Question

Unusually high concentration of metals in drinking water can pose a health hazard. Twenty couples of data were taken from different locations of X Lake measuring zinc concentration in bottom water and surface water.

Table 1. Zinc Concentration (mg/l) in bottom and surface water in different regions and areas of X Lake

Location	Region	Area	Zinc_Bottom	Zinc_Surface	Zinc_Mean	TZ3_Mean
Loc 1	North	Area A	0.51	0.49	0.50	2.67
Loc 2	North	Area A	0.46	0.45	0.46	2.45
Loc 3	North	Area A	0.52	0.52	0.52	2.66
Loc 4	North	Area A	0.51	0.50	0.50	2.56
Loc 5	North	Area A	0.58	0.57	0.58	3.03
Loc 6	North	Area A	0.57	0.55	0.56	3.21
Loc 7	North	Area A	0.59	0.58	0.59	3.50
Loc 8	North	Area B	0.46	0.44	0.45	2.40
Loc 9	North	Area B	0.46	0.44	0.45	2.35
Loc 10	North	Area B	0.45	0.43	0.44	2.56
Loc11	East	Area B	0.41	0.40	0.41	1.33
Loc 12	East	Area B	0.41	0.39	0.40	1.56
Loc 13	East	Area B	0.41	0.39	0.40	1.23
Loc 14	East	Area B	0.47	0.45	0.46	1.12
Loc 15	East	Area C	0.41	0.39	0.40	1.23
Loc 16	East	Area C	0.37	0.36	0.37	1.10
Loc 17	East	Area C	0.41	0.39	0.40	1.20
Loc 18	East	Area C	0.40	0.39	0.39	1.10
Loc 19	East	Area C	0.39	0.37	0.38	1.12
Loc 20	East	Area C	0.46	0.44	0.45	1.30

QESTION

Including “Region” factor as a dummy variable to your previous model, fit a multiple regression model to explain zinc concentration (Zinc_Mean). Obtain the ANOVA table of the regression. Comment on overall significance of the model and the coefficients of each variable. Do you think adding “Region” next to “TZ3_Mean” improved the model? explain

Answer 1

Here we are given the unusually high concentration of metals in drinking water which can cause a health hazard. 20 samples are given and we have to fit a multiple regression model to explain Zinc concentration by treating Region factor as a dummy variable.

We write 1 for north and 0 for east. Now we perform the following steps in MINITAB:

1. Enter the given values in different columns.

2. Go to “Stat” then “Regression” then “Regression” then “Fit regression model”.

3. Enter Zinc_Mean in “response” and Zinc_Bottom, Zinc_Surface, TZ3_Mean, Region in “continuous predictor”.

4. Click OK.

Hence we get the following output:

Analysis of Variance

Source          DF    Adj SS    Adj MS F-Value P-Value
Regression       4 0.084499 0.021125 1613.75    0.000
Zinc_Bottom    1 0.000014 0.000014     1.08    0.316
Zinc_Surface   1 0.001185 0.001185    90.51    0.000
TZ3_Mean       1 0.000031 0.000031     2.38    0.144
Region         1 0.000022 0.000022     1.72    0.210
Error           15 0.000196 0.000013
Lack-of-Fit   14 0.000196 0.000014        *        *
Pure Error     1 0.000000 0.000000
Total           19 0.084695

Coefficients

Term              Coef SE Coef T-Value P-Value    VIF
Constant       0.01524 0.00863     1.77    0.098
Zinc_Bottom     0.0922   0.0888     1.04    0.316 49.49
Zinc_Surface    0.8657   0.0910     9.51    0.000 54.28
TZ3_Mean       0.00727 0.00472     1.54    0.144 21.90
Region        -0.00720 0.00550    -1.31    0.210 11.54

Regression Equation

Zinc_Mean = 0.01524 + 0.0922 Zinc_Bottom + 0.8657 Zinc_Surface + 0.00727 TZ3_Mean
            - 0.00720 Region

Interpretation of coefficients:

1) If 1 unit changes in Zinc_Bottom 0.0922 units increase in Zinc_Mean.

2) If 1 unit changes in Zinc_Surface 0.8657 units increase in Zinc_Mean.

3) If 1 unit changes in TZ3_Mean 0.00727 units increase in Zinc_Mean.

4) If 1 unit changes in Region 0.00720 units decrease in Zinc_Mean.

We know if the p value of the coefficients is <= 0.05 then they are significant.

Here, the pvalue of Zinc_Surface is only less than 0.05 so it is significant while the rest are insignificant.

So we can say overall the model is not very significant.

Now we run the regression excluding the region factor and run the following steps in MINITAB:

1. Enter the given values in different columns.

2. Go to “Stat” then “Regression” then “Regression” then “Fit regression model”.

3. Enter Zinc_Mean in “response” and Zinc_Bottom, Zinc_Surface, TZ3_Mean in “continuous predictor”.

4. Click OK.

The r square obtained is 0.9974

The r square obtained in the model including Region factor is 0.9977

Hence we can conclude that if the Region is included in the model it improves the model since the r square value is more.