Hypothesis Testing
Executive Summary
Hypothesis testing is the formal statistical framework used to test claims about population parameters. By comparing sample statistics against null expectations, researchers determine if observed differences are statistically significant or just random noise.
Key Takeaways
- significance check: Hypothesis testing evaluates if sample evidence deviates sufficiently from a null hypothesis to reject it.
- test statistics: Z-score or T-score measures the number of standard errors separating the sample mean from the null mean.
- p-value criterion: The p-value measures the probability of observing our results if the null hypothesis is true. Reject when p < alpha.
Formula
z = (x̄ - μ₀) / (s / √n)
One-Sample Hypothesis Testing
Variable Glossary
| Symbol | Meaning & Description |
|---|---|
| z / t | Standardized test statistic |
| x̄ | Sample Mean |
| μ₀ | Null Hypothesized Mean |
| SE | Standard Error = s / √n |
Step-by-Step Worked Example
1. Mapped Variables
Sample Mean (x̄)
x̄105
Null Mean (μ₀)
μ₀100
Standard Deviation (s)
s15
Sample Size (n)
n30
2. Equation Substitution
Equation with standard inputs
z = (105 - 100) / (15 / √30)
3. Calculation Steps
Step 1: Calculate Standard Error (SE)
SE = s / √n = 15 / √30 = 2.7386
Step 2: Solve test statistic
z = (x̄ - μ₀) / SE = (105 - 100) ÷ 2.7386 = 1.8257
Step 3: Determine p-value outcome
Test statistic = 0.06789. p-value = 1.8257
Final Resolved P-Value0.06789
Live Simulation
Hypothesis Testing Calculator Sandbox
Tweak variables below to see the formula calculate instantly.
Sample Mean (x̄)105
Null Mean (μ₀)100
Standard Deviation (s)15
Calculating...
Common Mistakes to Avoid
Accepting the Null Hypothesis: If a p-value is greater than alpha (e.g. 0.05), we do not "prove" the null hypothesis is true. We simply "fail to reject" it due to insufficient evidence.
Frequently Asked Questions
Interactive Tool
Hypothesis Testing Calculator
Perform one-sample z/t hypothesis testing and compute exact p-values.