Hypothesis Testing
Hypothesis testing follows the same conceptual framework regardless of the specific situation:
Step 1: Set Up Your Hypothesis
Choose any hypothesis you want to test that makes a specific numerical claim:
H₀: parameter = specific value
From this hypothesis, you get μ₀ (or the claimed value).
Examples:
- H₀: μ = 50 → μ₀ = 50
- H₀: p = 0.3 → p₀ = 0.3
- H₀: β₁ = 0 → claimed slope = 0
Step 2: Collect Your Data
Gather sample data and calculate your sample statistic (mean, proportion, slope, etc.).
Step 3: Calculate Test Statistic
Compute either a t-statistic OR a z-statistic using the general formula:
Test Statistic = (Sample Estimate - Hypothesized Value) / Standard Error of Estimate
Choose your test statistic:
Use z-statistic when:
- You know the population standard deviation (σ)
- Large sample size
z = (x̄ - μ₀) / (σ/√n)
Use t-statistic when:
- You don't know the population standard deviation
- You estimate it with sample standard deviation (s)
t = (x̄ - μ₀) / (s/√n)
Step 4: Compare to Appropriate Distribution
The Rule:
- t-statistic → Compare to t-distribution
- z-statistic → Compare to normal distribution
Step 5: Look Up P-Value in Tables
Use your test statistic and appropriate parameters:
For t-statistic:
- Need: t-value and degrees of freedom
- Look up in t-distribution table
- Degrees of freedom depends on test type
For z-statistic:
- Need: z-value only
- Look up in standard normal table
The table directly gives you the p-value.
Step 6: Interpret P-Value
The p-value answers: "If my hypothesis were true, what's the probability of getting a test statistic this extreme or more extreme?"
P-value = P(getting result this extreme or more | H₀ is true)
Interpretation:
- Low p-value (close to 0): Data is inconsistent with hypothesis → Reject H₀
- High p-value (close to 1): Data is consistent with hypothesis → Don't reject H₀
Step 7: Make Decision
Compare p-value to significance level α (commonly α = 0.05):
If p ≤ α:
Reject H₀
(statistically significant)
If p > α:
Fail to reject H₀
(not statistically significant)
Key Insights
- Universal Process: Same framework applies to any hypothesis about any parameter
- One Test Statistic: You calculate either t OR z, never both
- Tables Do the Work: You don't calculate p-values by hand
- P-value Scale: 0 ≤ p ≤ 1
- p ≈ 0: Data very inconsistent with H₀
- p ≈ 1: Data very consistent with H₀
- Conservative Approach: We assume H₀ is true until strong evidence suggests otherwise