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.
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