Concrete A researcher wants to determine a model that can be used to predict the 28-day strength of a concrete mixture. The following data represent the 28-day and 7-day strength (in pounds per square inch) of a certain type of concrete along with the concrete's slump. Slump is a measure of the uniformity of the concrete, with a higher slump indicating a less uniform mixture.
|
Slump (inches)
|
7-Day psi
|
28-Day psi
|
|
4.5
|
2330
|
4025
|
|
4.25
|
2640
|
4535
|
|
3
|
3360
|
4985
|
|
4
|
1770
|
3890
|
|
3.75
|
2590
|
3810
|
|
2.5
|
3080
|
4685
|
|
4
|
2050
|
3765
|
|
5
|
2220
|
3350
|
|
4.5
|
2240
|
3610
|
|
5
|
2510
|
3875
|
|
2.5
|
2250
|
4475
|
(a) Construct a correlation matrix between slump, 7-day psi, and 28-day psi. Is there any reason to be concerned with multicollinearity based on the correlation matrix?
(b) Find the least-squares regression equation y = b0 + b1x1 + b2x2, where x1 is slump, x2 is 7-day strength, and y is the response variable, 28-day strength.
(c) Draw residual plots and a boxplot of the residuals to assess the adequacy of the model.
(d) Interpret the regression coefficients for the least-squares regression equation.
(e) Determine and interpret R2 and the adjusted R2.
(f) Test H0: β1 = β2 = 0 versus H1: at least one of the βi ≠ 0 at the a = 0.05 level of significance.
(g) Test the hypotheses H0: β1 = 0 versus H1: β1 ≠ 0 and H0: β2 = 0 versus H1: β2 ≠ 0 at the α = 0.05 level of significance.
(h) Predict the mean 28-day strength of all concrete for which slump is 3.5 inches and 7-day strength is 2450 psi.
(i) Predict the 28-day strength of a specific sample of concrete for which slump is 3.5 inches and 7-day strength is 2450 psi.
(j) Construct 95% confidence and prediction intervals for concrete for which slump is 3.5 inches and 7-day strength is 2450 psi. Interpret the results.