Hypothesis Testing for Variance and Standard Deviation: Chi-Square Tests

Hypothesis Testing with One Sample

Section 7.5: Hypothesis Testing for Variance and Standard Deviation

This section introduces the use of the chi-square test for evaluating claims about population variance and standard deviation. The chi-square test is applicable only when the population is normally distributed and is used to determine whether sample data provide sufficient evidence to reject or support claims about variance or standard deviation.

• Variance (\(\sigma^2\)): A measure of the spread of data values in a population.

• Standard Deviation (\(\sigma\)): The square root of the variance, representing the average distance from the mean.

• Chi-Square Test: A statistical test used to compare the sample variance to a claimed population variance.

Finding Critical Values for a Chi-Square Test

Critical values are essential for determining the rejection region in hypothesis testing. The chi-square distribution is used, and the critical values depend on the degrees of freedom and the significance level.

• Level of Significance (\(\alpha\)): The probability of rejecting the null hypothesis when it is true.

• Degrees of Freedom (d.f.): Calculated as \(n - 1\), where \(n\) is the sample size.

• Right-Tailed Test: Use the critical value corresponding to d.f. and \(\alpha\).

• Left-Tailed Test: Use the critical value corresponding to d.f. and \(1 - \alpha\).

• Two-Tailed Test: Use the critical values corresponding to d.f. and \(\alpha/2\) and \(1 - \alpha/2\).

Example: Finding a Critical Value for a Right-Tailed Test

Suppose \(n = 26\) and \(\alpha = 0.05\). The degrees of freedom are \(25\). Using the chi-square table, find the critical value for a right-tailed test.

• Step 1: Calculate d.f.: \(n - 1 = 25\).

• Step 2: Use Table 6 in Appendix B to find the critical value for d.f. = 25 and \(\alpha = 0.05\).

Example: Finding a Critical Value for a Left-Tailed Test

Suppose \(n = 11\) and \(\alpha = 0.01\). The degrees of freedom are \(10\). Use Table 6 to find the critical value for d.f. = 10 and area 0.99.

• Step 1: Calculate d.f.: \(n - 1 = 10\).

• Step 2: Use Table 6 for d.f. = 10 and area 0.99.

Example: Finding Critical Values for a Two-Tailed Test

Suppose \(n = 9\) and \(\alpha = 0.05\). The degrees of freedom are \(8\). Use Table 6 for d.f. = 8 and areas 0.025 and 0.975.

• Step 1: Calculate d.f.: \(n - 1 = 8\).

• Step 2: Use Table 6 for d.f. = 8 and areas 0.025 and 0.975.

The Chi-Square Test for Variance or Standard Deviation

The chi-square test is used to test claims about population variance or standard deviation. The test statistic is calculated as follows:

• Test Statistic: $\chi^2 = \frac{(n-1)s^2}{\sigma^2}$

• Where \(s^2\) is the sample variance, \(\sigma^2\) is the claimed population variance, and \(n\) is the sample size.

• The test statistic follows a chi-square distribution with \(n-1\) degrees of freedom.

Steps for Hypothesis Testing for Variance or Standard Deviation

1. State the hypotheses: Null hypothesis (\(H_0\)) and alternative hypothesis (\(H_1\)).

2. Determine the significance level (\(\alpha\)) and degrees of freedom (d.f.).

3. Find the critical value(s) from the chi-square table.

4. Calculate the test statistic: $\chi^2 = \frac{(n-1)s^2}{\sigma^2}$

5. Compare the test statistic to the critical value(s) and make a decision.

Example: Hypothesis Test for the Population Variance

A dairy processing company claims that the variance of the amount of fat in whole milk is no more than 0.25. A sample of 41 milk containers has a variance of 0.27. At \(\alpha = 0.05\), is there enough evidence to reject the company’s claim?

• Step 1: State hypotheses: \(H_0: \sigma^2 \leq 0.25\), \(H_1: \sigma^2 > 0.25\).

• Step 2: Calculate d.f.: \(41 - 1 = 40\).

• Step 3: Find critical value for d.f. = 40 and \(\alpha = 0.05\).

• Step 4: Calculate test statistic: $\chi^2 = \frac{(41-1) \times 0.27}{0.25}$

• Step 5: Compare test statistic to critical value and make a decision.

Conclusion: If the test statistic is not in the rejection region, fail to reject the null hypothesis.

Example: Hypothesis Test for the Standard Deviation

A company claims that the standard deviation of the time to transfer a call is less than 1.4 minutes. A sample of 25 calls has a standard deviation of 1.1 minutes. At \(\alpha = 0.10\), is there enough evidence to support the claim?

• Step 1: State hypotheses: \(H_0: \sigma \geq 1.4\), \(H_1: \sigma < 1.4\).

• Step 2: Calculate d.f.: \(25 - 1 = 24\).

• Step 3: Find critical value for d.f. = 24 and area 0.90.

• Step 4: Calculate test statistic: $\chi^2 = \frac{(25-1) \times (1.1)^2}{(1.4)^2}$

• Step 5: Compare test statistic to critical value and make a decision.

Conclusion: If the test statistic is in the rejection region, reject the null hypothesis and support the claim.

Example: Hypothesis Test for the Population Variance (Two-Tailed)

A sporting goods manufacturer claims that the variance of the strength in a fishing line is 15.9. A sample of 15 spools has a variance of 21.8. At \(\alpha = 0.05\), is there enough evidence to reject the manufacturer’s claim?

• Step 1: State hypotheses: \(H_0: \sigma^2 = 15.9\), \(H_1: \sigma^2 \neq 15.9\).

• Step 2: Calculate d.f.: \(15 - 1 = 14\).

• Step 3: Find critical values for d.f. = 14 and areas 0.025 and 0.975.

• Step 4: Calculate test statistic: $\chi^2 = \frac{(15-1) \times 21.8}{15.9}$

• Step 5: Compare test statistic to critical values and make a decision.

Conclusion: If the test statistic is not in the rejection regions, fail to reject the null hypothesis.

Summary Table: Chi-Square Test Types

Test Type	Critical Value(s)	Rejection Region
Right-Tailed	\(\chi^2_{\alpha}\)	\(\chi^2 > \chi^2_{\alpha}\)
Left-Tailed	\(\chi^2_{1-\alpha}\)	\(\chi^2 < \chi^2_{1-\alpha}\)
Two-Tailed	\(\chi^2_{\alpha/2}\), \(\chi^2_{1-\alpha/2}\)	\(\chi^2 < \chi^2_{1-\alpha/2}\) or \(\chi^2 > \chi^2_{\alpha/2}\)

Important Notes

• The chi-square test for variance or standard deviation requires the population to be normally distributed.

• Always check the assumptions before applying the test.

• Use technology or statistical tables to verify critical values and test statistics.