Question

Problem 3

Professor Bell suspected that there might be a relationship between shyness and loneliness. For 40 participants he measured shyness using the Ottawa Shyness Questionnaire and measured loneliness using the Carleton Loneliness scale. For both measures, a higher value indicates more shyness or more loneliness. The data are on the right. Did he find support for his hypothesis?

Person	Shyness	Loneliness
1	12	6
2	6	8
3	4	8
4	13	7
5	7	5
6	13	7
7	17	13
8	9	4
9	3	7
10	19	15
11	18	3
12	5	10
13	9	4
14	15	9
15	14	8
16	15	9
17	19	15
18	7	3
19	15	9
20	6	9
21	8	4
22	16	11
23	14	8
24	18	12
25	4	3
26	7	6
27	7	9
28	12	6
29	20	15
30	17	7
31	8	10
32	14	8
33	18	7
34	11	6
35	19	15
36	11	6
37	4	8
38	10	5
39	13	15
40	13	11

For hypothesis testing questions, use the four-step procedure outlined in class and assume α = 0.05, unless instructed to do otherwise in the problem. For all problems, provide a conclusion for each question in everyday English.

For ANOVA question(s), include a post-hoc test and an effect size calculation. For correlation questions, include a scatterplot that incorporates a regression line.

To obtain full credit, show any calculations that are required via formulas in Excel. Use the sample assignment 3 as a model.

(Please show all your work THANKS!)

Answer 1

		x	y	x^2	y^2	xy	(x - x-bar)^2	(y - y-bar)^2	(x - x-bar)(y - y-bar)
		12	6	144	36	72	0.0625	5.1756	-0.5688
		6	8	36	64	48	33.0625	0.0756	1.5813
		4	8	16	64	32	60.0625	0.0756	2.1313
		13	7	169	49	91	1.5625	1.6256	-1.5938
		7	5	49	25	35	22.5625	10.7256	15.5563
		13	7	169	49	91	1.5625	1.6256	-1.5938
		17	13	289	169	221	27.5625	22.3256	24.8063
		9	4	81	16	36	7.5625	18.2756	11.7563
		3	7	9	49	21	76.5625	1.6256	11.1563
		19	15	361	225	285	52.5625	45.2256	48.7563
		18	3	324	9	54	39.0625	27.8256	-32.9688
		5	10	25	100	50	45.5625	2.9756	-11.6438
		9	4	81	16	36	7.5625	18.2756	11.7563
		15	9	225	81	135	10.5625	0.5256	2.3563
		14	8	196	64	112	5.0625	0.0756	-0.6188
		15	9	225	81	135	10.5625	0.5256	2.3563
		19	15	361	225	285	52.5625	45.2256	48.7563
		7	3	49	9	21	22.5625	27.8256	25.0563
		15	9	225	81	135	10.5625	0.5256	2.3563
		6	9	36	81	54	33.0625	0.5256	-4.1688
		8	4	64	16	32	14.0625	18.2756	16.0313
		16	11	256	121	176	18.0625	7.4256	11.5813
		14	8	196	64	112	5.0625	0.0756	-0.6188
		18	12	324	144	216	39.0625	13.8756	23.2813
		4	3	16	9	12	60.0625	27.8256	40.8813
		7	6	49	36	42	22.5625	5.1756	10.8063
		7	9	49	81	63	22.5625	0.5256	-3.4438
		12	6	144	36	72	0.0625	5.1756	-0.5688
		20	15	400	225	300	68.0625	45.2256	55.4813
		17	7	289	49	119	27.5625	1.6256	-6.6938
		8	10	64	100	80	14.0625	2.9756	-6.4688
		14	8	196	64	112	5.0625	0.0756	-0.6188
		18	7	324	49	126	39.0625	1.6256	-7.9688
		11	6	121	36	66	0.5625	5.1756	1.7063
		19	15	361	225	285	52.5625	45.2256	48.7563
		11	6	121	36	66	0.5625	5.1756	1.7063
		4	8	16	64	32	60.0625	0.0756	2.1313
		10	5	100	25	50	3.0625	10.7256	5.7313
		13	15	169	225	195	1.5625	45.2256	8.4063
		13	11	169	121	143	1.5625	7.4256	3.4063
n =	40
Sums =		470	331	6498	3219	4248	975.5	479.975	358.75
Means =		11.75	8.275
SS(x) = Σ[(x - x-bar)^2] OR Σ(x^2) - [(Σx)^2 /n] =				975.5000
SS(y) = Σ[(y - y-bar)^2] OR Σ(y^2) - [(Σy)^2 /n] =				479.9750
SS(xy) = Σ[(x - x-bar)(y - y-bar)] OR Σ(xy) - [(Σx)(Σy)/n] =				358.7500
Correlation coefficient, r = SSxy / √[SS(x) * SS(y)] =				0.5243

Data:    

n = 40   

R = 0.5243   

Hypotheses:    

Ho: There is no significant correlation, that is ρ = 0    

Ha: There is significant correlation, that is ρ ≠ 0    

Decision Rule:    

t (Two-tailed), α =   0.05

Degrees of freedom =   40 - 2 = 38

Lower Critical t- score =   -2.024394147

Upper Critical t- score = 2.024394147

Reject Ho if |t| >   2.024394147

Test Statistic:    

SE = √{(1 - R^2)/DOF} =   √((1 - 0.5243^2)/38) = 0.138136994

t = R/SE = 0.5243/0.138136994358956 = 3.795507514

p- value = 0.000515404   

Decision (in terms of the hypotheses):    

Since 3.795507514 > 2.024394147 we reject Ho

Conclusion (in terms of the problem):    

Conclusion: There is evidence of a significant correlation between shyness and loneliness..

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