In hypothesis testing using the chi-square ($\chi^2$) test, the decision to reject or not reject the null hypothesis $H_0$ depends on the comparison between the test statistic ($\chi^2_0$) and the critical value ($\chi^2_{\alpha}$) for a given significance level $\alpha$.

• Reject $H_0$ if the test statistic $\chi^2_0$ is greater than the critical value $\chi^2_{\alpha}$, meaning that the result is significant, and the null hypothesis is unlikely to be true.
• Do not reject $H_0$ if the test statistic $\chi^2_0$ is less than the critical value $\chi^2_{\alpha}$, meaning that there is insufficient evidence to reject the null hypothesis.

Let's match this logic to your answer options:

1. Reject $H_0$ because $\chi^2_0 < \chi^2_{0.01}$: This is incorrect, because when the test statistic is less than the critical value, we should not reject the null hypothesis.
2. Do not reject $H_0$ because $\chi^2_0 > \chi^2_{0.01}$: This is also incorrect. If the test statistic is greater than the critical value, we should reject the null hypothesis.
3. Reject $H_0$ because $\chi^2_0 > \chi^2_{0.01}$: This is correct, as when the test statistic exceeds the critical value, we reject the null hypothesis.
4. Do not reject $H_0$ because $\chi^2_0 < \chi^2_{0.01}$: This is correct, as when the test statistic is less than the critical value, we do not reject the null hypothesis.

So, the correct conclusion depends on whether $\chi^2_0$ is greater than or less than $\chi^2_{0.01}$. If $\chi^2_0 > \chi^2_{0.01}$, we reject $H_0$; otherwise, we do not reject $H_0$.

Would you like more details or have any questions?

Here are 5 related questions for further exploration:

1. How is the chi-square critical value $\chi^2_{\alpha}$ determined?
2. What are the assumptions required to perform a chi-square test?
3. How does the sample size affect the chi-square test statistic?
4. Can the chi-square test be used for continuous data?
5. What happens if the expected frequency in a chi-square test is too small?

Tip: Always verify the degrees of freedom before using the chi-square distribution table to find the critical value.