Hypothesis testing only clicks once you’ve run the five steps yourself, repeatedly, on new scenarios. The Statistics Made Simple Practice Questions workbook has dozens of worked hypothesis tests with full mark schemes.
Statistics · Hypothesis Testing
Hypothesis Testing
A complete step-by-step guide — from stating your hypotheses to reaching a conclusion, with worked examples from economics and social science.
Does a minimum wage increase reduce employment? Does a new teaching method improve test scores? These are empirical questions — and hypothesis testing is the statistical framework that lets us answer them rigorously using data, even when we cannot observe the entire population.
📘 Key Terms
Null Hypothesis (H₀): The default position — usually that there is no effect, no difference, or no relationship. We assume H₀ is true until evidence suggests otherwise.
Alternative Hypothesis (H₁ or Hₐ): The claim we want to test for — that there is an effect, difference, or relationship. We reject H₀ in favour of H₁ only when evidence is strong enough.
Chapters 10 and 11 of Statistics Made Simple cover the full hypothesis-testing framework, from stating H₀ to interpreting the result in context.
The 5 Steps of Hypothesis Testing
Step 1: State the Hypotheses
H₀: μ = μ₀ (no effect) | H₁: μ ≠ μ₀ (two-tailed) or H₁: μ > μ₀ / H₁: μ < μ₀ (one-tailed)
Step 2: Choose Significance Level (α)
Common choices: α = 0.05 (5%) or α = 0.01 (1%). This is the maximum acceptable probability of incorrectly rejecting a true H₀.
Step 3: Calculate the Test Statistic
For a z-test: z = (x̄ − μ₀) / (σ/√n). For a t-test (σ unknown): t = (x̄ − μ₀) / (s/√n) with (n−1) degrees of freedom.
Step 4: Find the Critical Value or P-Value
Look up the critical value from the Z or t table, OR calculate the p-value from the test statistic. Compare against α.
Step 5: State the Conclusion
If p ≤ α (or |test stat| ≥ critical value): Reject H₀. If p > α: Fail to reject H₀. Always state the conclusion in context.
Worked Example: Minimum Wage Study
Context: Before a minimum wage increase, mean employment in the sector was 500 workers per firm (μ₀ = 500). After the increase, a sample of n = 36 firms shows x̄ = 488 workers and s = 42.
H₀: μ = 500 (no change) | H₁: μ < 500 (employment fell) | α = 0.05
t-statistic: t = (488 − 500) / (42/√36) = −12/7 = −1.71 | df = 35
Critical value (one-tailed, df=35, α=0.05): −1.690
Conclusion: −1.71 < −1.690, so we reject H₀. There is significant evidence at the 5% level that employment fell after the minimum wage increase.
💡 Key Insight
Rejecting H₀ does not prove H₁ is true — it means the data are inconsistent with H₀ at your chosen significance level. The legal analogy is useful: courts don’t find defendants ‘innocent,’ they find them ‘not guilty.’ Similarly, in statistics we don’t ‘accept H₀’ — we either reject it or fail to reject it.
⚠️ Common Error
Avoid stating ‘we accept H₀.’ The correct phrase is ‘we fail to reject H₀.’ Failing to find evidence against something is not the same as proving it is true. Also, never choose α after seeing the data — significance levels must be set in advance to maintain the integrity of the test.
Statistics Made Simple
The method and the practice, together.
Both books use matching chapter numbering, so you can read a technique like this one and work matched drills side by side — 1,569 questions in total, all with fully worked answers.
Q1. A researcher believes that mean weekly hours worked is more than 37 hours. A sample of n = 25 workers gives x̄ = 38.8 and s = 5. Conduct a hypothesis test at α = 0.05. [7 marks]
Answer: H₀: μ = 37 | H₁: μ > 37 (one-tailed). t = (38.8 − 37)/(5/√25) = 1.8/1 = 1.80. df = 24. Critical value (one-tailed, α = 0.05, df = 24) = 1.711. Since 1.80 > 1.711, we reject H₀. There is significant evidence at the 5% level that mean weekly hours worked exceeds 37 hours.
Every worked hypothesis test in the Statistics Made Simple bundle was verified by independent computation before publication, so the method you practise is one you can trust under exam conditions.
References
1. Moore, D.S., McCabe, G.P. and Craig, B.A. (2021) Introduction to the Practice of Statistics. W.H. Freeman.
2. Field, A. (2018) Discovering Statistics Using IBM SPSS Statistics. SAGE.
3. Triola, M.F. (2022) Elementary Statistics. Pearson.
4. Wonnacott, T.H. and Wonnacott, R.J. (1990) Introductory Statistics for Business and Economics. Wiley.
5. Freedman, D., Pisani, R. and Purves, R. (2007) Statistics. W.W. Norton.