Hypothesis Test Calculator Online – Free | 8gwifi.org

Hypothesis Test Calculator

Perform Z-tests, T-tests, and proportion tests with automatic statistical decision making

Test Type

Z-Test for Mean: When population σ is known (large sample: n ≥ 30)
T-Test for Mean: When population σ is unknown (small sample: n < 30)
Z-Test for Proportion: Test if sample proportion differs from claimed proportion
Two-Proportion Z-Test: Compare proportions from two independent samples
Sample 1
Sample 2

Results

Select test type and enter data

Understanding Hypothesis Testing

Hypothesis testing is a statistical method for making decisions about population parameters based on sample data.

The Five Steps of Hypothesis Testing

  1. State the hypotheses: H₀ (null) and H₁ (alternative)
  2. Choose significance level: α (typically 0.05)
  3. Calculate test statistic: Z or T value
  4. Find p-value: Probability of observing results if H₀ is true
  5. Make decision: Reject or fail to reject H₀

Types of Hypotheses

  • Null Hypothesis (H₀): No effect, no difference, status quo
  • Alternative Hypothesis (H₁): What we're testing for
  • Two-tailed: μ ≠ μ₀ (difference in either direction)
  • Right-tailed: μ > μ₀ (increase)
  • Left-tailed: μ < μ₀ (decrease)

Test Statistics

Z-test for mean: Z = (x̄ - μ₀) / (σ / √n)
T-test for mean: t = (x̄ - μ₀) / (s / √n)
Z-test for proportion: Z = (p̂ - p₀) / √(p₀(1-p₀) / n)
Two-proportion Z-test: Z = (p̂₁ - p̂₂) / √(p̂(1-p̂)(1/n₁ + 1/n₂))

Decision Making

Condition Decision Interpretation
p-value ≤ α Reject H₀ Statistically significant result
p-value > α Fail to reject H₀ Not statistically significant

Type I and Type II Errors

  • Type I Error (α): Rejecting H₀ when it's actually true (false positive)
  • Type II Error (β): Failing to reject H₀ when it's actually false (false negative)
  • Power (1-β): Probability of correctly rejecting false H₀

When to Use Each Test

Test Use When
Z-test (mean) σ is known OR n ≥ 30
T-test (mean) σ is unknown AND n < 30
Z-test (proportion) np₀ ≥ 5 AND n(1-p₀) ≥ 5
Two-proportion Comparing two independent proportions

Real-World Applications

  • Medicine: Testing if new drug is more effective than placebo
  • Business: A/B testing website conversion rates
  • Quality Control: Testing if defect rate exceeds threshold
  • Education: Comparing teaching methods
  • Marketing: Testing if ad campaign increased sales
Important: "Fail to reject H₀" is NOT the same as "accepting H₀." We never prove the null hypothesis true; we only find insufficient evidence against it.

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Hypothesis Testing: FAQ

Which test should I choose?

Use z‑test for large samples/known σ, t‑test for small samples unknown σ, proportion tests for binary data, and chi‑square/F tests for categorical/variance comparisons.

One‑tailed or two‑tailed?

Decide before looking at data based on your research question: directional claims use one‑tailed; non‑directional use two‑tailed.

How to interpret p‑value and α?

If p ≤ α, reject H₀; if p > α, do not reject H₀. Statistical significance does not guarantee practical importance.

Assumptions matter?

Yes: independence, distributional assumptions, variance equality, etc. Check diagnostics or use robust alternatives when violated.

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