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Hypothesis Testing Calculator

Free hypothesis testing calculator. One-sample z/t, two-sample t, paired t, proportion tests. p-valu

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Why This Statistical Analysis Matters

Why: Statistical calculator for analysis.

How: Enter inputs and compute results.

H₀
STATISTICSInference & Tests

Hypothesis Testing — Comprehensive Tool

One-sample z/t, two-sample t, paired t, one/two-proportion z. Test stat, p-value, CI, decision, effect size.

Z-Test (σ known) →T-Test →

Real-World Scenarios — Click to Load

Test Configuration

hypothesis_test_results
CALCULATED
Decision
FAIL TO REJECT
t-statistic
2.0540
p-value
0.0606
95% CI
[165.013, 170.987]
SE
1.4606
Cohen's d
0.3750
Power
50.0%
Critical value
±2.045
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Hypothesis Test Result
One-sample t-test
t = 2.054
p = 0.0606Not significantPower: 50%
numbervibe.com/calculators/statistics/hypothesis-testing-calculator

95% Confidence Interval

165.0128Estimate: 168.0000170.9872

Red line = null value. CI includes null → not significant.

Distribution: Rejection Region & p-value

Effect Size vs Benchmarks

Calculation Breakdown

CONFIGURATION
Test Type
One-sample t-test (x̄=168, μ₀=165, s=8, n=30)
DECISION
Critical value
±2.0452
α=0.05, two-tailed
p-value
0.0606
2(1 − Φ(|stat|))
DECISION
FAIL TO REJECT H₀
CONFIDENCE INTERVAL
95% CI
[165.0128, 170.9872]
EFFECT SIZE & POWER
Standard Error
1.4606
Cohen's d
0.3750 (Small)
Power
50.0%

For educational and informational purposes only. Verify with a qualified professional.

Key Takeaways

  • • One-sample z: z = (x̄ − μ₀) / (σ/√n). Use when σ is known.
  • • One-sample t: t = (x̄ − μ₀) / (s/√n), df = n−1. Use when σ unknown.
  • • Two-sample t: t = (x̄₁−x̄₂) / (sₚ√(1/n₁+1/n₂)), df = n₁+n₂−2.
  • • Paired t: t = d̄ / (s_d/√n), df = n−1. For before/after or matched pairs.
  • • One-proportion z: z = (p̂−p₀) / √(p₀(1−p₀)/n).
  • • Two-proportion z: z = (p̂₁−p̂₂) / √(p̂(1−p̂)(1/n₁+1/n₂)).
  • • Decision: Reject H₀ if p-value < α or |test stat| ≥ critical value.
  • • Effect size: Cohen's d for means; h for proportions.

Did You Know?

📊The p-value is the probability of observing a test statistic as extreme (or more) than the one you got, assuming H₀ is true.Source: NIST Handbook
🔬Student's t was developed by William Gosset at Guinness for small-sample quality control.Source: Statistical History
📈Cohen's d: 0.2 = small, 0.5 = medium, 0.8 = large effect.Source: Cohen, 1988
🗳️For proportion tests, np and n(1−p) should be ≥ 5 for the normal approximation.Source: Penn State STAT 500
⚖️Two-tailed tests split α in half. One-tailed tests have more power when the direction is known.Source: OpenIntro Statistics
📐Power = 1 − β = P(reject H₀ | H₁ true). Aim for 80% power when designing studies.Source: Khan Academy

Expert Tips

z vs t: The Decision Rule

Use z when σ is known. Use t when σ is estimated from your sample. For n > 120, the difference is negligible.

One-Tailed vs Two-Tailed

Only use one-tailed tests when the direction was specified BEFORE seeing data. Post-hoc switching inflates Type I error.

Interpreting Non-Significance

"Fail to reject H₀" does NOT mean H₀ is true. Check power — low power means you may lack sample size.

Paired vs Independent

Use paired t when observations are matched (before/after). Use two-sample t for independent groups. Paired designs have more power.

When to Use Each Test

TestWhen to Use
One-sample zKnown σ, or n large
One-sample tUnknown σ, small/moderate n
Two-sample tCompare two independent group means
Paired tBefore/after, matched pairs
One-proportion zTest proportion vs p₀
Two-proportion zCompare two proportions (A/B)

Frequently Asked Questions

When do I use z vs t?

Use z when σ is known. Use t when estimating σ from the sample.

What does p-value mean?

Probability of getting your result (or more extreme) if H₀ were true. Small p-value = evidence against H₀.

What is effect size?

Cohen's d for means, h for proportions. Tells you how big the effect is.

Why use confidence intervals?

CI gives the range of plausible values. If CI excludes the null value, the test is significant.

What is statistical power?

Power = P(reject H₀ when H₁ is true). Aim for 80% when planning studies.

Paired vs two-sample t?

Paired: same subjects measured twice. Two-sample: two independent groups.

What are Type I and Type II errors?

Type I (α): reject H₀ when true. Type II (β): fail to reject when H₁ true. Power = 1 − β.

Hypothesis Testing by the Numbers

1.96
z* for α=0.05, two-tailed
2.576
z* for α=0.01, two-tailed
80%
Recommended minimum power
0.05
Standard significance level

Disclaimer: This calculator is for educational purposes. For research, verify assumptions and use established statistical software.

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