Question

This data talks about a solution of a chemical after sitting for a certain amount of time and provides the time it was left sitting.

Concentration of a chemical solution (y)	Time after solution was made (x)
0.07	9
0.09	9
0.08	9
0.16	7
0.17	7
0.21	7
0.49	5
0.58	5
0.53	5
1.22	3
1.15	3
1.07	3
2.84	1
2.57	1
3.1	1

1. Run regression analysis on this. Make sure to output the residuals and residual plots.
2. Plot the predicted values versus the residual values in a scatter plot.
3. What assumption is violated in this data?
4. What solution can be applied here to fix the issues? Try your solution and see if it fixes the problems with your residuals.

Answer 1

a)

Regression Analysis: y versus x

The regression equation is
y = 2.58 - 0.324 x

Predictor Coef SE Coef T P
Constant 2.5753 0.2487 10.35 0.000
x -0.32400 0.04330 -7.48 0.000

S = 0.474314 R-Sq = 81.2% R-Sq(adj) = 79.7%

Analysis of Variance

Source DF SS MS F P
Regression 1 12.597 12.597 55.99 0.000
Residual Error 13 2.925 0.225
Total 14 15.522

b)

y   x   Residual   predicted values
0.07   9   0.410667   -0.34067
0.09   9   0.430667   -0.34067
0.08   9   0.420667   -0.34067
0.16   7   -0.147333   0.30733
0.17   7   -0.137333   0.30733
0.21   7   -0.097333   0.30733
0.49   5   -0.465333   0.95533
0.58   5   -0.375333   0.95533
0.53   5   -0.425333   0.95533
1.22   3   -0.383333   1.60333
1.15   3   -0.453333   1.60333
1.07   3   -0.533333   1.60333
2.84   1   0.588667   2.25133
2.57   1   0.318667   2.25133
3.10   1   0.848667   2.25133

The scatter plot of the predicted values versus the residual values is shown at the above figure.

c) From the scatter plot of the predicted values versus the residual values in b), we know that this violates the shapeless without a clear picture pattern, no obvious outliers, and be generally symmetrically distributed around the 0 line without particularly large residuals. Hence, we can conclude that the assumption of independent and identically distribution on residuals for the linear regression model is violated.

d) One of the solutions that can be applied here to fix the issues is that the log-transformation of the dependent variable Concentration of a chemical solution (y).

From the above scatter plot of residual VS fitted values, we know that it improves the randomness of the scatter plot and may satisfy the assumption of the normal distribution of the residual. Hence, it fixes the problem of residuals.