AP Statistics Unit 4 Cheatsheet: Inference for Quantitative Data: Means

Mean and SD:

• μ_x̄ = μ
• σ_x̄ = σ/√n (when values independent)

When is x̄ normal?

• Population normal: x̄ is normal, any n
• Population not normal: x̄ ≈ normal if n ≥ 30 (more if extremely skewed)

Conditions to verify:

• Randomization: random sample
• 10% condition: n ≤ 10% of N
• Sample data: population normal OR n ≥ 30 OR n < 30 with no strong skew/outliers

When to use t: Population σ unknown; use sample s instead

Shape:

• Symmetric, bell-shaped
• Wider tails than standard normal
• Identified by degrees of freedom (df)
• As df increases, approaches standard normal

Degrees of freedom: df = n − 1 for one-sample

Mean and SD of x̄₁ − x̄₂:

• μ_(x̄₁−x̄₂) = μ₁ − μ₂
• σ_(x̄₁−x̄₂) = √(σ₁²/n₁ + σ₂²/n₂)

Normality conditions:

• Both populations normal → difference is normal
• Both n₁ ≥ 30 and n₂ ≥ 30 → difference ≈ normal

Conditions for inference:

• Randomization: two independent random samples (or randomized experiment)
• 10% condition: n₁ ≤ 10%N₁ and n₂ ≤ 10%N₂ (waived for experiments)
• Sample data: check each group separately — that group's n ≥ 30, or its population is stated to be approximately normal, or (if n < 30) a graph of that sample shows no strong skewness or outliers