Data show that men between the ages of 20 and 29 in a general population have a mean height of 69.3 inches, with a standard deviation of 2.4 inches. A baseball analyst wonders whether the standard deviation of heights of major-league baseball players is less than 2.4 inches. The heights (in inches) of 20 randomly selected players are shown in the table.
Click the icon to view the data table.
Test the notion at the $\alpha =0.01$ level of significance.

Calculate the value of the test statistic.
${\chi }^2=◻$ (Round to three decimal places as needed.)
Data table
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Step 1 ：The problem is asking to test the hypothesis that the standard deviation of heights of major-league baseball players is less than 2.4 inches. This is a one-tailed chi-square test for variance.

Step 2 ：The test statistic for a chi-square test is calculated as: $${\chi }^2=\frac{(n-1)s^2}{{\sigma }^2}$$ where n is the sample size, s is the sample standard deviation, and σ is the population standard deviation.

Step 3 ：First, we need to calculate the sample standard deviation from the given data. The data is [72, 74, 71, 71, 76, 70, 77, 75, 72, 72, 77, 73, 75, 70, 73, 74, 75, 73, 74, 73]. The sample standard deviation (s) is calculated to be approximately 2.084.

Step 4 ：Then we can substitute the values into the formula to find the test statistic. The population standard deviation (σ) is given as 2.4 and the sample size (n) is 20.

Step 5 ：Substituting the values into the formula, we get $${\chi }^2=\frac{(20-1)\ast {2.084}^2}{{2.4}^2}$$

Step 6 ：Solving the above expression, we get ${\chi }^2=14.331597222222221$

Step 7 ：Rounding to three decimal places, the final answer is $\boxed{{\displaystyle 14.332}}$