A researcher carried out a series of experiments to gather information on the co

Question

A researcher carried out a series of experiments to gather information on the coefficient of permeability of concrete. In one experiment, the outflow of water from the pores of a concrete specimen after it had been under saturating water pressure for a period of time was recorded for different combinations of concrete permeability and porosity. The multiple regression should be based upon a linear relationship with both variables entered simultaneously. The data for the experiment is found in the following table. (15 points total)

Find the least squares prediction equation. (1 point)

Interpret the coefficient estimates. (6 points)

Conduct a test of overall model utility at alpha equal to 0.05 and interpret the results in one sentence. (1 point)

Is there evidence that concrete porosity is a useful predictor of coefficient of permeability for an alpha of 0.05? Explain in one sentence. (1 point)

Is there evidence that the estimated water outflow-time slope is a useful predictor of coefficient of permeability for an alpha of 0.05? Explain in one sentence. (1 point)

Find r2 and interpret its value. (2 points)

Find the estimate of the standard deviation for the coefficient of permeability. (1 point)

Find a 95% prediction interval for y when porosity is 0.05 and water outflow-time is 0.30. (2 points)

Coefficient of permeability of concrete Coefficient of Permeability Porosity Water Outflow–Time Slope Coefficient 1.00 0.05 0.903 1.00 0.035 0.722 1.00 0.025 0.590 0.10 0.05 0.345 0.10 0.035 0.282 0.10 0.025 0.233 0.01 0.05 0.103 0.01 0.035 0.091 0.01 0.025 0.078

Explanation / Answer

Find the least squares prediction equation. (1 point)

y=0.132-9.307X1+1.558X2

Interpret the coefficient estimates. (6 points)

this is the change in y due to unit change in x if other factors are constant.

Conduct a test of overall model utility at alpha equal to 0.05 and interpret the results in one sentence. (1 point)

P-value=0.0005 is less than alpha=0.05, so model is significant

Is there evidence that concrete porosity is a useful predictor of coefficient of permeability for an alpha of 0.05? Explain in one sentence. (1 point)

P-value =0.115 of X1 concrete porosity is more than alpha=0.05, so it is not significant

Is there evidence that the estimated water outflow-time slope is a useful predictor of coefficient of permeability for an alpha of 0.05? Explain in one sentence. (1 point)

P-value =0.0001 of X2 water outflow-time slope is less than alpha=0.05, so it is significant

Find r2 and interpret its value. (2 points)

0.9227, 92% of variation is explained by the regression model

Find the estimate of the standard deviation for the coefficient of permeability. (1 point)

sqrt(MS residual)=sqrt(0.0231)=0.152

Find a 95% prediction interval for y when porosity is 0.05 and water outflow-time is 0.30. (2 points)

estimated y=0.132-9.307*0.05+1.5576*0.3=0.1339

following information has been generated using ms-excel

Coefficient of Permeability (y) Porosity (x1) Water Outflow–Time Slope Coefficient (x2) Regression Statistics 1 0.05 0.903 Multiple R 0.960607335 1 0.035 0.722 R Square 0.922766452 1 0.025 0.59 Adjusted R Square 0.897021936 0.1 0.05 0.345 Standard Error 0.152141034 0.1 0.035 0.282 Observations 9 0.1 0.025 0.233 0.01 0.05 0.103 ANOVA 0.01 0.035 0.091 df SS MS F Significance F 0.01 0.025 0.078 Regression 2 1.659318634 0.829659317 35.84322401 0.000461 Residual 6 0.138881366 0.023146894 Total 8 1.7982 Coefficients Standard Error t Stat P-value Lower 95% Upper 95% Intercept 0.132020763 0.190051297 0.694658571 0.51325603 -0.33302 0.597059534 X Variable 1 -9.30712231 5.057025288 -1.840434204 0.115308749 -21.6812 3.066972774 X Variable 2 1.557563041 0.183961568 8.466784987 0.000148343 1.107425 2.00770078

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