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 who are, we can follow these steps:

1. **Define the proportions**:
   - \( p_1 \): Proportion of fatalities among occupants not wearing seat belts.
   - \( p_2 \): Proportion of fatalities among occupants wearing seat belts.

   Given:
   - \( n_1 = 2975 \) (not wearing seat belts)
   - \( x_1 = 35 \) (fatalities not wearing seat belts)
   - \( n_2 = 7864 \) (wearing seat belts)
   - \( x_2 = 16 \) (fatalities wearing seat belts)

   Calculate the proportions:
   \[
   p_1 = \frac{x_1}{n_1} = \frac{35}{2975} \approx 0.0118
   \]
   \[
   p_2 = \frac{x_2}{n_2} = \frac{16}{7864} \approx 0.0020
   \]

2. **Calculate the difference in proportions**:
   \[
   p_1 - p_2 = 0.0118 - 0.0020 = 0.0098
   \]

3. **Calculate the standard error (SE) of the difference**:
   \[
   SE = \sqrt{\frac{p_1(1 - p_1)}{n_1} + \frac{p_2(1 - p_2)}{n_2}}
   \]
   \[
   SE = \sqrt{\frac{0.0118(1 - 0.0118)}{2975} + \frac{0.0020(1 - 0.0020)}{7864}}
   \]
   \[
   SE \approx \sqrt{\frac{0.0118 \times 0.9882}{2975} + \frac{0.0020 \times 0.9980}{7864}} \approx \sqrt{0.00000386 + 0.00000051} \approx \sqrt{0.00000437} \approx 0.0021
   \]

4. **Calculate the 95% confidence interval**:
   The critical value for a 95% confidence interval from the standard normal distribution (Z-distribution) is approximately 1.96.

   The confidence interval (CI) is given by:
   \[
   (p_1 - p_2) \pm Z \times SE
   \]
   \[
   CI = 0.0098 \pm 1.96 \times 0.0021
   \]
   \[
   CI = 0.0098 \pm 0.0041
   \]
   \[
   CI = (0.0057, 0.0139)
   \]

Thus, we can be 95% confident that front-seat occupants not wearing seat belts are between **0.57% and 1.39%** more likely to be killed in a car crash compared to those wearing seat belts.

### Questions for Further Exploration:
1. What are the implications of these findings for public safety policies?
2. How does the sample size impact the confidence interval?
3. What assumptions are made in this analysis?
4. How would the confidence interval change with different sample sizes?
5. Are there any other factors that could influence the likelihood of fatalities in car crashes?

### Tip:
When interpreting confidence intervals, remember that they provide a range of values that likely contain the true parameter, but do not guarantee that the parameter lies within this range for every sample.

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 analysis of fatalities among car crash occupants based on seat belt usage. This investigation reveals that front-seat occupants not wearing seat belts are between 0.57% and 1.39% more likely to be killed compared to those wearing seat belts, using a 95% confidence interval.

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