Question

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

step 2 of 3: compute the value of the test statistic. round your answer to three decimal places.
answer

Explanation:

Step1: Calculate sample means

Let $x_1$ be Judge Hughes' data and $x_2$ be Judge Wilson's data.
$n_1 = n_2=12$
$\bar{x}_1=\frac{30 + 18+18+\cdots+12}{12}=\frac{240}{12} = 20$
$\bar{x}_2=\frac{18 + 18+18+\cdots+30}{12}=\frac{318}{12}=26.5$

Step2: Calculate sample variances

$s_1^2=\frac{\sum_{i = 1}^{n_1}(x_{1i}-\bar{x}_1)^2}{n_1 - 1}$
$\sum_{i=1}^{12}(x_{1i}-20)^2=(30 - 20)^2+(18 - 20)^2+\cdots+(12 - 20)^2=384$
$s_1^2=\frac{384}{11}\approx34.909$
$s_2^2=\frac{\sum_{i = 1}^{n_2}(x_{2i}-\bar{x}_2)^2}{n_2 - 1}$
$\sum_{i=1}^{12}(x_{2i}-26.5)^2=(18 - 26.5)^2+(18 - 26.5)^2+\cdots+(30 - 26.5)^2=291$
$s_2^2=\frac{291}{11}\approx26.455$

Step3: Calculate pooled - variance

$s_p^2=\frac{(n_1 - 1)s_1^2+(n_2 - 1)s_2^2}{n_1 + n_2-2}$
$=\frac{(12 - 1)\times34.909+(12 - 1)\times26.455}{12 + 12-2}$
$=\frac{11\times(34.909 + 26.455)}{22}$
$=\frac{11\times61.364}{22}=30.682$

Step4: Calculate the test - statistic

The test - statistic for a two - sample t - test (equal variances) is given by:
$t=\frac{(\bar{x}_1-\bar{x}_2)-(\mu_1-\mu_2)}{s_p\sqrt{\frac{1}{n_1}+\frac{1}{n_2}}}$
Since $\mu_1-\mu_2 = 0$ (null hypothesis),
$t=\frac{20 - 26.5}{ \sqrt{30.682\times(\frac{1}{12}+\frac{1}{12})}}$
$=\frac{- 6.5}{\sqrt{30.682\times\frac{1}{6}}}$
$=\frac{-6.5}{\sqrt{5.114}}$
$=\frac{-6.5}{2.261}\approx - 2.875$

Answer:

$-2.875$