Chapter 12.2, Problem 22E

Calcium phosphate cement is gaining increasing attention for use in bone repair applications. The article “Short-Fibre Reinforcement of Calcium Phosphate Bone Cement” (J. of Engr. in Med., 2007: 203–211) reported on a study in which polypropylene fibers were used in an attempt to improve fracture behavior. The following data on x = fiber weight (%) and y = compressive strength (MPa) was provided by the article’s authors.

x	0.00	0.00	0.00	0.00	0.00	1.25	1.25	1.25	1.25
y	9.94	11.67	11.00	13.44	9.20	9.92	9.79	10.99	11.32

x	2.50	2.50	2.50	2.50	2.50	5.00	5.00	5.00	5.00
y	12.29	8.69	9.91	10.45	10.25	7.89	7.61	8.07	9.04

x	7.50	7.50	7.50	7.50	10.00	10.00	10.00	10.00
y	6.63	6.43	7.03	7.63	7.35	6.94	7.02	7.67

1. a. Fit the simple linear regression model to this data. Then determine the proportion of observed variation in strength that can be attributed to the model relationship between strength and fiber weight. Finally, obtain a point estimate of the standard deviation of e, the random deviation in the model equation.
2. b. The average strength values for the six different levels of fiber weight are 11.05, 10.51, 10.32, 8.15, 6.93, and 7.24, respectively. The cited paper included a figure in which the average strength was regressed against fiber weight. Obtain the equation of this regression line and calculate the corresponding coefficient of determination. Explain the difference between the r² value for this regression and the r² value obtained in (a).