Intro to Business Statistics: Estimation Confidence Interval for Population Variance ChiSquare

What This Is

A confidence interval for population variance (chi-square) is a statistical method used to estimate the variability of a population based on a sample. This is crucial in business decisions, such as determining the quality of a product, assessing the risk of a financial investment, or evaluating the effectiveness of a marketing campaign. For instance, a manufacturing company wants to know if the variability of their product's weight exceeds a certain threshold to ensure it meets customer expectations.

Key Formulas & Symbols

• χ² = Σ (x_i - x̄)² / σ² where x_i = individual data points, x̄ = sample mean, σ² = sample variance, and Σ denotes the sum of the squared differences.
• σ² = Σ (x_i - x̄)² / (n - 1) where n is the sample size.
• χ² = (n - 1) * s² / σ² where s² is the sample variance.
• χ² distribution: a probability distribution used to calculate the test statistic.
• Degrees of Freedom (df) = n - 1 where n is the sample size.
• Critical Value (χ² crit): the value from the chi-square distribution used to determine the rejection region.
• Test Statistic (χ²): the calculated value used to determine the p-value.
• p-value: the probability of observing the data (or more extreme) if the null hypothesis is true.
• Null Hypothesis (H₀): the statement that there is no significant difference in population variance.
• Alternative Hypothesis (H_a): the statement that there is a significant difference in population variance.

Step-by-Step Procedure

1. State hypotheses: Formulate the null and alternative hypotheses (e.g., H₀: σ² = σ₀² vs. H_a: σ² ≠ σ₀²).
2. Choose test: Select the chi-square test for population variance.
3. Compute test statistic: Calculate the test statistic using the formula χ² = Σ (x_i - x̄)² / σ² or χ² = (n - 1) * s² / σ².
4. Find p-value or critical value: Determine the p-value or critical value from the chi-square distribution using the calculated test statistic and degrees of freedom.
5. Compare to α: Compare the p-value to the significance level (α = 0.05) or compare the test statistic to the critical value.
6. Conclude: Make a decision based on the comparison (e.g., reject H₀ if p-value < α or test statistic > critical value).

Common Mistakes

• Mistake: Using the Z-test when the population standard deviation is unknown.
• Correction: Use the t-test or chi-square test when the population standard deviation is unknown.
• 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 (e.g., normality of the data, independence of observations).
• Correction: Verify that the data meet the assumptions before conducting the test.

Quick Practice Problems

1. A company wants to know if the variability of their product's weight exceeds 10 units. A sample of 25 products has a sample variance of 15 units². What is the 95% confidence interval for the population variance? Answer: (8.45, 34.55) Explanation: Calculate the test statistic using the formula χ² = (n - 1) * s² / σ², then find the critical value from the chi-square distribution with df = 24 and α = 0.05.
2. A financial analyst wants to determine if the variability of stock prices exceeds 100 units². A sample of 30 stocks has a sample variance of 120 units². What is the p-value of the test? Answer: 0.003 Explanation: Calculate the test statistic using the formula χ² = Σ (x_i - x̄)² / σ², then find the p-value from the chi-square distribution with df = 29.
3. A quality control engineer wants to know if the variability of a manufacturing process exceeds 5 units². A sample of 20 products has a sample variance of 6 units². What is the 99% confidence interval for the population variance? Answer: (2.45, 9.55) Explanation: Calculate the test statistic using the formula χ² = (n - 1) * s² / σ², then find the critical value from the chi-square distribution with df = 19 and α = 0.01.

Last-Minute Cram Sheet

1. χ² distribution: used to calculate the test statistic for population variance.
2. Degrees of Freedom (df) = n - 1: used to determine the critical value from the chi-square distribution.
3. Critical Value (χ² crit): used to determine the rejection region.
4. Test Statistic (χ²): calculated value used to determine the p-value.
5. p-value: probability of observing the data (or more extreme) if the null hypothesis is true.
6. Null Hypothesis (H₀): statement that there is no significant difference in population variance.
7. Alternative Hypothesis (H_a): statement that there is a significant difference in population variance.
8. Assumptions of the chi-square test: normality of the data, independence of observations.
9. ⚠️ 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.
10. ⚠️ Use the chi-square test when the population standard deviation is unknown.