Question

# Test the given claim. Identify the null​ hypothesis, alternative​ hypothesis, test​ statistic, P-value, and then state...

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 20212021 passenger cars in a particular​ region, 216216 had only rear license plates. Among 382382 commercial​ trucks, 5555 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.

a)

Given,

sample n1 = 2021 , n2 = 5555

x1 = 216 , x2 = 382

sample proportion p1^ = x1/n1 = 216/2021 = 0.1069

p2^ = x2/n2 = 382/5555 = 0.0688

Null hypothesis Ho : p1 = p2

Alternative hypothesis Ha : p1 < p2

Pooled proportion p^ = (x1+x2)/(n1+n2)

substitute values

= (216+382)/(2021+5555)

= 0.0789

consider,

test statistic z = (p1^ - p2^)/sqrt(p^(1-p^)*(1/n1 + 1/n2))

substitute values

= (0.1069 - 0.0688)/sqrt(0.0789(1-0.0789)*(1/2021 + 1/5555))

z = 5.44

P value = P(z < 5.44)

= 1 [since from z table]

= 1

Here we observe that, p value > 0.1, so we fail to reject Ho.

There is no sufficient evidence.

b)

Let us consider,

Here at 90% CI, z value is 1.645

90% Interval = (p1^ - p2^) /- z*sqrt(p^(1-p^)(1/n1 + 1/n2))

substitute values

= (0.1069 - 0.0688) +/- 1.645*sqrt(0.0789(1-0.0789)*(1/2021 + 1/5555))

= 0.0381 +/- 0.0115

= (0.0266 , 0.0496)

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