Problem
An electronic company's records show that the mean number of circuit boards rejected per batch is 5.5 with a variance of 1.2 . Due to a change in the molding process, factory management has hired you to perform a hypothesis test to check if the variance, $\sigma^{2}$, has increased. To do so, you test a random sample of 12 batches produced using the new process; the number of circuit boards rejected per batch has a sample variance of 2.2 . Under the assumption that the number of circuit boards rejected per batch using the new process follows a normal distribution, you will perform a chi-square test. Find $\chi^{2}$, the value of the test statistic for your chi-square test. Round your answer to three or more decimal places.
Solution
Step 1 ：Given that the sample size \(n = 12\), the sample variance \(s^2 = 2.2\), and the population variance \(\sigma^2 = 1.2\)
Step 2 ：The chi-square test statistic is calculated using the formula: \(\chi^2 = (n - 1) * s^2 / \sigma^2\)
Step 3 ：Substitute the given values into the formula: \(\chi^2 = (12 - 1) * 2.2 / 1.2\)
Step 4 ：Simplify the equation: \(\chi^2 = 11 * 2.2 / 1.2\)
Step 5 ：Further simplify the equation: \(\chi^2 = 24.2 / 1.2\)
Step 6 ：Finally, calculate the value of the test statistic: \(\chi^2 = 20.167\)
Step 7 ：So, the value of the test statistic for your chi-square test is \(\boxed{20.167}\)
