Skill: distributions

Distributions Practice

Compute binomial probabilities, standardize with z-scores, and use the empirical rule and normal tables.

1 Practice

Type your answer and press Check answer (or Enter). A wrong answer earns a hint; a second miss earns a stronger hint; a third miss walks you through the full solution. Scores and streaks are session-only.

Score 0 · Streak 0 · Problem 0

Score counts first-try correct answers. Streak counts consecutive correct answers.

Problem

Enter integers or decimals. Round as asked; keep the sign on z-scores.

2 What you’ll practice

Every problem is generated fresh from templates across several question types — a problem never repeats until its whole pool is used up.

Binomial probability

C(n,k) p^k (1−p)^(n−k).

Ex: n=5, k=4, p=0.7 → 0.360

Binomial mean/SD

μ = np, σ = √(np(1−p)).

Ex: n=10, p=0.4 → 4, 1.55

z-scores

Count SDs from the mean.

Ex: x=82, μ=74, σ=4 → z=2

Empirical rule

68/95/99.7 within 1/2/3 σ.

Ex: μ±2σ → 95%

z-table

Table = area LEFT of z.

Ex: Φ(1.65) → 0.9505

x from z

Undo: x = μ + zσ.

Ex: z=2, μ=100, σ=15 → 130

3 Watch out for these

The mistakes students make most often on these problems.

Empirical rule on skewed data

68–95–99.7 describes the normal curve only — not incomes or waiting times.

Dropping the z-sign

z = −1.5 means 1.5 SDs below the mean. The sign is information.

Table gives right area

z-tables give area to the LEFT. For P(Z > z), use 1 − Φ(z).

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