Minutes 23 29 49 64 74 87 96 97 109 119 149 145 154 166 Units 1 2 34456 678 9 91
ID: 3053337 • Letter: M
Question
Minutes 23 29 49 64 74 87 96 97 109 119 149 145 154 166 Units 1 2 34456 678 9 910 10 When estimating a linear regression model for the data, keep in man the objective is to predict Y given values of X. Also note that the relationship between a response variable Y and a predictor variable X can be written as follows. EXCEL (and other statistical software) can be used to estimate the following from of this equation. Without mathematical proof the parameters needed to estimate the "predicted value" of the response variable - Y- in equation (2) are as follows. r (3) In both equations (3) and (4), Y -y1V2. n and X-x1,X2,xn When discussing the regression coefficients, note the formulas, annotate them, and show the answers. Show how you solved (3) and (4) by producing a table containing the figures neededExplanation / Answer
Minutes(x)
Units(y)
y-mean(y)
x-mean(x)
(y-mean(y))(x-mean(x))
(x-mean(x))^2
23
1
-5
-74.2143
371.0714286
5507.76
29
2
-4
-68.2143
272.8571429
4653.189
49
3
-3
-48.2143
144.6428571
2324.617
64
4
-2
-33.2143
66.42857143
1103.189
74
4
-2
-23.2143
46.42857143
538.9031
87
5
-1
-10.2143
10.21428571
104.3316
96
6
0
-1.21429
0
1.47449
97
6
0
-0.21429
0
0.045918
109
7
1
11.78571
11.78571429
138.9031
119
8
2
21.78571
43.57142857
474.6173
149
9
3
51.78571
155.3571429
2681.76
145
9
3
47.78571
143.3571429
2283.474
154
10
4
56.78571
227.1428571
3224.617
166
10
4
68.78571
275.1428571
4731.474
Total
1361
84
0
0
1768
27768.36
Mean
97.2142857
6
0
0
126.2857143
1983.454
Beta1 = 1768/27768.36 = 0.06367
Beta0 = 6 – 0.06367*97.2142857 = -0.18959
If we carry out Regression (from DATA ANALYSIS TOOLPACK) on the given data, we get the following output :
SUMMARY OUTPUT
Regression Statistics
Multiple R
0.993699
R Square
0.987437
Adjusted R Square
0.98639
Standard Error
0.345466
Observations
14
ANOVA
df
SS
MS
F
Significance F
Regression
1
112.5678
112.5678
943.2009
8.92E-13
Residual
12
1.432159
0.119347
Total
13
114
Coefficients
Standard Error
t Stat
P-value
Lower 95%
Upper 95%
Lower 95.0%
Upper 95.0%
Intercept
-0.18959
0.221682
-0.85525
0.409163
-0.6726
0.29341
-0.6726
0.29341
Minutes(x)
0.06367
0.002073
30.71158
8.92E-13
0.059153
0.068187
0.059153
0.068187
The coefficients here match with the calculated coefficients.
Minutes(x)
Units(y)
y-mean(y)
x-mean(x)
(y-mean(y))(x-mean(x))
(x-mean(x))^2
23
1
-5
-74.2143
371.0714286
5507.76
29
2
-4
-68.2143
272.8571429
4653.189
49
3
-3
-48.2143
144.6428571
2324.617
64
4
-2
-33.2143
66.42857143
1103.189
74
4
-2
-23.2143
46.42857143
538.9031
87
5
-1
-10.2143
10.21428571
104.3316
96
6
0
-1.21429
0
1.47449
97
6
0
-0.21429
0
0.045918
109
7
1
11.78571
11.78571429
138.9031
119
8
2
21.78571
43.57142857
474.6173
149
9
3
51.78571
155.3571429
2681.76
145
9
3
47.78571
143.3571429
2283.474
154
10
4
56.78571
227.1428571
3224.617
166
10
4
68.78571
275.1428571
4731.474
Total
1361
84
0
0
1768
27768.36
Mean
97.2142857
6
0
0
126.2857143
1983.454