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.