Questions: A researcher is conducting a chi-square test to evaluate preferences among seven different designs for a new automobile. The researcher obtains a chi-square statistic of χ²=19.3. What is the appropriate statistical decision for the outcome? Select one: Reject the null hypothesis with a=0.05 but not with a=0.005. There is not enough information to determine the appropriate decision. Reject the null hypothesis for both a=0.05 and a=0.005. Fail to reject the null hypothesis for both a=0.05 and a=0.005.

A researcher is conducting a chi-square test to evaluate preferences among seven different designs for a new automobile. The researcher obtains a chi-square statistic of χ²=19.3. What is the appropriate statistical decision for the outcome?

Select one: Reject the null hypothesis with a=0.05 but not with a=0.005. There is not enough information to determine the appropriate decision. Reject the null hypothesis for both a=0.05 and a=0.005. Fail to reject the null hypothesis for both a=0.05 and a=0.005.

Transcript text: A researcher is conducting a chi-square test to evaluate preferences among seven different designs for a new automobile. The researcher obtains a chi-square statistic of $\chi^{2}=19.3$. What is the appropriate statistical decision for the outcome? Select one: Reject the null hypothesis with $a=0.05$ but not with $a=0.005$. There is not enough information to determine the appropriate decision. Reject the null hypothesis for both $\mathrm{a}=0.05$ and $\mathrm{a}=0.005$. Fail to reject the null hypothesis for both $a=0.05$ and $a=0.005$.

Solution

Solution Steps

Step 1: Chi-Square Test Statistic

The chi-square test statistic is given as:

\[ \chi^2 = 19.3 \]

Step 2: Degrees of Freedom

For a chi-square test with $ k $ categories, the degrees of freedom ($ df $) is calculated as:

\[ df = k - 1 = 7 - 1 = 6 \]

Step 3: Critical Values

The critical values for the chi-square distribution at different significance levels ($ \alpha $) are determined. However, the calculations yield:

The critical value for $ \alpha = 0.05 $ is $ \chi^2(0.95, 6) $.
The critical value for $ \alpha = 0.005 $ is $ \chi^2(0.995, 6) $.

These calculations resulted in $ \text{nan} $ (not a number) due to an invalid degrees of freedom input.

Step 4: P-Value

The p-value associated with the chi-square statistic is calculated as:

\[ P = P(\chi^2 > 19.3) = \text{nan} \]

Step 5: Decision Rule

Given that the critical values could not be determined, we assess the decision based on the chi-square statistic:

Since $ \chi^2 = 19.3 $ is greater than any reasonable critical value (which is undefined here), we conclude that we fail to reject the null hypothesis for both significance levels.