Question

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 2135 passenger cars in a particular​ region, 244 had only rear license plates. Among 339 commercial​ trucks, 51 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 0.01 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. A. Upper H 0​: p 1equalsp 2

Answer 1

Answer:

Being π1: proportion of passenger cars with only rear plates, and π2: proportion of commercial trucks with only rear plates, the null and alternative hypothesis are:

$H_0: \pi_1-\pi_2=0\\\\H_a:\pi_1-\pi_2$

The test statistic is z=-1.895.

The P-value is 0.029.

The P-value (0.03) is bigger than the significance level (0.01), so the effect is not significant. The null hypothesis failed to be rejected.

There is no enough evidence to claim that commercial trucks owners violate laws requiring front license plates at a higher rate than owners of passenger cars.

Step-by-step explanation:

The claim is that commercial trucks owners violate laws requiring front license plates at a higher rate than owners of passenger cars. We have to test the hypotheses on the difference between proportions.

Being π1: proportion of passenger cars with only rear plates, and π2: proportion of commercial trucks with only rear plates, the null and alternative hypothesis are:

$H_0: \pi_1-\pi_2=0\\\\H_a:\pi_1-\pi_2$

The significance level is 0.01.

The passenger cars, with a sample size n1=2135, have a proportion of:

$p_1=x_1/N_1=244/2135=0.114$

The commercial trucks, with a sample size n2=339, have a proportion of:

$p_2=x_2/N_2=51/339=0.150$

The weighted proportion, needed to estimate the standard error, is calculated as:

$p=\dfrac{x_1+x_2}{N_1+N_2}=\dfrac{244+51}{2135+339}=\dfrac{295}{2474}=0.119$

The standard error for the difference between proportions can be calculated as:

$\sigma=\sqrt{\dfrac{p(1-p)}{N_1}+\dfrac{p(1-p)}{N_2}}=\sqrt{\dfrac{0.119*0.881}{2135}+\dfrac{0.119*0.881}{339}}\\\\\\\sigma=\sqrt{ 0.00036 }= 0.019$

Then, the z-statistic is:

$z=\dfrac{p_1-p_2}{\sigma}=\dfrac{0.114-0.150}{0.019}=\dfrac{-0.036}{0.019}= -1.895$

The P-value for this left tailed test is:

$P-value=P(z$

The P-value (0.03) is bigger than the significance level (0.01), so the effect is not significant. The null hypothesis failed to be rejected.

There is no enough evidence to claim that commercial trucks owners violate laws requiring front license plates at a higher rate than owners of passenger cars.