To test $H_{0}: \sigma=1.1$ versus $H_{1}: \sigma \neq 1.1$, a random sample of size $n=19$ is obtained from a population that is known to be normally distributed.
(a) If the sample standard deviation is determined to be $s=1.9$, compute the test statistic.
$\chi_{0}^{2}=53.702$ (Round to three decimal places as needed.)
(b) If the researcher decides to test this hypothesis at the $\alpha=0.10$ level of significance, determine the critical values.
The critical values are $\chi_{0.05}^{2}=\square$ and $\chi_{0.95}^{2}=$
(Round to three decimal places as needed.)
(c) Draw a chi-squared distribution and depict the critical region. Choose the correct graph below.
A.
B.
c.
D.
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(d) Will the researcher reject the null hypothesis? Why? Choose the correct answer below.
A. Yes, because the test statistic is in the critical region.
B. Yes, because the test statistic is not in the critical region.
C. No, because the test statistic is not in the critical region.
D. No, because the test statistic is in the critical region.

Step 1 ：The test statistic for a hypothesis test about a population standard deviation or variance is a chi-square statistic. The formula for the test statistic is: \(\chi^{2} = \frac{(n-1)s^{2}}{\sigma^{2}}\)

Step 2 ：Substitute the given values into the formula: \(\chi^{2} = \frac{(19-1)1.9^{2}}{1.1^{2}}\)

Step 3 ：Calculate the test statistic: \(\chi^{2} = 53.702\)

Step 4 ：The critical values for a chi-square distribution with \(n-1 = 18\) degrees of freedom and a significance level of \(\alpha = 0.10\) are \(\chi_{0.05}^{2}\) and \(\chi_{0.95}^{2}\). These values can be found in a chi-square distribution table or calculated using a statistical software.

Step 5 ：The critical values are \(\chi_{0.05}^{2} = 27.587\) and \(\chi_{0.95}^{2} = 10.117\)

Step 6 ：The chi-square distribution is skewed to the right. The critical region for a two-tailed test at the 0.10 level of significance is the area to the left of \(\chi_{0.05}^{2} = 27.587\) and to the right of \(\chi_{0.95}^{2} = 10.117\).

Step 7 ：The test statistic \(\chi^{2} = 53.702\) is in the critical region because it is greater than the critical value \(\chi_{0.05}^{2} = 27.587\).

Step 8 ：Therefore, the researcher will reject the null hypothesis because the test statistic is in the critical region. The correct answer is A. Yes, because the test statistic is in the critical region.