Strength of a concrete mixture

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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