Test the given claim. Identify the null hypothesis, alternative hypothesis, test statistic, P-value, and then state the conclusion about the null hypothesis, as well as the final conclusion that addresses the original claim. Among 2082 passenger cars in a particular region, 233 had only rear license plates. Among 332 commercial trucks, 45 had only rear license plates. A reasonable hypothesis is that commercial trucks owners violate laws requiring front license plates at a higher rate than owners of passenger cars. Use a 0.05 significance level to test that hypothesis.
a. Test the claim using a hypothesis test.
b. Test the claim by constructing an appropriate confidence interval.
a. Identify the null and alternative hypotheses for this test. Let population 1 correspond to the passenger cars and population 2 correspond to the commercial trucks. Let a success be a vehicle that only has a rear license plate.
Question $4(0 / 1)$
A.
\[
\begin{array}{l}
H_{0}: p_{1}< p_{2} \\
H_{1}: p_{1}=p_{2}
\end{array}
\]
B.
\[
\begin{array}{l}
H_{0}: p_{1}=p_{2} \\
H_{1}: p_{1}< p_{2}
\end{array}
\]
c.
\[
\begin{array}{l}
H_{0}: p_{1}=p_{2} \\
H_{1}: p_{1} \neq p_{2}
\end{array}
\]
D.
\[
\begin{array}{l}
H_{0}: p_{1}=p_{2} \\
H_{1}: p_{1}> p_{2}
\end{array}
\]
Identify the test statistic.
(Type an integer or a decimal. Round to two decimal places as needed.)
an example
Get more help *
Clear all
Check answer

Step 1 ：Identify the null and alternative hypotheses for this test. Let population 1 correspond to the passenger cars and population 2 correspond to the commercial trucks. Let a success be a vehicle that only has a rear license plate. The null and alternative hypotheses are: \[H_{0}: p_{1}=p_{2}\] \[H_{1}: p_{1}

Step 2 ：Calculate the sample proportions and the pooled sample proportion. For passenger cars (population 1), the sample size n1 is 2082 and the number of successes x1 is 233, so the sample proportion p1 is \(\frac{233}{2082} = 0.112\). For commercial trucks (population 2), the sample size n2 is 332 and the number of successes x2 is 45, so the sample proportion p2 is \(\frac{45}{332} = 0.136\). The pooled sample proportion p is \(\frac{233 + 45}{2082 + 332} = 0.115\).

Step 3 ：Calculate the test statistic using the formula for a two-proportion z-test: \[z = \frac{p1 - p2}{\sqrt{p(1 - p) * [(1/n1) + (1/n2)]}}\] Substituting the calculated values gives \[z = \frac{0.112 - 0.136}{\sqrt{0.115(1 - 0.115) * [(1/2082) + (1/332)]}} = -1.25\].

Step 4 ：Calculate the P-value. The P-value is the probability of observing a test statistic as extreme as the one calculated, assuming the null hypothesis is true. The P-value is 0.105.

Step 5 ：Since the P-value is greater than the significance level of 0.05, we do not reject the null hypothesis. Therefore, we do not have sufficient evidence to support the claim that commercial trucks owners violate laws requiring front license plates at a higher rate than owners of passenger cars.

Step 6 ：The final answer is: The null and alternative hypotheses are: \[H_{0}: p_{1}=p_{2}\] \[H_{1}: p_{1}