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These examples are illustrative reference cases for common hypothesis-testing scenarios.
| Scenario | Inputs | Statistic | P-value | Decision at α = 0.05 |
|---|---|---|---|---|
| Z test for mean | n = 36, x̄ = 104, μ₀ = 100, σ = 15 | z = 1.6000 | 0.1096 | Fail to reject |
| T test for mean | n = 25, x̄ = 52, μ₀ = 50, s = 6 | t = 1.6667 | 0.1086 | Fail to reject |
| Z test for proportion | n = 100, x = 58, p₀ = 0.50 | z = 1.6000 | 0.1096 | Fail to reject |
| Chi-square variance test | n = 20, s² = 81, σ₀² = 64 | χ² = 24.0469 | 0.1921 | Fail to reject |
z = (x̄ - μ₀) / (σ / √n)
Two-sided p-value = 2 × min[Φ(z), 1 - Φ(z)]
Confidence interval = x̄ ± z(1-α/2) × σ / √n
t = (x̄ - μ₀) / (s / √n), with df = n - 1
Two-sided p-value = 2 × min[Ft(t), 1 - Ft(t)]
Confidence interval = x̄ ± t(1-α/2, df) × s / √n
p̂ = x / n
z = (p̂ - p₀) / √[p₀(1-p₀)/n]
Approximate interval = p̂ ± z(1-α/2) × √[p̂(1-p̂)/n]
χ² = (n - 1)s² / σ₀², with df = n - 1
Two-sided p-value = 2 × min[Fχ²(χ²), 1 - Fχ²(χ²)]
Variance interval = ((n-1)s² / χ²upper, (n-1)s² / χ²lower)
The p-value measures how unusual your sample result would be if the null hypothesis were true. Smaller values indicate stronger evidence against the null model.
Use a z test when the population standard deviation is known or when a normal approximation is explicitly justified. Use a t test when the standard deviation is estimated from the sample.
A two-sided test checks whether the parameter differs in either direction from the hypothesized value. It detects both increases and decreases.
Power is the probability of correctly rejecting a false null hypothesis. Higher power means the test is more likely to detect a real effect.
Type II error is the probability of failing to reject the null hypothesis when it is actually false. Power equals one minus β.
Confidence bounds provide an interval of plausible parameter values based on your sample. They complement the p-value and help show effect magnitude.
The chi-square variance test depends strongly on normality. If your data are far from normal, the p-value and interval may be unreliable.
No. It helps compute the main quantities, but study design, assumptions, data quality, and practical significance still require careful interpretation.
Important Note: All the Calculators listed in this site are for educational purpose only and we do not guarentee the accuracy of results. Please do consult with other sources as well.