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 chart on the right with $\chi_0^2 = 31.4100$ and $\chi^2$ axis up to 55)
(c) choose the correct answer below.
\\(\bigcirc\\) a. reject the null hypothesis because the standardized test statistic is not in the rejection region.
\\(\bigcirc\\) b. do not reject the null hypothesis because the standardized test statistic is in the rejection region.
\\(\bigcirc\\) c. do not reject the null hypothesis because the standardized test statistic is not in the rejection region.
\\(\bigcirc\\) d. reject the null hypothesis because the standardized test statistic is in the rejection region.
(d) choose the correct answer below.
\\(\bigcirc\\) a. reject the null hypothesis because the standardized test statistic is not in the rejection region.
\\(\bigcirc\\) b. do not reject the null hypothesis because the standardized test statistic is in the rejection region.
\\(\bigcirc\\) c. do not reject the null hypothesis because the standardized test statistic is not in the rejection region.
\\(\bigcirc\\) d. reject the null hypothesis because the standardized test statistic is in the rejection region

Explanation:

To determine whether to reject the null hypothesis, we compare the test statistic \(\chi^2\) with the critical value \(\chi_0^2 = 31.4100\) (from the graph, the rejection region is to the right of \(\chi_0^2\)).

Part (c)

Step 1: Identify the test statistic and critical value

The test statistic for part (c) is \(\chi^2 = 1.974\), and the critical value is \(\chi_0^2 = 31.4100\).

Step 2: Compare the test statistic with the critical value

Since \(1.974 < 31.4100\), the test statistic is not in the rejection region (which is to the right of \(31.4100\)). By the decision rule, if the test statistic is not in the rejection region, we do not reject the null hypothesis.

Part (d)

Step 1: Identify the test statistic and critical value

The test statistic for part (d) is \(\chi^2 = 31.492\), and the critical value is \(\chi_0^2 = 31.4100\).

Step 2: Compare the test statistic with the critical value

Since \(31.492 > 31.4100\), the test statistic is in the rejection region (to the right of \(31.4100\)). By the decision rule, if the test statistic is in the rejection region, we reject the null hypothesis.

Answer:

(c) C. Do not reject the null hypothesis because the standardized test statistic is not in the rejection region.
(d) D. Reject the null hypothesis because the standardized test statistic is in the rejection region.