Consider a manufacturing process for making pistons from metal ingots in which each ingot produces enough material for 1000pistons. Sometimes, a piston cracks whilst cooling after being forged. Previous research has shown that, in a batch of 1000pistons, the average number of pistons that develop a crack during cooling is dependent on the purity of the ingots. Ingots of known purity X were forged into pistons, and the average number Y of cracked pistons per batch was recorded in the table below. Each batch is made up of a random sample of pistons of the same purity. Ingots purity I; 0.960.940.950.980.980.97 Average number y; Of cracked pistons 3.795.074.432.892.063.16 Below is the data from the table reproduced in a format which you can copy-paste easily. 0. 96 0. 94 0. 95 0. 98 0. 99 0.97 3. 79 5. 07 4. 43 2. 89 2. 06 3. 16 One has I = 0.965 and ( 1)2 = 0.00175. The following linear regression model is fit to these data: Y = Bo + BIX+E. Use the following Matlab regression output to answer the questions below. (Note that you may need to adjust the width of your browser window for the output to be displayed properly) Linear regression model: Y1 +X Estimated Coefficients: Estimate SE tStat pValue ( Intercept) 59. 537 2. 862 20. 81 3. 15e-05 X -58. 000 2. 965 -19. 56 4. 03e-05 Number of observations: 6, Error degrees of freedom: 4 Root Mean Squared Error: 0. 1240 R-squared: 0. 9897a) Use the Matlab regression output to answer this part. i) [1 mark] What proportion of variability in the response Y is explained by the predictor X? (Enter your answer correct to at least 4 decimal places) ii) [1 mark] How would you describe the quality of the fit of the above linear model? o Poor fit O Average fit o Good fit o Excellent fit b) [1 mark] Determine the observed sample correlation coefficient between X and Y. (Enter your answer correct to at least 4 decimal places) c) [2 marks] Assume a is the standard deviation of the error term & Give an estimate of a (with units). (Enter your numerical answer correct to at least 4 decimal places) d) [1 mark] Give an estimate of the expected change in the average number of cracked pistons per batch Y for a change of 0.01in the ingots purity X. (Enter your answer correct to at least 4 decimal places)