Statistics 2: Statistical Inference: practice problems and exam questions

From the normal distribution to confidence intervals and hypothesis tests: the inferential statistics that follow an introductory course.

Practise this course for freePractise this courseTake a mock exam5 free problems every day. No card needed.

Cram pack (PDF)Formula sheets, method steps and practice problems with full solutions, ready to print.

Topics in this course

  1. Normal distributionarea in one tail · area between two values · percentilescalculations
  2. Confidence intervalsz interval · t interval · sample sizecalculations
  3. Hypothesis tests & confidence intervalsz statistic · t statistic · confidence intervalscalculations
  4. Linear regression & correlationslope and intercept · correlation coefficient · predictioncalculations
  5. Chi-square testsgoodness of fit · test of independence · critical value decisioncalculations

Exam tips for Statistics 2: Statistical Inference

  • Standardise first: almost every question starts with a z-score or a t-score.
  • Use t instead of z when σ is unknown and the sample is small.
  • A 95% confidence interval and a two-sided test at 5% always agree with each other.

Worked examples

Normal distributionEasy
μ = 3236.8P = ?

The delivery times are normally distributed with mean 32 min and standard deviation 6 min. What fraction is above 36.8 min? Find P(X > x) as a decimal.

μ = 32σ = 6x = 36.8

Find P(X > x)

Show hints
  1. Convert to a z-score: z = (x − μ)/σ.
  2. z = (36.8 − 32)/6 = 0.8.
  3. The area to the right is 1 − Φ(z).
Show the answer and the step-by-step solution

Normal distribution: convert a value x to a z-score, z = (x − μ)/σ, which counts standard deviations from the mean. A z-table (or Φ(z)) gives the area to the left of z, which is P(X < x). The area to the right is 1 − Φ(z), and the area between two values is the difference of their two areas.

z-score

z = (36.8 − 32)/6 = 0.8

Probability

P = 1 − Φ(0.8) = 1 − 0.7881 = 0.2119

Answer

P = 0.2119

That is 21.19%.

Confidence intervalsMedium
x̄E₁ = 7.895%, n = 50x̄E₂ = ?95%, n = 100

A 95% confidence interval for a mean, based on 50 observations and a known σ, has a margin of error of 7.8. What is the margin of error E₂ if the same study uses 100 observations instead?

95%n₁ = 50E₁ = 7.8n₂ = 100

Find E₂

Show hints
  1. E = z·σ/√n, and σ stays the same.
  2. E₂/E₁ = (z₂/z₁)·√(n₁/n₂).
  3. z₁ = 1.96, z₂ = 1.96.
Show the answer and the step-by-step solution

Confidence interval for a mean: x̄ ± margin of error. With a known population σ the margin is z·σ/√n; with only the sample standard deviation s, use t with n − 1 degrees of freedom: t·s/√n. Critical values: z = 1.645 (90%), 1.96 (95%), 2.576 (99%). To get a margin E or smaller, you need n ≥ (z·σ/E)², rounded up.

Ratio of margins

E₂/E₁ = (1.96/1.96)·√(50/100) = 0.7071

New margin

E₂ = 7.8·0.7071 = 5.515

Answer

E₂ = 5.515

How practising works

  • “Practise this course” mixes problems from all its topics, like a real exam.
  • Every problem has new values, so you learn the method instead of an answer.
  • Stuck? Open a small hint first, and the full explanation only when you need it.
  • Mock exams are timed and have no hints, and you get your score and a full review at the end.

Calculators for this course

Z-score and normal probability calculatorConfidence interval calculatorSample size calculatorLinear regression calculatorChi-square goodness-of-fit calculatorNormal distribution calculatort-test calculatorP-value calculatorCorrelation coefficient calculator (Pearson and Spearman)One-way ANOVA calculator

Related courses

AP Statistics practice problemsResearch Methods and Statistics for PsychologyIntroductory StatisticsCFA Level I quantitative methods and fixed income practiceEvidence-Based PracticeEngineering MathematicsFE Civil exam practiceCalculus 1
Practise this course for freePractise this courseTake a mock exam5 free problems every day. No card needed.