396 CHAPTER 7 Hypothesis Testing with One Sample The Chi-Square Test To test a variance s 2 or a standard deviation s of a population that is normally distributed, you can use the chi-square test. The chi-square test for a variance or standard deviation is not as robust as the tests for the population mean m or the population proportion p. So, it is essential in performing a chi-square test for a variance or standard deviation that the population be normally distributed. The results can be misleading when the population is not normal. The chi-square test for a variance S 2 or standard deviation S is a statistical test for a population variance or standard deviation. The chi-square test can only be used when the population is normal. The test statistic is s2 and the standardized test statistic x 2 = 1n - 12s2 s 2 Standardized test statistic for s 2 or s follows a chi-square distribution with degrees of freedom d.f. = n - 1. Chi-Square Test for a Variance S 2 or Standard Deviation S In Step 8 of the guidelines below, the decision rule uses rejection regions. You can also test a claim using P-values (see Exercises 31–34). Using the Chi-Square Test for a Variance S 2 or a Standard Deviation S In Words In Symbols 1. Verify that the sample is random and the population is normally distributed. 2. State the claim mathematically State H0 and Ha. and verbally. Identify the null and alternative hypotheses. 3. Specify the level of significance. Identify a. 4. Identify the degrees of freedom. d.f. = n - 1 5. Determine the critical value(s). Use Table 6 in Appendix B. 6. Determine the rejection region(s). 7. Find the standardized test statistic x 2 = 1n - 12s2 s 2 and sketch the sampling distribution. 8. Make a decision to reject or fail to If x 2 is in the rejection region, reject the null hypothesis. then reject H0. Otherwise, fail to reject H0. 9. Interpret the decision in the context of the original claim. GUIDELINES For Step 5 of the guidelines, in addition to using Table 6 in Appendix B, you can use technology to find the critical value(s). Also, some technology tools allow you to perform a hypothesis test for a variance (or a standard deviation) using only the descriptive statistics.
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