A psychologist wanted to assess the relationship between depression and self-esteem. She believes that as depression increases, self-esteem decreases. She asked a group of 25 participants to complete both Beck’s Depression Inventory (Beck, Ward, Mendelson, Mock, & Erbaugh, 1961) and Rosenberg’s Self-Esteem Scale (1965). The data for each participant is as follows:
Depression (X) |
Self-Esteem (Y) |
60 |
11 |
27 |
34 |
15 |
33 |
30 |
24 |
56 |
14 |
52 |
16 |
40 |
23 |
49 |
33 |
24 |
27 |
38 |
26 |
48 |
17 |
58 |
10 |
22 |
39 |
51 |
18 |
41 |
22 |
56 |
17 |
49 |
16 |
54 |
13 |
52 |
12 |
41 |
23 |
35 |
28 |
20 |
35 |
55 |
15 |
16 |
32 |
28 |
20 |
Descriptive Statistics
For Depression:
For Self-Esteem:
Correlation
Regression
For dipression
i) The scale of measurement of the variable: Quantitative data (numerical values are provided).
ii) The data is normally distributed. We plot normal probability in Minitab.
iii) Since we have numerical measurements we can calculate mean and variance (standard deviation).
iv) Since the data is numerical and normally distributed, the mean and variance are the correct measures to calculate.
v) Mean=sum of all observations/total no of observation=1017/25=40.68
Variance=(sum of all squared observations/total no of observation)-(mean)^2=196.2176
For Self-esteem
i) The scale of measurement of the variable: Quantitative data (numerical values are provided).
ii) The data is normally distributed. We plot normal probability in Minitab.
iii) Since we have numerical measurements we can calculate mean and variance (standard deviation).
iv) Since the data is numerical and normally distributed, the mean and variance are the correct measures to calculate.
v) Mean=sum of all observations/total no of observation=558/25=22.32
Variance=(sum of all squared observations/total no of observation)-(mean)^2=69.0176
Correlation
i) Predictor variable: Self-esteem
ii) Criterion variable: Dipression
X^2 | Y^2 | XY | |
3600 | 121 | 660 | |
729 | 1156 | 918 | |
225 | 1089 | 495 | |
900 | 576 | 720 | |
3136 | 196 | 784 | |
2704 | 256 | 832 | |
1600 | 529 | 920 | |
2401 | 1089 | 1617 | |
576 | 729 | 648 | |
1444 | 676 | 988 | |
2304 | 289 | 816 | |
3364 | 100 | 580 | |
484 | 1521 | 858 | |
2601 | 324 | 918 | |
1681 | 484 | 902 | |
3136 | 289 | 952 | |
2401 | 256 | 784 | |
2916 | 169 | 702 | |
2704 | 144 | 624 | |
1681 | 529 | 943 | |
1225 | 784 | 980 | |
400 | 1225 | 700 | |
3025 | 225 | 825 | |
256 | 1024 | 512 | |
784 | 400 | 560 | |
Total | 46277 | 14180 | 20238 |
X | Y | |
Mean | 40.68 | 22.32 |
Varince | 196.2176 | 69.0176 |
Covariance | -98.4576 | |
Correlation | -0.84606 |
Null hypothesis H0:
ρ = 0
Alternative hypothesis HA:
ρ ≠ 0 or HA: ρ < 0 or
HA: ρ > 0
Test statistic: , where r is correlation and n is total no of observation
t=-14.5849
|t|=14.5849
p-value=0.0000
Since the p-value< 5%.
We may reject H0.
Regression
SUMMARY OUTPUT | ||||||||
Regression Statistics | ||||||||
Multiple R | 0.846058 | |||||||
R Square | 0.715815 | |||||||
Adjusted R Square | 0.703459 | |||||||
Standard Error | 4.617286 | |||||||
Observations | 25 | |||||||
ANOVA | ||||||||
df | SS | MS | F | Significance F | ||||
Regression | 1 | 1235.096 | 1235.096 | 57.93314 | 9.97E-08 | |||
Residual | 23 | 490.3445 | 21.31933 | |||||
Total | 24 | 1725.44 | ||||||
Coefficients | Standard Error | t Stat | P-value | Lower 95% | Upper 95% | Lower 95.0% | Upper 95.0% | |
Intercept | 42.73231 | 2.836353 | 15.06593 | 2.09E-13 | 36.86487 | 48.59976 | 36.86487 | 48.59976 |
X Variable 1 | -0.50178 | 0.065925 | -7.61138 | 9.97E-08 | -0.63815 | -0.3654 | -0.63815 | -0.3654 |
Here intercept: b0=42.73231
slope: b1= -0.50178
a) x=28: Y=28.68254
b) x=42: Y=21.6576
c) x=54: Y=15.6363
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