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!)
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..
[Please give me a Thumbs Up if you are satisfied with my answer. If you are not, please comment on it, so I can edit the answer. Thanks.]
Problem 3 Professor Bell suspected that there might be a relationship between shyness and loneliness. For...
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