Exercise 1 (18 marks) The following table presents shear strengths (in kN/mm) and weld diameters (in mm) for a sample of spot welds. Diameter Strength 4.2 51 4.4 54 4.6 69 4.8 81 5.0 75 5.2 79 5.4 89 5.6 101 5.8 98 6.0 102 a) Construct a scatterplot of strength (y) versus diameter (x). Verify that a linear model is appropriate. (done by hand) b) Compute the least-squares line for predicting strength from diameter. (show all your calculations) c) Compute the fitted values (i.e. the predicted values) and the residual for each point. Which point has the residual with the largest magnitude? (can be done in excel) d) If the diameter is increased by 0.3 mm, by how much would you predict the strength to increase or decrease? e) Predict the strength for a diameter of 5.5 mm. f) Can the least-squares line be used to predict the strength for a diameter of 8mm? If so, predict the strength. If not, explain why not. g) For what diameter would you predict a strength of 95 kN/mm? h) Compute the regression sum of squares, the error sum of squares, and the total sum of squares. i) Divide the regression sum of squares by the total sum of squares. What is the relationship between this quantity and the correlation coefficient? Exercise 2 (19 marks) Following are measurements of tensile strength in ksi (x) and Brinell hardness (y) for 10 specimens of cold drawn copper. Assume that tensile strength and Brinell hardness follow a bivariate normal distribution. 106.2 106.3 105.3 106.1 105.4 106.3 104.7 105.4 105.5 105.1 . 35.0 37.2 39.8 35.8 41.3 40.7 38.7 40.2 38.1 41.6 a) Find a 95% confidence interval for P, the population correlation between tensile strength and Brinell hardness. b) Can you conclude that P < 0.3? c) Can you conclude that P = 0? Exercise 3 (10 marks) The article "Optimization of Medium Composition for Lipase Production by Candida rugosa NCIM 3462 Using Response Surface Methodology" describes a series of experiments in which lipase was produced from a bacterial culture. In each experiment, the rate of lipase production (in " mol per ml enzyme per minute) and the cell mass (in g/L) were measured. The results are presented in the following table: Cell Lipase Cell Lipase Cell Lipase Cell Lipase mass (x) (y) mass (x) (y) mass (x) (y) mass (x) (y) 4.5 2.06 3.98 2.1 3.65 2.2 4.15 3.75 4.68 2.1 4.72 2.75 4.23 2.3 4.3 3.15 5.4 3.15 3.41 2.8 4.1 2.4 4.9 5.1 5.45 4.1 4.8 4.6 5.03 4.75 5.23 5.04 4.2 2.2 3.6 2.5 4.19 3.15 5.4 4.96 4.12 3.2 4.95 4.1 4.4 3.9 4.85 5 4 2.85 3.25 2.15 3.92 3.2 5.1 4.92 4.41 4.5 4.4 4.4 3.5 2.1 4.94 4.98 Analyze the above data using SAS. Create a scatter plot, find the least squares line (highlight it in your output and write it out), find Pearson's correlation coefficient (highlight it in your output and explain it), complete a test of linearity (based on your SAS output but show any necessary hypotheses, decision and conclusions by hand), and find the coefficient of determination (highlight it in you output and explain it). Note: Print out your SAS code and output. Exercise 4 (10 marks) A chemical company wishes to study the effect of extraction time on the efficiency of an extraction operation. Here are the descriptive statistics that were generated in SAS based on the extraction time (in minutes) and the extraction efficiency (y): Variable time efficiency The SAS System The CORR Procedure 2 Variables: time efficiency Covariance Matrix, DF = 9 time efficiency time efficiency 138.8888889 106.1111111 106.1111111 119.1666667 Simple Statistics N Mean Std Dev Sum Minimum Maximum 10 32.00000 11.78511 320.00000 15.00000 49.00000 10 63.50000 10.91635 635.00000 46.00000 80.00000 (a) Compute the sample correlation coefficient. (b) Is there evidence of a significant correlation between extraction time and extraction efficiency? Run the appropriate test at = 0.05. (c) Compute the equation of the line that best describes the relationship between extraction time and extraction efficiency. [Note: Extraction efficiency is the dependent variable.]