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

20912091

passenger cars in a particular​ region,

227227

had only rear license plates. Among

344344

commercial​ trucks,

4848

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

0.100.10

significance level to test that hypothesis.

a. Test the claim using a hypothesis test.

b. Test the claim by constructing an appropriate confidence interval.

Answer 1

a) As we are testing here whether commercial trucks owners violate laws requiring front license plates at a higher rate than owners of passenger cars, this is a test for comparison of proportions. The null and alternate hypothesis here are given as:

The sample proportions are computed here as:
p₁ = 227/2091 = 0.1086
p₂ = 48/344 = 0.1395

The pooled proportion here is computed as:
P = (227 + 48) / (2091 + 344) = 0.1129

The standard error now is computed as:

The test statistic thus is computed here as:

As this is a lower test, the p-value here is computed from the standard normal tables as:

p = P( Z < -1.6779) = 0.0467

Therefore 0.0467 is the required p-value here.

As the p-value here is 0.0467 < 0.1, therefore the test is significant and we can reject the null hypothesis here and conclude that we have sufficient evidence here that commercial trucks owners violate laws requiring front license plates at a higher rate than owners of passenger cars, this is a test for comparison of proportions.

b) From standard normal tables, we have:

P( -1.645 < Z < 1.645) = 0.9

Therefore the confidence interval for the difference in proportion is computed here as:

As the whole confidence interval lies below 0, we have sufficient evidence here that commercial trucks owners violate laws requiring front license plates at a higher rate than owners of passenger cars, this is a test for comparison of proportions.