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 2156 passenger cars in a particular​ region, 233 had only rear license plates. Among 338 commercial​ trucks, 49 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.

Answer 1

Two-Proportion Z test

The following information is provided: (a) Sample 1 - The sample size is N1 = 2156, the number of favorable cases is X1 = 233 and the sample proportion is p^1​=X1/N1​=233/2156​=0.1081 (b) Sample 2 - The sample size is N2 = 338, the number of favorable cases is X2 = 49 and the sample proportion is p^2​=X2/N2​=49/338​=0.145 and the significance level is α=0.05 Pooled Proportion The value of the pooled proportion is computed as (1) Null and Alternative Hypotheses The following null and alternative hypotheses need to be tested: Ho: p1 = p2 Ha: p1 < p2 This corresponds to a Left-tailed test, for which a z-test for two population proportions needs to be conducted. (2a) Critical Value Based on the information provided, the significance level is α=0.05, therefore the critical value for this Left-tailed test is Zc​=-1.6449. This can be found by either using excel or the Z distribution table. (2b) Rejection Region The rejection region for this Left-tailed test is Z<-1.6449 (3) Test Statistics The z-statistic is computed as follows: (4) The p-value The p-value is the probability of obtaining sample results as extreme or more extreme than the sample results obtained, under the assumption that the null hypothesis is true. In this case, the p-value is p =P(Z<-1.9918)=0.0232 (5) The Decision about the null hypothesis (a) Using traditional method Since it is observed that Z=-1.9918 < Zc​=-1.6449, it is then concluded that the null hypothesis is rejected. (b) Using p-value method Using the P-value approach: The p-value is p=0.0232, and since p=0.0232≤0.05, it is concluded that the null hypothesis is rejected. (6) Conclusion It is concluded that the null hypothesis Ho is rejected. Therefore, there is enough evidence to claim that the population proportion p1 is less than p2, at the 0.05 significance level.

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