Question

*in r studio file

4. The director of admissions of a small college selected 120 students at random from the new freshman class in a study to determine whether a student’s grade point average (GPA) at the end of the freshman year (y) can be predicted from the ACT test score (x). Estimate the simple linear regression of, y = β0 + β1x + ε, using gpa.txt data and answer the following questions. (4 pts each)

(a) Report the least squares estimates of β0 and β1 and state the estimated regression function.

(b) Plot the estimated regression function and the data. Does the estimated regression function appear to fit the data well?

gpa.txt

y x
3.897 21
3.885 14
3.778 28
2.540 22
3.028 21
3.865 31
2.962 32
3.961 27
0.500 29
3.178 26
3.310 24
3.538 30
3.083 24
3.013 24
3.245 33
2.963 27
3.522 25
3.013 31
2.947 25
2.118 20
2.563 24
3.357 21
3.731 28
3.925 27
3.556 28
3.101 26
2.420 28
2.579 22
3.871 26
3.060 21
3.927 25
2.375 16
2.929 28
3.375 26
2.857 22
3.072 24
3.381 21
3.290 30
3.549 27
3.646 26
2.978 26
2.654 30
2.540 24
2.250 26
2.069 29
2.617 24
2.183 31
2.000 15
2.952 19
3.806 18
2.871 27
3.352 16
3.305 27
2.952 26
3.547 24
3.691 30
3.160 21
2.194 20
3.323 30
3.936 29
2.922 25
2.716 23
3.370 25
3.606 23
2.642 30
2.452 21
2.655 24
3.714 32
1.806 18
3.516 23
3.039 20
2.966 23
2.482 18
2.700 18
3.920 29
2.834 20
3.222 23
3.084 26
4.000 28
3.511 34
3.323 20
3.072 20
2.079 26
3.875 32
3.208 25
2.920 27
3.345 27
3.956 29
3.808 19
2.506 21
3.886 24
2.183 27
3.429 25
3.024 18
3.750 29
3.833 24
3.113 27
2.875 21
2.747 19
2.311 18
1.841 25
1.583 18
2.879 20
3.591 32
2.914 24
3.716 35
2.800 25
3.621 28
3.792 28
2.867 25
3.419 22
3.600 30
2.394 20
2.286 20
1.486 31
3.885 20
3.800 29
3.914 28
1.860 16
2.948 28

Answer 1

using minitab>stat>Regression

we have

Regression Analysis: y versus x

Analysis of Variance

Source DF Adj SS Adj MS F-Value P-Value
Regression 1 3.588 3.5878 9.24 0.003
x 1 3.588 3.5878 9.24 0.003
Error 118 45.818 0.3883
Lack-of-Fit 19 6.486 0.3414 0.86 0.632
Pure Error 99 39.332 0.3973
Total 119 49.405

Model Summary

S R-sq R-sq(adj) R-sq(pred)
0.623125 7.26% 6.48% 3.63%

Coefficients

Term Coef SE Coef T-Value P-Value VIF
Constant 2.114 0.321 6.59 0.000
x 0.0388 0.0128 3.04 0.003 1.00

Regression Equation

y = 2.114 + 0.0388 x

(a) least squares estimates of β0 and β1 are 2.114 and 0.0388 respectively and the estimated regression function is

y = 2.114 + 0.0388 x

(b) plot of the estimated funtion is given below ,  yes , the estimated regression function appear to fit the data well