Intro to Business Statistics: Hypothesis Testing ChiSquare Test for Variance

What This Is

The Chi-Square Test for Variance is a statistical method used to determine if a population variance is equal to a known value or if it differs significantly from a hypothesized value. A retail chain wants to know if the variance of daily sales exceeds $10,000, which would indicate a need to adjust inventory levels and staffing.

Key Formulas & Symbols

• χ² = Σ (x_i - μ)^2 / σ^2 where x_i = individual data points, μ = population mean, σ^2 = population variance.
• χ² = (n-1) * s^2 / σ^2 where n = sample size, s^2 = sample variance.
• χ² distribution: a theoretical distribution used to determine the probability of observing a given χ² value.
• Degrees of Freedom (df): n - 1, where n is the sample size.
• Critical Value (χ²_c): the χ² value that separates the rejection region from the non-rejection region.
• p-value: the probability of observing a χ² value at least as extreme as the one observed, assuming the null hypothesis is true.
• Null Hypothesis (H₀): the hypothesis that the population variance is equal to a known value (e.g., σ^2 = 10000).
• Alternative Hypothesis (H_a): the hypothesis that the population variance is not equal to the known value (e.g., σ^2 ≠ 10000).

Step-by-Step Procedure

1. State hypotheses: Define the null and alternative hypotheses.
2. Choose test: Select the Chi-Square Test for Variance.
3. Compute test statistic: Calculate the χ² value using the formula χ² = Σ (x_i - μ)^2 / σ^2 or χ² = (n-1) * s^2 / σ^2.
4. Find p-value or critical value: Determine the p-value or critical value using a χ² distribution table or calculator.
5. Compare to α: Compare the p-value or critical value to the significance level (α = 0.05).
6. Conclude: Reject the null hypothesis if the p-value is less than α or the χ² value exceeds the critical value.

Common Mistakes

• Mistake: Using the Z-test instead of the Chi-Square Test for Variance when the population variance is unknown.
• Correction: Use the Chi-Square Test for Variance when the population variance is unknown, as it is a more robust test.
• Mistake: Misinterpreting the p-value as the probability that the null hypothesis is true.
• Correction: The p-value is the probability of observing the data (or more extreme) if the null hypothesis is true.
• Mistake: Failing to check the assumptions of the Chi-Square Test for Variance (e.g., normality of residuals).
• Correction: Check the assumptions before conducting the test to ensure the results are valid.

Quick Practice Problems

1. A manufacturing company wants to know if the variance of production time exceeds 4 hours. A sample of 25 production times has a sample variance of 3.5 hours^2. The population variance is unknown. What is the p-value? Answer: 0.023, The p-value is calculated using the χ² distribution with 24 degrees of freedom.
2. A retail chain wants to know if the variance of daily sales is equal to $10,000. A sample of 36 daily sales has a sample variance of $12,000. The population variance is known to be $10,000. What is the critical value? Answer: 34.17, The critical value is determined using a χ² distribution table with 35 degrees of freedom and α = 0.05.
3. A quality control team wants to know if the variance of defect rates is equal to 0.05. A sample of 49 defect rates has a sample variance of 0.07. The population variance is unknown. What is the χ² value? Answer: 2.44, The χ² value is calculated using the formula χ² = (n-1) * s^2 / σ^2.

Last-Minute Cram Sheet

1. χ² = Σ (x_i - μ)^2 / σ^2 is used when the population variance is known.
2. χ² = (n-1) * s^2 / σ^2 is used when the population variance is unknown.
3. χ² distribution is used to determine the probability of observing a given χ² value.
4. Degrees of Freedom (df) = n - 1, where n is the sample size.
5. Critical Value (χ²_c) separates the rejection region from the non-rejection region.
6. p-value is the probability of observing the data (or more extreme) if the null hypothesis is true.
7. Null Hypothesis (H₀) states that the population variance is equal to a known value.
8. Alternative Hypothesis (H_a) states that the population variance is not equal to the known value.
9. α = 0.05 is the default significance level.
10. ⚠️ p-value is NOT the probability that H₀ is true – it’s the probability of observing the data (or more extreme) if H₀ is true.