Final answer:
To test if the population variance exceeds the specified variance in manufacturing, a right-tailed chi-square test for a single variance is used with the null hypothesis H0: σ² = 0.0004 and alternative hypothesis Ha: σ² > 0.0004. The sample variance, degrees of freedom, and alpha level determine if the null hypothesis is rejected in favor of the alternative.
Step-by-step explanation:
To test whether the population variance specification is being violated when manufacturing parts with tight tolerances, we can use a chi-square test for a single variance. Given that parts specification calls for a maximum variance of 0.0004 and the sample variance is 0.0005 for 26 parts with an alpha level of 0.05, we set up our hypotheses:
- H0 (null hypothesis): σ² = 0.0004
- Ha (alternative hypothesis): σ² > 0.0004
The null hypothesis (H0) states that the population variance is equal to the specified variance of 0.0004. The alternative hypothesis (Ha) suggests that the population variance is greater than the specification. Because our alternative hypothesis indicates a 'greater than' condition, this is a right-tailed test.
For a chi-square test, the test statistic would be calculated using the formula χ² = (n - 1)s² / σ², where s² is the sample variance and σ² is the specified population variance. The degrees of freedom (df) for this test would be n - 1, which equals 25. We then compare the calculated chi-square statistic to the critical value at 0.05 significance level with 25 degrees of freedom to make a decision about the null hypothesis.