The article “Optimization of Enterocin P Production by Batch Fermentation of Ent

Question

The article “Optimization of Enterocin P Production by Batch Fermentation of Enterococcus faecium P13 at Constant pH” (C. Herran, J. Martinez, et al., Applied Microbiology and Biotechnology, 2001:378–383) described a study involving the growth rate of the bacterium Enterococcus faecium in media with varying pH. The log of the maximum growth rate for various values of pH are presented in the following table:

ln Growth rate 2.12 1.51 0.89 0.33 0.05 0.11 0.39 0.25

pH 4.7 5.0 5.3 5.7 6.0 6.2 7.0 8.5

a. Fit the linear model: ln Growth rate = 0+1 pH+. For each coefficient, find the P-value for the null hypothesis that the coefficient is equal to 0. In addition, compute the analysis of variance (ANOVA) table.

b. Fit the quadratic model: ln Growth rate = 0 +1 pH+2 pH2 +. For each coefficient, find the P-value for the null hypothesis that the coefficient is equal to 0. In addition, compute the ANOVA table.

c. Fit the cubic model: ln Growth rate = 0 + 1 pH + 2 pH2 + 3 pH3 + . For each coefficient, find the P-value for the null hypothesis that the coefficient is equal to 0. In addition, compute the ANOVA table.

d. Which of these models do you prefer, and why?

Explanation / Answer

(A) Here the excel results with ANOVA and regression table alongwith p - value below.

The given resuls are

P - value for pH = 0.05855 > 0.05 so not significant in nature.  

ln (Growth rate) = -3.4641 + 0.47196 * pH

for  quadratic model

Here,

ln (growth rate) = -20.485 + 5.832884 * pH - 0.40645 (pH)2

Both p - values is less than 0.05 so regression is significant in nature.

For cubic model:

ln(Grwth rate) = -32.0268 + 11.38274 * (pH) - 1.27604 * (pH)2 + 0.044329 * (pH)3

P- values are not significant in nature here for all the independent variable.

(d) Here we will prefer the quadratic model as F - value is highest for it and p - values are also significant here.

SUMMARY OUTPUT Regression Statistics Multiple R 0.68938497 R Square 0.47525163 Adjusted R Square 0.38779357 Standard Error 0.65730547 Observations 8 ANOVA df SS MS F Significance F Regression 1 2.34778 2.34778 5.43405 0.05855 Residual 6 2.59230 0.43205 Total 7 4.94009 Coefficients Standard Error t Stat P-value Lower 95% Upper 95% Lower 95.0% Upper 95.0% Intercept -3.46413 1.24675 -2.77852 0.03206 -6.51483 -0.41343 -6.51483 -0.41343 pH 0.47196 0.20246 2.33111 0.05855 -0.02345 0.96737 -0.02345 0.96737

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