Sampling Distributions of the Sample Mean (under $H_0$ and $H

Parameters

Significance Level (α) 0.05

Probability of rejecting H₀ when it is true (Type I Error).

Null Mean (μ₀) 0.0

The hypothesized mean under H₀.

Alternative Mean (μ₁) 2.0

Assumed true mean for power calculation.

Cohen's d (Effect Size) Small (d = 0.40)

d = |μ₁ - μ₀| / σ = |2.0 - 0.0| / 5.0 = 0.40

What it means: Cohen's d measures the distance between μ₀ and μ₁ in units of standard deviation (σ).

Small: d ≈ 0.2 Medium: d ≈ 0.5 Large: d ≈ 0.8

Population Std Dev (σ) 5.0

Variability in the population data.

Sample Size (n) 30

Standard Error (SE) = σ / √n = 0.91

Test Type

Two-tailed One-tailed

Show Distributions

Null (H₀)

Alternative (H₁)

Null (H₀)

Alternative (H₁)

α (Type I)

β (Type II)

Power (1-β)

True Negative

Confidence Level (1-α)

95.0%

Correctly failing to reject H₀ when it is true.

False Positive

Type I Error (α)

5.0%

Incorrectly rejecting H₀ when it is actually true.

False Negative

Type II Error (β)

--.-%

Failing to reject H₀ when H₁ is actually true.

True Positive

Power (1-β)

--.-%

Correctly rejecting H₀ when H₁ is actually true.