Question

compute the standardized test statistic, $\chi^2$, to test the claim $\sigma^2 \leq 22.4$ if $n = 20$, $s^2 = 43.4$, and $\alpha = 0.01$.\
\
\bigcirc a. 9.322\
\bigcirc b. 33.41\
\bigcirc c. 12.82\
\bigcirc d. 36.813

Explanation:

Step1: Recall the formula for chi - square test statistic

The formula for the chi - square test statistic \(\chi^{2}\) when testing a claim about a population variance \(\sigma^{2}\) is \(\chi^{2}=\frac{(n - 1)s^{2}}{\sigma^{2}}\), where \(n\) is the sample size, \(s^{2}\) is the sample variance, and \(\sigma^{2}\) is the hypothesized population variance.

Step2: Identify the given values

We are given that \(n = 20\), \(s^{2}=43.4\), and the hypothesized population variance \(\sigma^{2}=22.4\) (from the claim \(\sigma^{2}\leq22.4\), we use \(\sigma^{2} = 22.4\) in the formula).

Step3: Substitute the values into the formula

First, calculate \(n-1\): \(n - 1=20 - 1=19\).
Then, substitute into the formula: \(\chi^{2}=\frac{(19)\times(43.4)}{22.4}\).
Calculate the numerator: \(19\times43.4 = 824.6\).
Then divide by the denominator: \(\frac{824.6}{22.4}\approx36.813\).

Answer:

D. 36.813