Question

h_a: σ≤41
h_a: σ>41
c. h_0: σ<41
h_a: σ≥41
d. h_0: σ≥41
h_a: σ<41
identify the standardized test statistic.
18.709 (round to three decimal places as needed.)
identify the critical value(s).
7.015
(round to three decimal places as needed. use a comma to separate answers as needed.)
choose the correct conclusion below.
a. fail to reject h_0. there is enough evidence at the 1% level of significance to support the claim.
b. reject h_0. there is enough evidence at the 1% level of significance to support the claim.
c. fail to reject h_0. there is not enough evidence at the 1% level of significance to support the claim.
d. reject h_0. there is not enough evidence at the 1% level of significance to support the claim.

Explanation:

Step1: Recall Hypothesis Testing Rules

In hypothesis testing, we compare the test statistic to the critical value. If the test statistic is in the rejection region (beyond the critical value for a right - tailed test, or in the appropriate region for other tails), we reject \(H_0\); otherwise, we fail to reject \(H_0\). Here, we have a test about the standard deviation \(\sigma\), and the test statistic is \(18.709\) and the critical value is \(7.015\). Wait, actually, for a chi - square test (used for testing standard deviation or variance), the test statistic formula is \(\chi^{2}=\frac{(n - 1)s^{2}}{\sigma_{0}^{2}}\), and the critical value is from the chi - square distribution. But in terms of comparing the test statistic and critical value: if the test statistic is greater than the critical value (for a right - tailed test) or less than (for a left - tailed test), we make a decision. Wait, in this case, the alternative hypothesis is \(H_a:\sigma\lt41\), which is a left - tailed test for variance (since variance is \(\sigma^{2}\)). The test statistic for variance is \(\chi^{2}=\frac{(n - 1)s^{2}}{\sigma_{0}^{2}}\). But in terms of the values given: the test statistic is \(18.709\) and the critical value is \(7.015\). Wait, no, maybe I mixed up. Wait, actually, for a left - tailed test, we reject \(H_0\) if the test statistic is less than the critical value. But here, \(18.709>7.015\). Wait, no, maybe the test is a right - tailed? Wait, no, the null hypothesis is \(H_0:\sigma\geq41\) and alternative is \(H_a:\sigma\lt41\), so it's a left - tailed test for \(\sigma\), which is a left - tailed test for \(\sigma^{2}\) as well (since \(\sigma\) and \(\sigma^{2}\) are related monotonically). The test statistic for \(\sigma^{2}\) is \(\chi^{2}=\frac{(n - 1)s^{2}}{\sigma_{0}^{2}}\). If the test statistic is less than the critical value (left - tailed critical value), we reject \(H_0\). But here, \(18.709>7.015\), so we fail to reject \(H_0\)? Wait, no, maybe I got the direction wrong. Wait, let's think again. The null hypothesis is \(H_0:\sigma\geq41\) (or \(H_0:\sigma^{2}\geq41^{2}\)) and alternative is \(H_a:\sigma\lt41\) (or \(H_a:\sigma^{2}\lt41^{2}\)). The test statistic for variance is \(\chi^{2}=\frac{(n - 1)s^{2}}{\sigma_{0}^{2}}\). If the sample standard deviation \(s\) is less than \(\sigma_0 = 41\), then \(s^{2}\lt\sigma_0^{2}\), so \(\chi^{2}=\frac{(n - 1)s^{2}}{\sigma_{0}^{2}}\lt n - 1\). But in our case, the test statistic is \(18.709\) and the critical value is \(7.015\). Wait, maybe the critical value is a left - tailed critical value. So, for a left - tailed test, we reject \(H_0\) when the test statistic is less than the critical value. Since \(18.709>7.015\), we fail to reject \(H_0\). And since we fail to reject \(H_0\), there is not enough evidence at the \(1\%\) level of significance to support the claim (because the claim is in the alternative hypothesis \(H_a:\sigma\lt41\)).

Step2: Analyze the Options

- Option A: Says "Fail to reject \(H_0\). There is enough evidence...", but if we fail to reject \(H_0\), we don't have enough evidence to support the alternative (the claim), so A is wrong.
- Option B: Says "Reject \(H_0\)", but our test statistic is not in the rejection region (since \(18.709>7.015\) for a left - tailed test, the rejection region is below the critical value), so we don't reject \(H_0\), so B is wrong.
- Option C: "Fail to reject \(H_0\). There is not enough evidence at the \(1\%\) level of significance to support the claim." This matches our analysis. Because we fail to reject \(H_0\), we can't support the alternative hypothesis (the claim) at the \(1\%\) significance level.
- Option D: Says "Reject \(H_0\)", which is wrong as we saw.

Answer:

C. Fail to reject \(H_0\). There is not enough evidence at the \(1\%\) level of significance to support the claim.