All core units covered — descriptive stats, probability, distributions, inference, and regression. Study concepts, then test yourself with 20 exam-style problems.
6Units
20Problems
20minTimer
★★★★★Difficulty
Unit 1 · Descriptive Statistics
Measures of Center & Spread
Descriptive statistics summarize the main features of a dataset using numerical measures and visualizations.
Mean: x̄ = (Σxᵢ) / n
Median: middle value when sorted
Variance: s² = Σ(xᵢ − x̄)² / (n−1)
Std Dev: s = √s²
IQR = Q3 − Q1
★ Memorize This
Use median & IQR for skewed data; use mean & SD for symmetric data
The Central Limit Theorem (CLT) is foundational: the distribution of sample means approaches normal as n increases, regardless of the population shape.
Sampling Dist of x̄: mean = μ, SE = σ/√n
CLT applies when n ≥ 30 (or population is normal)
Sampling Dist of p̂: mean = p, SE = √[p(1−p)/n]
Conditions for p̂: np ≥ 10 and n(1−p) ≥ 10
★ Memorize This
Larger n → smaller SE → sampling distribution is narrower
SE = σ/√n (standard error, not standard deviation)
CLT makes inference possible even for non-normal populations
Independence condition: sample size ≤ 10% of population
Unit 5 · Inference: Confidence Intervals & Hypothesis Testing
Confidence Intervals
A confidence interval gives a range of plausible values for a population parameter.
CI for μ (known σ): x̄ ± z* · (σ/√n)
CI for μ (unknown σ): x̄ ± t* · (s/√n), df = n−1
CI for p: p̂ ± z* · √[p̂(1−p̂)/n]
Margin of Error (ME) = z* · SE
Common z*: 90%→1.645, 95%→1.96, 99%→2.576
★ Memorize This
Wider CI = higher confidence OR smaller n OR larger σ
Use t* when σ is unknown and/or n is small
"We are 95% confident the true parameter lies in [a, b]" — correct interpretation
Margin of Error = half the width of the interval
To halve ME: multiply n by 4
📝 Example
x̄ = 82, s = 10, n = 25. Construct a 95% CI for μ.
df = 24, t* ≈ 2.064
ME = 2.064 × (10/5) = 4.128
CI: (77.87, 86.13)
Hypothesis Testing
Hypothesis testing uses sample data to evaluate claims about population parameters.
H₀: null hypothesis (no effect/difference)
Hₐ: alternative hypothesis
Test Statistic (z): z = (x̄ − μ₀) / (σ/√n)
Test Statistic (t): t = (x̄ − μ₀) / (s/√n)
p-value: probability of result as extreme, given H₀ true
Reject H₀ if p-value < α
★ Memorize This
Type I error (α): reject H₀ when H₀ is true (false positive)
Type II error (β): fail to reject H₀ when Hₐ is true (false negative)
Power = 1 − β = P(reject H₀ | Hₐ true)
Small p-value → strong evidence against H₀
Never "accept H₀" — only "fail to reject H₀"
📝 Example
H₀: μ = 100; Hₐ: μ ≠ 100. x̄ = 104, s = 12, n = 36, α = 0.05.
t = (104−100)/(12/6) = 4/2 = 2.0, df = 35
Two-tailed p-value ≈ 0.053 > 0.05
Decision: Fail to reject H₀
Unit 6 · Regression & Chi-Square
Linear Regression & Chi-Square Tests
Regression models relationships between variables; Chi-Square tests compare categorical frequencies.
Regression Line: ŷ = b₀ + b₁x
Slope: b₁ = r · (sy/sx)
Intercept: b₀ = ȳ − b₁x̄
Residual = y − ŷ
r² = proportion of variation in y explained by x
χ² = Σ (O − E)² / E
Chi-Square GOF df = k − 1
Chi-Square Independence df = (r−1)(c−1)
★ Memorize This
r is the correlation coefficient: −1 ≤ r ≤ 1
r close to ±1 = strong linear association; r near 0 = weak/no linear association
Correlation ≠ Causation
Residual plot should show no pattern (random scatter) for a good fit
Expected frequency in Chi-Square = (Row total × Column total) / Grand total
📝 Example
r = 0.8, sx = 5, sy = 10, x̄ = 3, ȳ = 20. Find the regression line.
b₁ = 0.8 × (10/5) = 1.6
b₀ = 20 − 1.6(3) = 15.2
Line: ŷ = 15.2 + 1.6x
Statistics Practice Problems — 20 Questions
20 Exam-Style Problems
Short-answer format · All units covered · Enter your answer and submit