A study was made of children who were hospitalized as a result of a car accident. 280 of the children were not wearing seat belts and 98 of these were seriously injured. 130 children wore seat belts and 26 were seriously injured.
Task
- Test the hypothesis that the rate of serious injury is the same for children who wear a seat belt or not.
- Clearly state your null and alternative hypotheses and your conclusion.
- Use a significance level of 5%.
Hypothesis Testing: Seat Belt Use and Injury Severity
(a) Testing the Difference in Serious Injury Rates
Study Summary:
| Group | Total Children | Seriously Injured |
|---|---|---|
| No Seat Belt | 280 | 98 |
| Wore Seat Belt | 130 | 26 |
Step 1: State Hypotheses
Null Hypothesis (H₀):
The rate of serious injury is the same for children who wear seat belts and those who do not.
\(p_1 = p_2\)Alternative Hypothesis (H₁):
The rates of serious injury differ between the two groups.
\(p_1 \ne p_2\)
Step 2: Calculate Sample Proportions
\[ \hat{p}_1 = \frac{98}{280} = 0.35, \quad \hat{p}_2 = \frac{26}{130} = 0.20 \]
Step 3: Compute Pooled Proportion
\[ \hat{p} = \frac{98 + 26}{280 + 130} = \frac{124}{410} \approx 0.3024 \]
Step 4: Compute Standard Error
\[ SE = \sqrt{ \hat{p}(1 - \hat{p}) \left( \frac{1}{n_1} + \frac{1}{n_2} \right) } = \sqrt{0.3024 \cdot 0.6976 \left( \frac{1}{280} + \frac{1}{130} \right)} \approx 0.0503 \]
Step 5: Compute z-Statistic
\[ z = \frac{\hat{p}_1 - \hat{p}_2}{SE} = \frac{0.35 - 0.20}{0.0503} \approx 2.98 \]
Step 6: Decision Rule
- At a 5% significance level, critical z-values are ±1.96
- Since \(|z| = 2.98 > 1.96\), reject the null hypothesis
Conclusion:
There is statistically significant evidence at the 5% level to suggest that the rate of serious injury differs between children who wear seat belts and those who do not. The group not wearing seat belts had a higher proportion of serious injuries.
Question (b): Cardiovascular Study Design Concept
An exercise physiologist is investigating whether multiple short bouts of exercise are as beneficial as one long bout for cardiovascular fitness. This question would typically be tested using:
- Experimental design with randomized groups
- A two-sample test (e.g., independent t-test) comparing fitness improvements
- Outcome measures might include VO₂ max, heart rate recovery, etc.
A full setup would include defining the null hypothesis (no difference in benefit) and selecting appropriate metrics for statistical comparison.