Question

Listed below are amounts of court income and salaries paid to the town justices. All amounts are in thousands of dollars. Construct a scatterplot, find the value of the linear correlation coefficient r, and find the P-value using a = 0.05. Is there sufficient evidence to conclude that there is a linear correlation between court incomes and justice salaries? Based on the results, does it appear that justices might profit by levying larger fines? Court Income 64.0 406.0 1566.0 1130.0 272.0 251.0 112.0 152.0 31.0. Justice Salary 29 46 91 56 45 62 25 26 18 What are the null and alternative hypotheses? O A. Ho: p=0 Hyp70 O B. Ho: p= 0 H:p> 0 0C. Hop+ 0 He: p= 0 OD. Ho: p= 0 H:p<0 Construct a scatterplot. Choose the correct graph below. OA. Justice Salary 100- OB A Justice Salary 100- ... OC. A Justice Salary 100- OD A Justice Salary 100- . 50- 50- 50- 50- . 0 . 800 1600 Court Income 800 1600 Court Income 800 1600 Court Income 800 1600 Court Income The linear correlation coefficient r is (Round to three decimal places as needed.)

Answer 1

	Court Income=x	Justice Salary=y	X-X	y-	(x-) (y-)	2 (x - 2)	2 (y-7)
	64	29	-378.667	-15.222	5764.148	143388.444	231.716
	406	46	-36.667	1.778	-65.185	1344.444	3.160
	1566	91	1123.333	46.778	52547.037	1261877.778	2188.160
	1130	56	687.333	11.778	8095.259	472427.111	138.716
	272	45	-170.667	0.778	-132.741	29127.111	0.605
	251	62	-191.667	17.778	-3407.407	36736.111	316.049
	112	25	-330.667	-19.222	6356.148	109340.444	369.494
	152	26	-290.667	-18.222	5296.593	84487.111	332.049
	31	18	-411.667	-26.222	10794.815	169469.444	687.605
Total	3984	398	0	0	85248.667	2308198	4267.556
Mean	442.667	44.222

Answer(1):

The Null and alternative hypothesis to test the significance of correlation are

H₀: ρ=0

H₁: ρ≠0

Hence the correct option is A.

Answer(2):

The scatter plot of the given data is as below

08 60 Justice Salary 40 20 0 500 1000 1500 Court Income

Hence the correct option is A.

Answer(c):

The coefficient of correlation between two variables can be obtained by

γ = Σ(y-y) * (x – x) Σ(x – x)? Σy - y)2

= 85248.667 2308198 * 4267.556

r = 85248.667 99248.996

r= 0.859

The linear correlation coefficient r is 0.859

Answer(d):

To check whether this correlation is significant or not we have to conduct a t-test for significance of correlation coefficient.

The test statistic to test this hypothesis is

t= r*vn-2 V1 -2

* V t 0.859 9-2 V1 -0.8592

+ 0.859* 2.645751 0.5121

2.2725 t= 0.5121

t= 4.438

The test statistic t is 4.438

Answer(e):

The p-value for above test is 0.003

Answer(f):

Because the p-value is less than the significance level 0.05, there is sufficient evidence to support the claim that there is linear correlation between court incomes and justice salaries for a significance level of α=0.05

Answer(g):

The correct option is

B. It does appear that justices might profit by levying larger fines.