Question

state whether each standardized test statistic \\(\chi^2\\) allows you to reject the null hypothesis. explain.
(a) \\(\chi^2 = 29.606\\) (b) \\(\chi^2 = 0\\)
(c) \\(\chi^2 = 1.974\\) (d) \\(\chi^2 = 31.492\\)
(there is a chi - square distribution graph with \\(\chi_0^2 = 31.4100\\) and the x - axis up to 55, and a shaded region on the right side of \\(\chi_0^2\\))
(b) choose the correct answer below.
\\(\circ\\) a. reject the null hypothesis because the standardized test statistic is not in the rejection region.
\\(\circ\\) b. do not reject the null hypothesis because the standardized test statistic is not in the rejection region.
\\(\circ\\) c. do not reject the null hypothesis because the standardized test statistic is in the rejection region.
\\(\circ\\) d. reject the null hypothesis because the standardized test statistic is in the rejection region.
(c) choose the correct answer below.
\\(\circ\\) a. reject the null hypothesis because the standardized test statistic is not in the rejection region.
\\(\circ\\) b. do not reject the null hypothesis because the standardized test statistic is in the rejection region.

Explanation:

Part (b)

Step1: Identify Rejection Region

The rejection region for a chi - square test (from the graph) is where \(\chi^{2}\geq\chi_{0}^{2} = 31.4100\). For \(\chi^{2}=0\), we check if \(0\geq31.4100\). Since \(0<31.4100\), the test statistic is not in the rejection region.

Step2: Decision Rule

If the test statistic is not in the rejection region, we do not reject the null hypothesis.

Step1: Identify Rejection Region

The rejection region is \(\chi^{2}\geq31.4100\). For \(\chi^{2}=1.974\), we check if \(1.974\geq31.4100\). Since \(1.974 < 31.4100\), the test statistic is not in the rejection region.

Step2: Decision Rule

If the test statistic is not in the rejection region, we do not reject the null hypothesis. But looking at the options, let's re - evaluate. Wait, there is a mistake in the initial analysis. Wait, the options for (c) are:
A. Reject the null hypothesis because the standardized test statistic is not in the rejection region.
B. Do not reject the null hypothesis because the standardized test statistic is in the rejection region.

Wait, first, the rejection region is \(\chi^{2}\geq31.41\). For \(\chi^{2}=1.974\), it is not in the rejection region. The correct decision is to not reject the null hypothesis. But option B says "because the standardized test statistic is in the rejection region" which is wrong. Wait, maybe there is a typo in the options, but based on the calculation:

Since \(\chi^{2}=1.974<31.41\) (not in rejection region), we do not reject the null hypothesis. But the given options for (c) have a mistake. However, if we assume that maybe the rejection region is on the left (but chi - square is non - negative and right - tailed for goodness - of - fit or test of variance), but no, chi - square distribution is right - tailed. So for \(\chi^{2}=1.974\), it is not in the rejection region. So the correct answer should be (but the options are misformatted). Wait, the user's (c) options:

A. Reject the null hypothesis because the standardized test statistic is not in the rejection region.

B. Do not reject the null hypothesis because the standardized test statistic is in the rejection region.

This is a problem with the options. But if we follow the logic, since the test statistic is not in the rejection region, we do not reject. But option B's reason is wrong. However, maybe it's a typo and option B's reason should be "not in the rejection region". But based on the given options, the intended answer (assuming a typo in option B's reason) would be that we do not reject. But since the options are as given, let's re - check.

Wait, maybe I made a mistake. Wait, the original problem's (c) options:

A. Reject the null hypothesis because the standardized test statistic is not in the rejection region.

B. Do not reject the null hypothesis because the standardized test statistic is in the rejection region.

This is inconsistent. But based on the chi - square test:

1. The rejection region for a right - tailed chi - square test is \(\chi^{2}\geq\chi_{\alpha}^{2}\). Here \(\chi_{\alpha}^{2}=31.41\).

1. For \(\chi^{2}=1.974\), it is less than 31.41, so it is not in the rejection region.

1. The decision: If the test statistic is not in the rejection region, we do not reject the null hypothesis. But option B's reason is wrong. However, since option A's action is wrong (we don't reject when not in rejection region), and option B's reason is wrong, but maybe it's a mistake in the question. But if we have to choose from the given options, there is an error. But assuming that the options are misprinted, and the correct answer should be that we do not reject the null hypothesis, but since option B's reason is wrong, maybe the intended answer is:

Wait, no, let's start over.

For part (c):

• Rejection region: \(\chi^{2}\geq31.41\) (from the graph, the shaded region is where \(\chi^{2}\geq31.41\))

• Test statistic: \(\chi^{2}=1.974\)

• Since \(1.974<31.41\), the test statistic is not in the rejection regi…

Answer:

B. Do not reject the null hypothesis because the standardized test statistic is not in the rejection region.

Part (c)