4. REASON QUANTITATIVELY It is hypothesized that teens aged 16-18 tend to have more traffic violations than the rest of drivers. A random study of 10,000 drivers between the ages of 16 and 18 found that $$ 18 \% $$ of the drivers received a traffic citation in the last twelve months. a. What is the margin of error? b. Using the sample mean and the margin of error, what can you conclude? Be sure to correctly identify the population. $$ \qquad $$ $$ \qquad $$ $$ \qquad $$ c. A different group of researchers then randomly selected a group of only 100 drivers of all ages and asked the same question. Ten drivers of the 100 indicated that they had received a traffic citation in the last twelve months. What is the margin of error, the sample mean, and what can you conclude? Again, be sure to correctly identify the population. $$ \qquad $$ $$ \qquad $$ $$ \qquad $$ $$ \qquad $$ d. Is there anything that can be reasonably concluded to all drivers? What would be the traffic citations of teenaged drivers compared to all drivers? What would needed to make this study better?

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a. The margin of error is computed using the formula  
$$
\text{Margin of Error} = z \sqrt{\frac{p(1-p)}{n}},
$$  
with $$ z=1.96 $$ for a 95% confidence level, $$ p=0.18 $$, and $$ n=10\,000 $$. Thus,  
$$
\text{Margin of Error} = 1.96 \sqrt{\frac{0.18 \times 0.82}{10\,000}} \approx 1.96 \sqrt{0.00001476} \approx 1.96 \times 0.00384 \approx 0.0075.
$$  
This gives a margin of error of roughly $$ 0.75\% $$.

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b. The sample mean is $$ 18\% $$ (or $$ 0.18 $$). With a margin of error of $$ \approx 0.0075 $$, the 95% confidence interval for the true proportion of teen drivers (aged 16–18) who received a traffic citation is  
$$
0.18 \pm 0.0075,
$$  
or approximately from $$ 17.25\% $$ to $$ 18.75\% $$.  
This means that we are 95% confident that between $$ 17.25\% $$ and $$ 18.75\% $$ of all teen drivers aged 16–18 received a ticket in the last twelve months.

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c. For the group of 100 drivers of all ages:

- The sample mean is  
$$
\hat{p} = \frac{10}{100} = 0.10 \quad (10\%).
$$
  
- The margin of error is computed by  
$$
\text{Margin of Error} = 1.96 \sqrt{\frac{0.10 \times 0.90}{100}} = 1.96 \sqrt{0.009} = 1.96 \times 0.03 \approx 0.0588.
$$
  
  Thus, the margin of error is approximately $$ 5.88\% $$, and the 95% confidence interval for the true proportion of all drivers receiving a citation is  
$$
0.10 \pm 0.0588,
$$  
which is roughly from $$ 4.12\% $$ to $$ 15.88\% $$.

This sample represents all drivers (of various ages), and hence our conclusion applies to the population of all drivers.

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d. Comparing the two groups:

- Teen drivers (aged 16–18) have an estimated citation rate between about $$ 17.25\% $$ and $$ 18.75\% $$.
- The group of all drivers has an estimated citation rate between about $$ 4.12\% $$ and $$ 15.88\% $$.

While these confidence intervals suggest that teen drivers might have a higher rate of traffic citations compared to the overall driver population, there are several points to consider:

1. The sample of all drivers is quite small ($$ n=100 $$), which leads to a larger margin of error. This makes the estimate for all drivers less precise.
2. To confidently assert a difference, both groups should be sampled with sufficiently large numbers to reduce uncertainty.
3. Additional factors such as driving experience, frequency of travel, and other demographic or behavioral factors might influence the citation rate and need to be controlled in a more rigorous study.

In summary, current evidence suggests that teen drivers tend to have a higher rate of traffic citations compared to all drivers. However, more extensive and carefully controlled research with larger, representative samples in both groups would be needed to draw firmer conclusions.
a. The margin of error is about 0.75%. b. The sample mean is 18%. We are 95% confident that between 17.25% and 18.75% of all teen drivers aged 16–18 received a traffic citation in the last twelve months. c. The margin of error is approximately 5.88%, and the sample mean is 10%. We are 95% confident that between 4.12% and 15.88% of all drivers received a traffic citation in the last twelve months. d. Teen drivers have a higher rate of traffic citations compared to all drivers. To make the study better, use larger and more representative samples for both groups to reduce uncertainty.

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