Problem
Workers at a certain soda drink factory collected data on the volumes (in ounces) of a simple random sample of 15 cans of the soda drink. Those volumes have a mean of $12.19 \mathrm{oz}$ and a standard deviation of $0.09 \mathrm{oz}$, and they appear to be from a normally distributed population. If the workers want the filling process to work so that almost all cans have volumes between $11.99 \mathrm{oz}$ and $12.55 \mathrm{oz}$, the range rule of thumb can be used to estimate that the standard deviation should be less than $0.14 \mathrm{oz}$. Use the sample data to test the claim that the population of volumes has a standard deviation less than $0.14 \mathrm{oz}$. Use a 0.01 significance level. Complete parts (a) through (d) below. b. Compute the test statistic. \[ \chi^{2}= \] (Round to three decimal places as needed.)
Solution
Step 1 ：We are given that the sample size (n) is 15, the sample standard deviation (s) is 0.09 oz, and the hypothesized population standard deviation (σ) is 0.14 oz.
Step 2 ：We can use the chi-square formula for testing a population variance or standard deviation, which is \(\chi^{2} = \frac{(n - 1)s^{2}}{\sigma^{2}}\).
Step 3 ：Substitute the given values into the formula to find the test statistic: \(\chi^{2} = \frac{(15 - 1) * (0.09)^{2}}{(0.14)^{2}}\).
Step 4 ：Calculate the value to get \(\chi^{2} = 5.785714285714285\).
Step 5 ：Round the test statistic to three decimal places to get \(\boxed{5.786}\).
