To solve this problem, we need to determine whether we should reject or not reject the null hypothesis [tex]\( H_0 \)[/tex] based on the chi-square Goodness-of-Fit test results.
1. Restate the hypotheses:
- Null hypothesis ([tex]\( H_0 \)[/tex]): The die has a uniform distribution.
- Alternative hypothesis ([tex]\( H_a \)[/tex]): The die does not have a uniform distribution.
2. Given information:
- Calculated chi-square value ([tex]\( \chi^2 \)[/tex] or [tex]\( \chi_0^2 \)[/tex]): 11.692
- Critical chi-square value at 1% significance level ([tex]\( \chi_{0.01}^2 \)[/tex]): 15.086
- Significance level ([tex]\( \alpha \)[/tex]): 0.01 (or 1%)
3. Decision rule for the chi-square test:
- If the calculated chi-square value is greater than the critical chi-square value, we reject [tex]\( H_0 \)[/tex].
- If the calculated chi-square value is less than or equal to the critical chi-square value, we do not reject [tex]\( H_0 \)[/tex].
4. Compare the calculated chi-square value to the critical chi-square value:
- Calculated chi-square value: 11.692
- Critical chi-square value: 15.086
5. Make the decision:
- Since the calculated chi-square value (11.692) is less than the critical chi-square value (15.086), we do not reject the null hypothesis [tex]\( H_0 \)[/tex].
6. Conclusion:
- We should not reject [tex]\( H_0 \)[/tex].
Therefore, the conclusions that can be drawn are:
- We should not reject [tex]\( H_0 \)[/tex].
