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 2160 passenger cars in a particular​ region, 243 had only rear license plates. Among 358 commercial​ trucks, 55 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.

Answer 1

For 0,90 of Confidence we reject H₀

For 0,95 CI we reject H₀

Explanation:

To evaluate a difference between two proportion with big sample sizes we proceed as follows

1.-Proportion 1

n = 2160

243 had rear license p₁ = 243/2160 p₁ = 0,1125

2.Proportion 2

n = 358

55 had rear license p₂ = 55/ 358 p₂ = 0,1536

Test Hypothesis

Null Hypothesis H₀ ⇒ p₂ = p₁

Alternative Hypothesis Hₐ ⇒ p₂ > p₁

With signficance level of 0,05 means z(c) = 1,64

T calculate z(s)

z(s) = ( p₂ - p₁ ) / √ p*q ( 1/n₁ + 1/n₂ )

p = ( x₁ + x₂ ) / n₁ + n₂

p = 243 + 55 / 2160 + 358

p = 0,1183 and then q = 1 - p q = 0,8817

z(s) = ( 0,1536 - 0,1125 ) / √ 0,1043 ( 1/ 2160 + 1 / 358)

z(s) = 0,0411 /√ 0,1043*0,003256

z(s) = 0,0411 / 0,01843

z(s) = 2,23

Then z(s) > z(c) 2,23 > 1,64

z(s) is in the rejection region we reject H₀

If we construct a CI for 0,95 α = 0,05 α/2 = 0,025

z (score ) is from z- table z = 1,96

CI = ( p ± z(0,025*SE)

CI = ( 0,1536 ± 1,96*√ 0,1043*0,003256 )

CI = ( 0,1536 ± 1.96*0,01843)

CI = ( 0,1536 ± 0,03612 )

CI = ( 0,11748 ; 0,18972 )

In the new CI we don´t find 0 value so we have enough evidence to reject H₀