An experiment was performed to test the effect of three process variables on the deviation in filling bottles with a certain soft drink up to the specified height. The variables investigated are percent carbonation, operating pressure in the filter, and the line speed. 16 Experiments were performed and the data can be found in the file filldeviation.csv. The purpose was to understand the variability in the filling heights.

1. Use the data and fit a multiple linear regression model in R for the deviation as dependent variable as a function of the three variables carbonation, pressure and linespeed as predictors. Use the centred variables for predictors i.e., subtract the mean for each independent variable/predictor. The model must contain the main effects and the two-order interaction terms for the predictors. Give the full summary table from the linear model fit.

2. Reduce the model to the model where all variables/terms in the model are significant at significance level α = 0.1. Give the full summary table from the linear model fit.

3. What is the R2 and adjusted R2 of the model?

4. What is the MSE of the model?

5. Make a qqnorm plot of the residuals, and interpret the plot.

6. Plot the studentized residuals against the fitted values of the model, and interpret the plot.

7. Perform a Shapiro-Wilk test to test for normality of the residuals, at significance level α = 0.05. What are your conclusions?

8. Make an interaction plot for variables carbonation and pressure, and interpret the plot.

9. In order to minimize the deviation in filling height of the the bottles, what would you recommend as optimum conditions for the variables in terms of their actual values (not centred)?

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carbonation, pressure, linespeed,deviation 10,25,200,-1 12,30,200,2 12,25,200,1 10,30,200,0 12,30,250,6 10,30,250,1 12,25,250,1 12,30,250,5 12,25,200,0 10,25,250,-1 10,30,250,1 12,25,250,2 10,30,200,-1 10,25,200,-3 12,30,200,3 10,25,250,0 carbonation, pressure, linespeed,deviation 10,25,200,-1 12,30,200,2 12,25,200,1 10,30,200,0 12,30,250,6 10,30,250,1 12,25,250,1 12,30,250,5 12,25,200,0 10,25,250,-1 10,30,250,1 12,25,250,2 10,30,200,-1 10,25,200,-3 12,30,200,3 10,25,250,0