- The decision rule states that if \(\chi^2_{calc} > \chi^2_{crit}\), the null hypothesis is rejected.
- Given \(\chi^2_{calc} = 4.31\) and \(\chi^2_{crit} = 3.84\), we observe that \(4.31 > 3.84\).
- Since the calculated value is greater than the critical threshold, the correct statistical action is to reject the null hypothesis.
- Final Answer: \(\boxed{\text{The null hypothesis should be rejected}}\)
Explanation
- Understand the Hypothesis Testing Framework
In statistical hypothesis testing using the Chi-Square (\(\chi^2\)) test, we compare a calculated test statistic, \(\chi^2_{calc}\), with a predetermined critical value, \(\chi^2_{crit}\), derived from the Chi-Square distribution table based on degrees of freedom and a chosen significance level (typically \(\alpha = 0.05\)). The critical value acts as a threshold that separates the regions of "failure to reject the null hypothesis" and "rejection of the null hypothesis."
- Apply the Decision Rule
The decision rule for a Chi-Square test is as follows:
- If \(\chi^2_{calc} > \chi^2_{crit}\), we fall into the rejection region, meaning the observed difference is statistically significant, and we reject the null hypothesis.
- If \(\chi^2_{calc} \leq \chi^2_{crit}\), we fall into the acceptance region, and we fail to reject the null hypothesis.
In this problem, we are given \(\chi^2_{calc} = 4.31\) and \(\chi^2_{crit} = 3.84\). Since \(4.31 > 3.84\), our calculated value exceeds the critical threshold.
- Evaluate the Statements and Conclude
Comparing the options provided:
- 'The null hypothesis should be accepted': Incorrect, as the calculated value exceeds the critical threshold.
- 'The null hypothesis should be changed': Incorrect, as this is not a standard statistical conclusion for a single test.
- 'The null hypothesis should be rejected': Correct, as the calculated \(\chi^2\) value is greater than the critical value.
- 'The degrees of freedom is 5': Incorrect, as the degrees of freedom cannot be determined from the provided values.
Therefore, the only correct statement is that the null hypothesis should be rejected.
Examples
In a clinical study testing if a new medication improves recovery time compared to a placebo, a researcher calculates a \(\chi^2\) statistic from the data. If the calculated value is \(4.31\) and the critical value at the \(5\%\) significance level is \(3.84\), the researcher can conclude that the results are statistically significant, allowing them to reject the null hypothesis and suggest that the medication has an effect. This process is essential for ensuring that research findings are backed by rigorous mathematical evidence.