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### Part 1: Hypotheses
Let:
- \( p_1 \) be the proportion of passenger cars with only rear license plates.
- \( p_2 \) be the proportion of commercial trucks with only rear license plates.

The hypotheses are:
- Null hypothesis (\( H_0 \)): \( p_1 = p_2 \)
- Alternative hypothesis (\( H_1 \)): \( p_1 < p_2 \)

Thus, the correct option is:
**D. \( H_0: p_1 = p_2 \) and \( H_1: p_1 < p_2 \)**

### Part 2: Test Statistic

The test statistic for comparing two proportions is given by:
\[ z = \frac{\hat{p}_1 - \hat{p}_2}{\sqrt{\hat{p}(1 - \hat{p}) \left(\frac{1}{n_1} + \frac{1}{n_2}\right)}} \]
where:
- \( \hat{p}_1 \) is the sample proportion of passenger cars with only rear license plates.
- \( \hat{p}_2 \) is the sample proportion of commercial trucks with only rear license plates.
- \( \hat{p} \) is the pooled sample proportion.
- \( n_1 \) is the sample size of passenger cars.
- \( n_2 \) is the sample size of commercial trucks.

First, we calculate the sample proportions:
\[ \hat{p}_1 = \frac{222}{2064} \approx 0.1076 \]
\[ \hat{p}_2 = \frac{50}{340} \approx 0.1471 \]

Next, calculate the pooled sample proportion:
\[ \hat{p} = \frac{222 + 50}{2064 + 340} = \frac{272}{2404} \approx 0.1131 \]

Now, calculate the standard error:
\[ \text{SE} = \sqrt{\hat{p}(1 - \hat{p}) \left(\frac{1}{n_1} + \frac{1}{n_2}\right)} = \sqrt{0.1131 \times 0.8869 \left(\frac{1}{2064} + \frac{1}{340}\right)} \]
\[ \text{SE} \approx \sqrt{0.1131 \times 0.8869 \left(0.0004846 + 0.0029412\right)} \]
\[ \text{SE} \approx \sqrt{0.1131 \times 0.8869 \times 0.0034258} \]
\[ \text{SE} \approx \sqrt{0.0003437} \]
\[ \text{SE} \approx 0.0185 \]

Finally, calculate the \( z \)-statistic:
\[ z = \frac{0.1076 - 0.1471}{0.0185} \approx -2.13 \]

### Part 3: P-value and Conclusion

Using the standard normal distribution, find the p-value for \( z = -2.13 \):
\[ P(z < -2.13) \approx 0.017 \]

Since the p-value (0.017) is less than the significance level (0.10), we reject the null hypothesis.

### Conclusion

1. **Null Hypothesis \( H_0 \)**: \( p_1 = p_2 \)
2. **Alternative Hypothesis \( H_1 \)**: \( p_1 < p_2 \)
3. **Test Statistic**: \( z \approx -2.13 \)
4. **P-value**: \( \approx 0.017 \)
5. **Conclusion about \( H_0 \)**: Reject the null hypothesis.

**Final Conclusion**: There is sufficient evidence to support the claim that commercial truck owners violate laws requiring front license plates at a higher rate than owners of passenger cars.

Would you like to test the claim by constructing an appropriate confidence interval?

### Questions You Might Ask
1. How do you calculate the pooled sample proportion?
2. What is the formula for the standard error in a two-proportion z-test?
3. How do you interpret a p-value in hypothesis testing?
4. What are the implications of rejecting the null hypothesis?
5. How can confidence intervals provide additional insights into hypothesis testing?

### Tip
When comparing proportions, always ensure your sample sizes are sufficiently large to justify the use of normal approximation in your test statistics.

Learn how to conduct hypothesis testing to compare the proportions of passenger cars and commercial trucks violating license plate laws. Includes detailed steps on calculating the z-statistic, standard error, and interpreting the p-value. Suitable for advanced undergraduate students and researchers.

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