How to Solve: Hypothesis Testing (One- and Two-Tailed Tests, p-Values, Critical Regions)

How to Solve: Hypothesis Testing (One- and Two-Tailed Tests, p-Values, Critical Regions)

GCSE / A-Level Maths

Introduction

"Mastering hypothesis testing lets you prove—with maths—whether a new drug works, a coin is biased, or a school’s exam results are truly improving. On your GCSE/A-Level exam, this topic is worth 10-15% of your stats paper—and one wrong step can cost you 5+ marks. Today, you’ll learn the exact method to solve any hypothesis test question, step by step."

What You Need To Know First

Before starting, you must understand: 1. Probability distributions (binomial, normal) and how to calculate probabilities. 2. Significance levels (α) – what 5% or 1% means in context. 3. Critical values – how to find them from tables or your calculator.

Key Vocabulary

Formulas To Know

Step-by-Step Method

Step 1: Define H₀ and H₁

• H₀: Always includes "=" (e.g., p = 0.5, μ = 10).
• H₁: Depends on the question:
• One-tailed (upper): H₁: p > 0.5 or μ > 10.
• One-tailed (lower): H₁: p < 0.5 or μ < 10.
• Two-tailed: H₁: p ≠ 0.5 or μ ≠ 10.

Step 2: State the Significance Level (α)

• Given in the question (e.g., 5%, 1%).
• If not given, assume 5%.

Step 3: Calculate the Test Statistic

• For binomial: Number of successes (e.g., 12 heads out of 20).
• For normal: Sample mean (x̄) or standardised z-score.

Step 4: Find the p-value or Critical Region

• p-value method:
• Calculate P(X ≥ test statistic) for upper-tailed, P(X ≤ test statistic) for lower-tailed, or 2 × P(X ≥ test statistic) for two-tailed.
• Critical region method:
• Find the critical value(s) from tables (e.g., for α = 5%, two-tailed, critical values are the 2.5th and 97.5th percentiles).

Step 5: Compare p-value to α or Test Statistic to Critical Region

• p-value ≤ α → Reject H₀.
• Test statistic in critical region → Reject H₀.

Step 6: Write Your Conclusion

• If you reject H₀: "There is sufficient evidence at the [α]% level to suggest [H₁]."
• If you do NOT reject H₀: "There is insufficient evidence at the [α]% level to suggest [H₁]."

Worked Examples

Example 1 – Basic (Binomial, One-Tailed)

Question: A coin is flipped 20 times, landing heads 14 times. Test at the 5% level whether the coin is biased towards heads.

Solution: 1. H₀: p = 0.5 (fair coin).
H₁: p > 0.5 (biased towards heads). 2. Significance level: α = 5%. 3. Test statistic: 14 heads. 4. p-value: P(X ≥ 14) where X ~ B(20, 0.5).
- From tables: P(X ≥ 14) = 1 – P(X ≤ 13) = 1 – 0.9423 = 0.0577. 5. Compare: 0.0577 > 0.05 → Do NOT reject H₀. 6. Conclusion: "There is insufficient evidence at the 5% level to suggest the coin is biased towards heads."

What we did and why: - We used a one-tailed test because the question asked "biased towards heads" (not just "biased"). - The p-value (0.0577) was just above 5%, so we couldn’t reject H₀.

Example 2 – Medium (Binomial, Two-Tailed)

Question: A dice is rolled 30 times, showing a 6 on 8 occasions. Test at the 1% level whether the dice is biased.

Solution: 1. H₀: p = 1/6 (fair dice).
H₁: p ≠ 1/6 (biased). 2. Significance level: α = 1% → 0.5% in each tail (two-tailed). 3. Test statistic: 8 sixes. 4. p-value: 2 × P(X ≥ 8) where X ~ B(30, 1/6).
- P(X ≥ 8) = 1 – P(X ≤ 7) = 1 – 0.8801 = 0.1199.
- p-value = 2 × 0.1199 = 0.2398. 5. Compare: 0.2398 > 0.01 → Do NOT reject H₀. 6. Conclusion: "There is insufficient evidence at the 1% level to suggest the dice is biased."

What we did and why: - Two-tailed test because the question asked "biased" (not "biased towards 6"). - We doubled the p-value because extreme results could be in either tail.

Example 3 – Exam-Style (Normal, One-Tailed)

Question: A factory claims its lightbulbs last 1000 hours on average. A sample of 50 bulbs has a mean lifetime of 990 hours with a standard deviation of 30 hours. Test at the 5% level whether the bulbs last less than claimed.

Solution: 1. H₀: μ = 1000 (bulbs last 1000 hours).
H₁: μ < 1000 (bulbs last less). 2. Significance level: α = 5%. 3. Test statistic: x̄ = 990, n = 50, σ = 30.
- z = (990 – 1000) / (30/√50) = -10 / 4.243 ≈ -2.36. 4. p-value: P(Z < -2.36) = 0.0091 (from normal tables). 5. Compare: 0.0091 < 0.05 → Reject H₀. 6. Conclusion: "There is sufficient evidence at the 5% level to suggest the bulbs last less than 1000 hours."

What we did and why: - Normal distribution because sample size > 30 (Central Limit Theorem). - One-tailed test because the question asked "less than."

Common Mistakes

Exam Traps

1-Minute Recap

"Here’s what you need to remember the night before your exam: 1. H₀ is always "="; H₁ is ">", "<", or "≠". 2. One-tailed test? Compare p-value to α. Two-tailed? Double the p-value or halve α. 3. p-value ≤ α? Reject H₀. Test statistic in critical region? Reject H₀. 4. Always write a conclusion—examiners love this! 5. Check the distribution: Binomial for small samples, normal for large (n > 30). Now go ace that exam!