In: Statistics and Probability
Use the data set which is labeled below and answer using regression output in excel
i. Find the correlation coefficients between Y and X1, X2 and X3 and test the significance of population correlation coefficient using the value of r calculated for X2 and X3.
ii. Estimate the regression equation for Y and X1, Y and X2 and report the results and explain the intercept and slope coefficients.
iii. Check the significance or insignificance of the independent variables? and explain the coefficient of determination and standard error of estimate.
iv. Find the predicted value of Y when X1= 20, X2= 30 and X3= 125. Write down first five residuals of the estimated equation.
Y | X1 | X2 | X3 |
33.4283 | 32.96268 | 16.92215 | 30.06201 |
28.97037 | 7.616413 | 56.9856 | 36.19529 |
14.42695 | 59.20676 | 38.73281 | 3.194128 |
68.30308 | 9.144719 | 71.65822 | 41.90527 |
36.60967 | 39.26304 | 49.15915 | 67.94662 |
34.50435 | 35.52693 | 20.51122 | 45.46315 |
16.17359 | 48.01627 | 61.08484 | 57.07025 |
12.12781 | 62.64879 | 17.2385 | 19.76266 |
41.56441 | 17.74422 | 49.82528 | 71.12354 |
57.93912 | 12.90088 | 16.6303 | 58.51167 |
30.42961 | 37.25575 | 4.753624 | 31.63265 |
8.420667 | 40.63094 | 53.26063 | 25.18525 |
52.69253 | 39.63063 | 63.03867 | 9.87991 |
26.43062 | 60.18702 | 51.28675 | 50.65404 |
5.769524 | 46.60604 | 68.49467 | 17.30534 |
61.96707 | 49.63369 | 8.369427 | 54.37901 |
28.9013 | 20.58473 | 15.58098 | 60.54347 |
52.05759 | 35.05463 | 42.73849 | 41.27033 |
50.60503 | 54.25425 | 64.52019 | 23.08661 |
56.78954 | 47.49495 | 11.34584 | 6.163854 |
2.251747 | 21.84793 | 54.51714 | 30.57219 |
39.44349 | 52.53212 | 13.18604 | 29.91498 |
40.79134 | 6.021271 | 48.14325 | 58.69213 |
20.77187 | 71.46217 | 25.71548 | 56.11673 |
50.56493 | 40.35691 | 8.810541 | 24.7174 |
23.3896 | 56.12342 | 61.39897 | 5.450941 |
30.27143 | 43.4915 | 71.39979 | 70.67351 |
25.29218 | 28.32875 | 41.38618 | 8.670186 |
51.05506 | 17.87121 | 18.974 | 16.11789 |
68.49913 | 45.93991 | 43.32441 | 28.17725 |
4.693472 | 2.131443 | 9.668264 | 24.55031 |
11.82482 | 68.55483 | 47.26548 | 36.2599 |
17.95587 | 69.09842 | 71.66713 | 74.44749 |
22.78585 | 14.81906 | 55.73354 | 19.93866 |
73.03726 | 4.417219 | 45.30497 | 59.92413 |
56.59795 | 11.38371 | 3.88699 | 31.4366 |
2.14481 | 52.81506 | 45.32948 | 10.02026 |
74.33164 | 30.45189 | 43.83236 | 65.27543 |
2.222785 | 30.96207 | 56.2504 | 61.61284 |
51.18427 | 10.98492 | 35.03903 | 53.99805 |
63.76717 | 18.99628 | 47.82244 | 39.93808 |
66.87503 | 25.50829 | 65.897 | 55.5219 |
65.7767 | 2.634938 | 26.67122 | 33.77139 |
1. The correlation coefficient is not significant.
Y | X1 | X2 | X3 | |
Y | 1.000 | |||
X1 | -.121 | 1.000 | ||
X2 | -.049 | -.230 | 1.000 | |
X3 | .063 | .063 | .036 | 1.000 |
2. The regression equation is:
Y = 59.3685 - 0.1341*X1
The relationship is insignificant. (p > .05)
For every increase in X1, the value of Y will decrease by 0.1341.
When X1 is constant, the value of Y will increase by 59.3685.
The regression equation is:
Y = 56.0864 - 0.0560*X2
The relationship is insignificant. (p > .05)
For every increase in X2, the value of Y will decrease by 0.0560.
When X2 is constant, the value of Y will increase by 56.0864.
3. X1 is not significant. (p > .05)
X2 is not significant. (p > .05)
X3 is not significant. (p > .05)
The coefficient of determination is 0.026.
The standard error of the estimate is 28.826.
4. The predicted value of Y is 57.82.
Observation | Y | Predicted | Residual |
1 | 21.87838 | 48.76989 | -26.89151 |
2 | 93.77804 | 56.24570 | 37.53234 |
3 | 72.82208 | 53.22351 | 19.59857 |
4 | 9.28010 | 58.60905 | -49.32895 |
5 | 31.10620 | 54.08195 | -22.97575 |
The Excel output is:
R² | 0.026 | |||||
Adjusted R² | 0.000 | |||||
R | 0.162 | |||||
Std. Error | 28.826 | |||||
n | 42 | |||||
k | 3 | |||||
Dep. Var. | Y | |||||
ANOVA table | ||||||
Source | SS | df | MS | F | p-value | |
Regression | 854.3926 | 3 | 284.7975 | 0.34 | .7945 | |
Residual | 31,575.9299 | 38 | 830.9455 | |||
Total | 32,430.3225 | 41 | ||||
Regression output | confidence interval | |||||
variables | coefficients | std. error | t (df=38) | p-value | 95% lower | 95% upper |
Intercept | 62.0474 | |||||
X1 | -0.1610 | 0.1830 | -0.880 | .3846 | -0.5314 | 0.2095 |
X2 | -0.0973 | 0.1892 | -0.514 | .6099 | -0.4803 | 0.2856 |
X3 | 0.0764 | 0.1639 | 0.466 | .6440 | -0.2555 | 0.4083 |
r² | 0.002 | |||||
r | -0.049 | |||||
Std. Error | 28.440 | |||||
n | 42 | |||||
k | 1 | |||||
Dep. Var. | Y | |||||
ANOVA table | ||||||
Source | SS | df | MS | F | p-value | |
Regression | 77.0951 | 1 | 77.0951 | 0.10 | .7591 | |
Residual | 32,353.2274 | 40 | 808.8307 | |||
Total | 32,430.3225 | 41 | ||||
Regression output | confidence interval | |||||
variables | coefficients | std. error | t (df=40) | p-value | 95% lower | 95% upper |
Intercept | 56.0864 | |||||
X2 | -0.0560 | 0.1814 | -0.309 | .7591 | -0.4227 | 0.3106 |
r² | 0.015 | |||||
r | -0.121 | |||||
Std. Error | 28.265 | |||||
n | 42 | |||||
k | 1 | |||||
Dep. Var. | Y | |||||
ANOVA table | ||||||
Source | SS | df | MS | F | p-value | |
Regression | 473.7874 | 1 | 473.7874 | 0.59 | .4458 | |
Residual | 31,956.5350 | 40 | 798.9134 | |||
Total | 32,430.3225 | 41 | ||||
Regression output | confidence interval | |||||
variables | coefficients | std. error | t (df=40) | p-value | 95% lower | 95% upper |
Intercept | 59.3685 | |||||
X1 | -0.1341 | 0.1742 | -0.770 |
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