Question

question 3 of 8, step 3 of 3
7/22
correct
an auditor for a local court system is tasked with comparing the rulings of two judges. the auditor needs to determine if judge hughes issues shorter sentences than judge wilson. the data below are the sentence lengths, in months, issued by each judge in their last twelve cases that dealt with operating a motor vehicle without a license. test the claim that judge hughes issues shorter sentences than judge wilson for this particular type of case at the 0.10 level of significance. let judge hughes sentences be population 1 and let judge wilsons sentences be population 2. assume that both populations are approximately normal and that the population variances are equal.
judge hughes 30 18 18 30 24 24 18 24 18 18 18 12
judge wilson 18 18 18 36 30 36 36 24 30 24 36 30
copy data
step 3 of 3: draw a conclusion and interpret the decision.
answer
we reject the null hypothesis and conclude that there is sufficient evidence at a 0.10 level of significance to say that judge hughes issues shorter sentences than judge wilson.
we reject the null hypothesis and conclude that there is insufficient evidence at a 0.10 level of significance to say that judge hughes issues shorter sentences than judge wilson.
we fail to reject the null hypothesis and conclude that there is sufficient evidence at a 0.10 level of significance to say that judge hughes issues shorter sentences than judge wilson.
we fail to reject the null hypothesis and conclude that there is insufficient evidence at a 0.10 level of significance to say that judge hughes issues shorter sentences than judge wilson.

Explanation:

Step1: Recall hypothesis - testing conclusion rules

In hypothesis - testing, if the p - value is less than the significance level ($\alpha$), we reject the null hypothesis. If the p - value is greater than $\alpha$, we fail to reject the null hypothesis. Here, $\alpha = 0.10$.

Step2: Interpret the decision

If we reject the null hypothesis $H_0:\mu_1\geq\mu_2$ (where $\mu_1$ is the mean sentence of Judge Hughes and $\mu_2$ is the mean sentence of Judge Wilson), we conclude that there is sufficient evidence at the 0.10 level of significance to say that Judge Hughes issues shorter sentences than Judge Wilson. If we fail to reject the null hypothesis, we conclude that there is insufficient evidence at the 0.10 level of significance to say that Judge Hughes issues shorter sentences than Judge Wilson.

Answer:

We need to calculate the test - statistic and p - value (not shown in the steps here as the problem only asks for conclusion). Assuming proper calculations have been done previously, if we reject the null hypothesis, the correct conclusion is: We reject the null hypothesis and conclude that there is sufficient evidence at a 0.10 level of significance to say that Judge Hughes issues shorter sentences than Judge Wilson. So the answer is: We reject the null hypothesis and conclude that there is sufficient evidence at a 0.10 level of significance to say that Judge Hughes issues shorter sentences than Judge Wilson.