To determine if smoking habits and seat belt usage are related, we will perform a Chi-Square test of independence using the given data. We will follow these steps:
1. Determine the null and alternative hypotheses:
- Null hypothesis ([tex]\(H_0\)[/tex]): Smoking habits and seat belt usage are independent.
- Alternative hypothesis ([tex]\(H_a\)[/tex]): Smoking habits and seat belt usage are dependent.
2. Determine the test statistic ([tex]\(\chi^2\)[/tex]):
- The test statistic [tex]\(\chi^2\)[/tex] is calculated based on the observed frequencies and the expected frequencies in the contingency table. The Chi-Square test will assess whether the observed frequencies deviate from the expected frequencies under the assumption of independence.
- In this case, the calculated test statistic [tex]\(\chi^2\)[/tex] is 0.00 (rounded to two decimal places).
3. Determine the p-value:
- The p-value is the probability of observing a test statistic as extreme as, or more extreme than, the observed value under the null hypothesis.
- Here, the calculated p-value is 1.0000 (rounded to four decimal places).
4. Make a decision:
- Compare the p-value to the significance level [tex]\(\alpha = 0.05\)[/tex]:
- If [tex]\( \text{p-value} < \alpha \)[/tex], reject the null hypothesis.
- If [tex]\( \text{p-value} \ge \alpha \)[/tex], fail to reject the null hypothesis.
- Since the p-value (1.0000) is greater than the significance level (0.05), we fail to reject the null hypothesis.
5. Conclusion:
- Based on the decision, we conclude that there is not sufficient evidence to support the claim that smoking habits and seat belt usage are related.
Therefore, the detailed step-by-step solution is as follows:
1. Hypotheses:
[tex]\[
H_0: \text{Smoking habits and seat belt usage are independent}
\][/tex]
[tex]\[
H_a: \text{Smoking habits and seat belt usage are dependent}
\][/tex]
2. Test Statistic:
[tex]\[
\chi^2 = 0.00
\][/tex]
3. P-value:
[tex]\[
\text{p-value} = 1.0000
\][/tex]
4. Decision:
- Fail to reject the null hypothesis.
5. Conclusion:
- There is not sufficient evidence to support the claim that smoking habits and seat belt usage are related.
This conclusion is derived from the Chi-Square test of independence applied to the given data, and it suggests that any observed association between smoking habits and seat belt usage could be due to random chance rather than a significant relationship.
