"

Interpretation Guide

Finding	Statistical Meaning	Practical Action
Significant Difference	Reject $H_{0}$	Consider effect size
No Significant Difference	Fail to reject $H_{0}$	Consider power
Large Effect	Practical importance	Implement changes
Small Effect	Statistical significance only	Consider costs

Hypothesis Testing Basics - General framework
Hypothesis Testing Key Concepts - Core concepts
Normal Distribution - Key distribution for proportion tests
Standard error - Understanding uncertainty
Confidence Interval for proportions - Complementary inference approach
sampling distribution - Foundation for tests