A simple random sample of front-seat occupants involved in car crashes is obtained. Among 2975 occupants not wearing seat belts, 35 were killed. Among 7864 occupants wearing seat belts, 16 were killed. Compute a 95% confidence interval to investigate whether occupants not wearing seat belts are more likely to be killed in a car crash.

We can be 95% confident that front-seat occupants not wearing seatbelts are between

To compute a 95% confidence interval for the difference in proportions of fatalities between occupants not wearing seat belts and those wearing seat belts, we can use the following steps:

1. **Calculate the proportions of fatalities**:
   - For occupants not wearing seat belts:
     \[
     p_1 = \frac{x_1}{n_1} = \frac{35}{2975} \approx 0.0118
     \]
   - For occupants wearing seat belts:
     \[
     p_2 = \frac{x_2}{n_2} = \frac{16}{7864} \approx 0.0020
     \]

2. **Difference in proportions**:
   \[
   \hat{p}_1 - \hat{p}_2 = p_1 - p_2 \approx 0.0118 - 0.0020 = 0.0098
   \]

3. **Calculate the standard error (SE)**:
   \[
   SE = \sqrt{\frac{p_1(1 - p_1)}{n_1} + \frac{p_2(1 - p_2)}{n_2}}
   \]
   Substituting the values:
   \[
   SE = \sqrt{\frac{0.0118(1 - 0.0118)}{2975} + \frac{0.0020(1 - 0.0020)}{7864}} \approx \sqrt{\frac{0.0118 \cdot 0.9882}{2975} + \frac{0.0020 \cdot 0.9980}{7864}}
   \]
   \[
   SE \approx \sqrt{\frac{0.0116}{2975} + \frac{0.0020}{7864}} \approx \sqrt{0.000003895 + 0.000000254} \approx \sqrt{0.000004149} \approx 0.002036
   \]

4. **Calculate the 95% confidence interval**:
   - The z-score for a 95% confidence level is approximately 1.96.
   \[
   CI = (\hat{p}_1 - \hat{p}_2) \pm z \cdot SE
   \]
   \[
   CI = 0.0098 \pm 1.96 \cdot 0.002036
   \]
   \[
   CI = 0.0098 \pm 0.003996
   \]
   \[
   CI \approx (0.005804, 0.013796)
   \]

Thus, we can be 95% confident that front-seat occupants not wearing seat belts are between approximately **0.0058 and 0.0138** more likely to be killed in a car crash compared to those wearing seat belts.

### Questions:
1. How does the confidence interval help interpret the risk of not wearing a seat belt?
2. What assumptions are made in calculating this confidence interval?
3. How would the results change if the sample size were larger?
4. What other factors could affect the likelihood of fatalities in car crashes?
5. How can this analysis be applied to policy-making regarding seat belt use?

### Tip:
Always consider the context of the data and the implications of the results when interpreting statistical analyses.

A simple random sample of front-seat occupants involved in car crashes is obtained. Among 2975 occupants not wearing seat belts, 35 were killed. Among 7864 occupants wearing seat belts, 16 were killed. Compute a 95% confidence interval to investigate whether occupants not wearing seat belts are more likely to be killed in a car crash.

Explore the calculation of a 95% confidence interval analyzing the likelihood of fatalities in car crashes among seat belt users versus non-users. With 35 fatalities among 2975 non-users and 16 among 7864 users, learn the statistical implications.

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