Types of Hypothesis Tests

Purpose and Rationale

Understanding different types of hypothesis tests is crucial because:

Appropriate Test Selection
- Ensures valid statistical analysis
- Prevents incorrect conclusions
- Maximizes statistical power
Research Design
- Guides data collection methods
- Influences sample size planning
- Affects study feasibility
Result Interpretation
- Ensures correct p-value calculation
- Guides proper decision making
- Affects confidence in conclusions

Why We Need Different Tests
- Different data types require different approaches
- Various research questions need specific methods
- Statistical assumptions vary by test type
How Test Selection Works
- Based on variable types
- Depends on number of groups
- Considers parameter of interest
- Accounts for available information

Type of Variable
- Quantitative (numerical)
  - Testing means ( $μ$ )
  - Examples: height, weight, price
- Categorical (groups/categories)
  - Testing proportions ( $p$ )
  - Examples: yes/no, success/failure
Number of Groups
- Single population
  - One-sample tests
- Two populations
  - Two-sample tests
- Multiple populations
  - ANOVA or multiple comparison tests
Parameter of Interest
- Single quantitative: Population mean ( $μ$ )
- Two quantitative: Difference in means ( $μ_{1} - μ_{2}$ )
- Single categorical: Population proportion ( $p$ )
- Two categorical: Difference in proportions ( $p_{1} - p_{2}$ )
Knowledge of Population Parameters
- Known $σ$ : Z-tests
- Unknown $σ$ : T-tests

Single Proportion Test
- Hypotheses: $H_{0} : p = p_{0}$ vs $H_{a} : p [<, >, \neq] p_{0}$
- Test Statistic: $Z = \frac{\hat{p} - p_{0}}{\sqrt{\frac{p_{0} (1 - p_{0})}{n}}}$
- Distribution: Standard Normal, $N (0, 1)$
- Conditions:
  - Random sample
  - $n p_{0} \geq 10$ and $n (1 - p_{0}) \geq 10$
Difference in Proportions Test
- Hypotheses: $H_{0} : p_{1} - p_{2} = 0$ vs $H_{a} : p_{1} - p_{2} [<, >, \neq] 0$
- Test Statistic: $Z = \frac{{\hat{p}}_{1} - {\hat{p}}_{2}}{\sqrt{{\hat{p}}_{p o o l} (1 - {\hat{p}}_{p o o l}) (\frac{1}{n_{1}} + \frac{1}{n_{2}})}}$
- Where ${\hat{p}}_{p o o l} = \frac{x_{1} + x_{2}}{n_{1} + n_{2}}$
- Conditions:
  - Independent random samples
  - $n_{1} {\hat{p}}_{1} \geq 10$ , $n_{1} (1 - {\hat{p}}_{1}) \geq 10$
  - $n_{2} {\hat{p}}_{2} \geq 10$ , $n_{2} (1 - {\hat{p}}_{2}) \geq 10$

Single Mean Test
- Hypotheses: $H_{0} : μ = μ_{0}$ vs $H_{a} : μ [<, >, \neq] μ_{0}$
- Conditions:
  - Random sample
  - Normal population OR $n \geq 30$
- Known $σ$ :
  - Test Statistic: $Z = \frac{\bar{x} - μ_{0}}{σ / \sqrt{n}}$
  - Distribution: $N (0, 1)$
- Unknown $σ$ :
  - Test Statistic: $T = \frac{\bar{x} - μ_{0}}{s / \sqrt{n}}$
  - Distribution: $t$ with $d f = n - 1$
Difference in Means Test
- Hypotheses: $H_{0} : μ_{1} - μ_{2} = 0$ vs $H_{a} : μ_{1} - μ_{2} [<, >, \neq] 0$
- Conditions:
  - Independent random samples
  - Normal populations OR $n_{1}, n_{2} \geq 30$
- Known $σ$ :
  - Test Statistic: $Z = \frac{{\bar{x}}_{1} - {\bar{x}}_{2}}{\sqrt{\frac{σ_{1}^{2}}{n_{1}} + \frac{σ_{2}^{2}}{n_{2}}}}$
  - Distribution: $N (0, 1)$
- Unknown $σ$ :
  - Test Statistic: $T = \frac{{\bar{x}}_{1} - {\bar{x}}_{2}}{\sqrt{\frac{s_{1}^{2}}{n_{1}} + \frac{s_{2}^{2}}{n_{2}}}}$
  - Distribution: $t$ with $d f = min (n_{1}, n_{2}) - 1$

Variable Type	Groups	Parameter	$σ$ Known?	Test	Distribution
Categorical	Single	$p$	N/A	Z-test	$N (0, 1)$
Categorical	Two	$p_{1} - p_{2}$	N/A	Z-test	$N (0, 1)$
Quantitative	Single	$μ$	Yes	Z-test	$N (0, 1)$
Quantitative	Single	$μ$	No	T-test	$t_{n - 1}$
Quantitative	Two	$μ_{1} - μ_{2}$	Yes	Z-test	$N (0, 1)$
Quantitative	Two	$μ_{1} - μ_{2}$	No	T-test	$t_{min (n_{1}, n_{2}) - 1}$

Before Analysis
- Identify variable types
- Determine number of groups
- Check parameter knowledge
- Verify conditions
During Analysis
- Use correct test statistic
- Apply proper distribution
- Calculate accurate p-value
- Check assumptions
After Analysis
- Report test used
- Justify selection
- Document conditions
- Interpret results