Problem
A simple random sample of 26 filtered $100-\mathrm{mm}$ cigarettes is obtained from a normally distributed population, and the tar content of each cigarette is measured. The sample has a standard deviation of $0.20 \mathrm{mg}$. Use a 0.05 significance level to test the claim that the tar content of filtered $100-\mathrm{mm}$ cigarettes has a standard deviation different from $0.30 \mathrm{mg}$, which is the standard deviation for unfiltered king-size cigarettes. Complete parts (a) through (d) below. b. Find the test statistic. $\chi^{2}=\square$ (Round to three decimal places as needed.)
Solution
Step 1 ：Given that the sample size (n) is 26, the sample standard deviation (s) is 0.20 mg, and the hypothesized population standard deviation (σ) is 0.30 mg.
Step 2 ：The test statistic for a chi-square test for a single variance is given by the formula: \(\chi^{2} = \frac{(n - 1)s^{2}}{\sigma^{2}}\)
Step 3 ：Substitute the given values into the formula: \(\chi^{2} = \frac{(26 - 1)(0.20)^{2}}{(0.30)^{2}}\)
Step 4 ：Simplify the expression to find the test statistic.
Step 5 ：Final Answer: The test statistic is \(\boxed{11.111}\)
