Question

We continue with the Concrete dataset. Concrete is a central product for most modern constructions and is used in homes, roads, and commercial structures and there are many other building applications. Frequently, there is an issue of strength (compressive strength) which is measured in megapascals (MPa). Several attributes contribute to the strength of concrete.

1. Download the data file https://docs.google.com/spreadsheets/d/1jVV26-UbjWhGEOi9Aww81JmSj3kM3JY6lQY662VK-pg/edit?usp=sharing

1)How much of the variability in strength is explained by the predictors?

Round to three decimals and use leading zeros if necessary.

2)How are the degrees of freedom regression computed?

The model of the data is described by ŷ = [a] + [b]*x1 + [c]*x₂ + [d]*x₃ + [e]*x₄ + [f]*x₅ + [g]*x₆+ [h]*x₇+ [i]*x₈

Consider the following regression analysis report:

Regression Statistics
Multiple R	0.977865
R Square	0.956221
Adjusted R Square	0.95492
Standard Error	32.85796
Observations	209
ANOVA
	df	SS	MS	F	Significance F
Regression	6	4763462	793910.3		2.8E-134
Residual		218088.4
Total	208	4981550
	Coefficients	Standard Error	t Stat	P-value	Lower 95%	Upper 95%
Intercept	-21.4174	3.613163	-5.92761	1.31E-08	-28.5418	-14.2931
MMIN	0.004862	0.00115	4.227276	3.58E-05	0.002594	0.007129
MMAX	0.003048	0.000403	7.560398	1.38E-12	0.002253	0.003842
CACH	0.0781	0.079442		0.326731	-0.07854	0.234743
CHMIN	-0.09145	0.468396	-0.19523	0.845408	-1.01502	0.832126
CHMAX	0.292503	0.132665	2.20483	0.028595	0.030918	0.554088
PRP	0.605678	0.037818	16.01576	8.28E-38	0.53111	0.680246

What is the value of F*?

Answer 1

1)How much of the variability in strength is explained by the predictors?

R Square

0.956221

R-squared is the coefficient of determination. It tells us how much of the variability in strength is explained by the predictors.

We have 95.62%variability in strength explained by the predictors.

DF Regression = number of predictors.

DF Residual= n - (DF Regression) - 1

DF Total = n-1

ANOVA
	df	SS	MS	F	Significance F
Regression	6	4763462	793910.3	=793910.3/1085.016915 =731.7031551	2.8E-134
Residual	=208-6-1 =201	218088.4	=218088.4/201 =1085.016915
Total	208	4981550

F*=731.7031551

The p-value is < .00001. The result is significant at p < .05.