Question

click the icon to view the chi-square critical values table.
identify the null and alternative hypotheses.
a. $h_0: \sigma \geq 39$
$h_a: \sigma < 39$
b. $h_0: \sigma \leq 39$
$h_a: \sigma > 39$
c. $h_0: \sigma > 39$
$h_a: \sigma \leq 39$
d. $h_0: \sigma < 39$
$h_a: \sigma \geq 39$
identify the standardized test statistic.
10.776 (round to three decimal places as needed.)
identify the critical value(s).
(round to three decimal places as needed. use a comma to separate answers as needed.)

Explanation:

Step1: Recall Chi - Square Test for Variance

When testing a hypothesis about a population standard deviation (or variance), the null hypothesis \(H_0\) and alternative hypothesis \(H_a\) are set up such that the null hypothesis contains the equality or the "no - change" or "status - quo" statement. For a left - tailed test (where we are testing if \(\sigma<\) a certain value), the null hypothesis is \(H_0:\sigma\geq k\) and the alternative hypothesis is \(H_a:\sigma < k\). Here, \(k = 39\). So option A (\(H_0:\sigma\geq39\), \(H_a:\sigma < 39\)) is correct.

Step2: Standardized Test Statistic

The standardized test statistic for a chi - square test for variance is given by the formula \(\chi^{2}=\frac{(n - 1)s^{2}}{\sigma_{0}^{2}}\), where \(n\) is the sample size, \(s\) is the sample standard deviation, and \(\sigma_{0}\) is the hypothesized population standard deviation. The value provided (10.776) is the calculated chi - square test statistic.

Step3: Critical Value for Left - Tailed Test

For a left - tailed chi - square test, we need to find the critical value \(\chi^{2}_{1-\alpha}\) with degrees of freedom \(df=n - 1\). However, since the sample size \(n\) is not given in the problem, we assume that we can use the chi - square critical values table. But from the context, if we assume a common significance level (e.g., \(\alpha = 0.05\)) and we need to find the critical value for a left - tailed test. But since the problem is about identifying the critical value, and we know that for a left - tailed test with \(H_0:\sigma\geq39\) and \(H_a:\sigma < 39\), we need to look up the chi - square critical value for \(df=n - 1\) and \(\alpha\) (significance level) in the left - tail. But since the sample size is not provided, we can't calculate it directly from the given information. But if we assume that the user is asking about the null - alternative hypothesis part first, the correct option for null and alternative hypothesis is A.

Answer:

A. \(H_0:\sigma\geq39\), \(H_a:\sigma < 39\)