An article in Biotechnology Progress (Vol. 17, 2001, pp. 366–368) reported an experiment to investigate and optimize nisin extraction in aqueous two-phase systems (ATPS). The nisin recovery was the dependent variable (y). The two regressor variables were concentration (%) of PEG 4000 (denoted as x1) and concentration (%) of Na2SO4 (denoted as x2). x1 x2 y
13 11 62.8739
15 11 76.1328
13 13 87.4667
15 13 102.3236
14 12 76.1872
14 12 77.5287

Requried:
a. Fit a multiple linear regression model to these data.
b. Estimate and the standard errors of the regression coefficients.
c. Test for significance of β1 and β2.

ŷ = 7.02895X1 + 12.6959X2 - 170.33728

Since the calculated F=32 falls in the critical region F ≥ 9.55 we reject our null hypothesis and conclude that there is association between at least one of the regressors and the dependent variable.

Step-by-step explanation:

Using stat calculator:

Part A:

x1         x2            y              Predicted Y          Residual

13        11         62.8739       60.693967        2.179933

15        11        76.1328         74.751867          1.380933

13        13        87.4667       86.085767         1.380933

15        13       102.3236      100.143667        2.179933

14         12      76.1872         80.418817         -4.231617

14        12        77.5287        80.418817         -2.890117

mean

14        12       80.418817     80.418817           0            

standard deviation

0.89 0.89      13.2812       12.979739       2.813451        

Calculation

Sum of X1 = 84

Sum of X2 = 72

Sum of Y = 482.5129

Mean X1 = 14

Mean X2 = 12

Mean Y = 80.4188

Sum of squares (SSX1) = 4

Sum of squares (SSX2) = 4

Sum of products (SPX1Y) = 28.1158

Sum of products (SPX2Y) = 50.7836

Sum of products (SPX1X2) = 0

Regression Equation = ŷ = b1X1 + b2X2 + a

b1 = ((SPx1y)*(SSx2)-(SPx1x2)*(SPx2y)) / ((SSx1)*(SSx2)-(SPx1x2)*(SPx1x2)) = 112.46/16 = 7.02895

b2 = ((SPx2y)*(SSx1)-(SPx1x2)*(SPx1y)) / ((SSx1)*(SSx2)-(SPx1x2)*(SPx1x2)) = 203.13/16 = 12.6959

a = MY - b1MX1 - b2MX2 = 80.42 - (7.03*14) - (12.7*12) = -170.33728

ŷ = 7.02895X1 + 12.6959X2 - 170.33728

X1-Mx1         X2-Mx2        Y-My               (X1-Mx1)²            (X2-Mx2)²

-1                     -1                 -17.545               1                       1

1                        -1                -4.286               1                       1

-1                       1                  7.048                1                      1

1                        1                  21.905              1                       1

0                       0                -4.232               0                      0

0                       0                 -2.89                0                     0            

                                                             SSX1: 4       SSX2: 4        

SPx1y            SPx2y         SPx1x2

17.545           17.545            1

-4.286            4.286            -1

-7.048             7.048          - 1

21.905             21.905        1

0                        0                0

0                       0                 0                      

SPX1Y:         SPX2Y:           SPX1X2: =0

= 28.116       =50.784                                    

Part B

Coefficient Table

             Coefficient           SE                     t- stat        

x1           -170.337283      33.519576          -5.081725

x2           7.028950         1.816075            3.870408

b            12.695900        1.816075           6.990847    

Part C:

Test for significance of β1 and β2.

State the null and alternate hypotheses as

H0: β1=β2=0

Ha:  At least one of the β1 and β2 is non zero.

The significance level is set at ∝= 0.05

The test statistic to use is

F= MSR/ MSE= MS regression/ MS Residual

which if H0 is true has F distribution with υ1= 2 and υ2= n- 3= 6-3= 3  degrees of freedom.

To set up the ANOVA table we find the necessary sum of squares .

Regression SS ( between y^ and y`) = SSR= 842.368060

Residual SS ( between yi and y^) = SSE=  39.577526

Total SS  = SSR+ SSE= 842.368060+39.577526= 881.945586

ANOVA table

Source                 DF      Sum of Square     Mean Square    F Statistic

Regression

(b/w ŷi and yi)2    842.368060421.18403031.926000

Residual

(b/w yi and ŷi)3    39.577526          13.192509                  

Total (b/w yi and yi)5      881.945586176.389117                              

The critical region is F≥F (0.05) (2,3)=9.55

Conclusion:

Since the calculated F=32 falls in the critical region F ≥ 9.55 we reject our null hypothesis and conclude that there is association between at least one of the regressors and the dependent variable.