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 $12.19 oz$ and a standard deviation of $0.09 oz$ , 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 $11.99 oz$ and $12.55 oz$ , the range rule of thumb can be used to estimate that the standard deviation should be less than $0.14 oz$ . Use the sample data to test the claim that the population of volumes has a standard deviation less than $0.14 oz$ . Use a 0.01 significance level. Complete parts (a) through (d) below.

b. Compute the test statistic.
$χ^{2} =$
(Round to three decimal places as needed.)

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Answer

Round the test statistic to three decimal places to get $5.786$ .

Steps

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 $χ^{2} = \frac{(n - 1) s^{2}}{σ^{2}}$ .

Step 3 ：Substitute the given values into the formula to find the test statistic: $χ^{2} = \frac{(15 - 1) * (0.09)^{2}}{(0.14)^{2}}$ .

Step 4 ：Calculate the value to get $χ^{2} = 5.785714285714285$ .

Step 5 ：Round the test statistic to three decimal places to get $5.786$ .

Answer

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