Skip to main content

Hypothesis testing practice problems and answers

📘 Example: F-Test for Two Variances

đŸŽ¯ Problem:

Two factories produce the same type of product. A quality inspector wants to know if their product weight variances are different.

The data:

  • Factory A:
    • Sample size
    • Sample variance
  • Factory B:
    • Sample size
    • Sample variance

At a 5% significance level, test whether the variances are significantly different.

✅ Step 1: Set Hypotheses

We are testing:

  • (variances are equal)
  • (variances are not equal)

✅ Step 2: Compute F-statistic

Use the formula:


F = \frac{\text{larger variance}}{\text{smaller variance}} = \frac{20}{10} = 2.00

Let:

  • Numerator df =
  • Denominator df =

✅ Step 3: Find Critical Values

We use a two-tailed test at Îą = 0.05 → Each tail has 0.025.

We need:

  • Upper critical value
  • Lower critical value

From F-distribution tables:

  • → so

  F_{0.025,14,11} = \frac{1}{3.21} \approx 0.311

✅ Step 4: Decision Rule

  • Reject if or
  • Our F = 2.00 → Lies between 0.311 and 3.29

✅ Step 5: Conclusion

Since is within the acceptance region, we do not reject the null hypothesis.

🔍 Conclusion: There is not enough evidence to say the variances are different. So, the product weight variation between the two factories is statistically similar.