Question 1: (20 marks) A cheese maker is investigating the relationship, if any, between the Taste Score of a particular mature cheddar cheese and the concentrations of Acetic Acid (Acetic), Hydrogen Sulphide (H25) and Lactic Acid (Lactic). Data are gathered on 30 samples of this cheese, with an excerpt of the data [units omitted) shown in the table below. Case Acetic H25 Lactic Taste 1 4.543 3.135 0.86 12.3 IN 5.159 5.043 1.53 20.9 3 5.366 5.438 1.57 39.0 5.759 7.496 1.81 47.9 # # # # (a) What type of data is provided here? Justify your answer within the context of data science. [2 marks] To assist with the investigation three different linear regression models were fitted using Taste as the response variable and Acetic, H25 and Lactic as explanatory variables. The resultant output from R Commander is provided on the following page. (b) Using the scatterplot matrix describe the relationship between each of the explanatory variables and the response variable. [3 marks] (c) If collinearity were to be a problem in this modelling, identify which two variables this would most likely relate to and justify your decision. [2 marks] (d) Using the output from the appropriate model, write down the fitted multiple linear regression equation to predict Taste from Acetic, H25 and Lactic. [2 marks] (e) A particular sample of this cheese not included in the study had the following readings: Acetic = 5.1, H25 = 3.2 and Lactic = 1.8. Calculate the predicted Taste score for the sample. [2 marks] (f) In Model 2 the coefficient associated with H25 is 3.953. Interpret this value. [2 marks] (g) Using the output from Model 1 conduct a test using a 5% level of significance to determine if this model is better than no model at all. (i.e. whether all the linear regression coefficients are equal to zero or not) State clearly the null and alternate hypotheses, the test statistic and its value, decision rule and the test result in terms of the original problem. [4 marks] (h) Compare the three models to determine which of these is best. Refer to relevant statistical measures from the output to support your decision. [2 marks] (i) Use the output of the appropriate model to calculate the correlation coefficient between the variables Taste and Lactic. [1 mark]Scatterplot matrix of Cheese Data B 10 Acetic 4.5 H2S Lactic 1.0 Taste 4.5 50 5.5 60 6.5 1.0 14 1.8 Model 1 Call : Im (formula = Taste > Acetic + 828 + Lactic, data = Cheese) Residuals : Min 10 Median 30 Max -17.408 -6. 627 -1. 012 4.909 25 . 443 Coefficients: Estimate Std. Error t value Pr (>|=1) (Intercept) -28 . 8544 19.7417 -1. 462 0. 15583 Acetic 0 . 3147 4. 4639 0. 071 0. 94434 H25 3. 9204 1. 2477 3. 142 0. 00416 * * Lactic 19. 6744 8. 6466 2. 275 0.03136 . Signif. codes: 0 ! * *# 1 0.001 1#*1 0.01 1#1 0.05 . " 0.1. . Residual standard error: 10.13 on 26 degrees of freedom Multiple R-squared: 0.6516, Adjusted R-squared: 0 . 6114 F-statistic: 16.21 on 3 and 26 DF, prvalue: 0. 000003839Model 2 Call : Im (formula = Taste ~ H23 + Lactic, data = Cheese) Residuals : Mir Median DE Max -17.362 -6.547 -1.161 4. 848 25 .606 Coefficients : Estimate Sed. Error t value (Intercept) -27. 622 9.001 -3.069 0 .00485 * H25 3. 953 1.125 3. 483 0 . 00170 Lactic 19.884 7.970 2. 495 0 . 01902 Signif. codes : ( 1 * * * 1 0.001 "#* 1 0.01 1*1 0.05 1. 0.1 " 1 Residual standard error: 9.946 on 27 degrees of freedom Multiple R-squared: 0. 6515, Adjusted R-squared: 0 . 6257 F-statistic: 25.24 on 2 and 27 DF, p-value: 0. 0000006602 Model 3 Call : Im (formula = Taste > Lactic, data = Cheese) Residuals : Min 10 Median Max -19.9508 -8.7039 -0.1123 8.9756 27. 4084 Coefficients : Estimate Std. Error value Pr ()[=1) (Intercept) -29.905 10 .613 -2. 818 0. 00877 . Lactic 37.761 7.209 5.238 0 . 0000145 * * * Signif. codes: 0 1* *# 1 0.001 1# #1 0.01 1#1 0.05 . " 0.1 . . 1 Residual standard error: 11.76 on 28 degrees of freedom Multiple R-squared: 0. 4949, Adjusted R-squared: 0. 4768 F-statistic: 27.43 on 1 and 28 DF, prvalue: 0. 00001448