The article "The Undrained Strength of Some Thawed Permafrost Soils"† contained the accompanying data on the following. y = shear strength of sandy soil (kPa) x1 = depth (m) x2 = water content (%) The predicted values and residuals were computed using the estimated regression equation ŷ = −145.41 − 14.24x1 + 12.70x2 + 0.079x3 − 0.236x4 + 0.441x5 where x3 = x12, x4 = x22, and x5 = x1x2. y x1 x2 Predicted y Residual 14.7 9.0 31.6 23.83 −9.13 48.0 36.5 27.1 47.07 0.93 25.6 36.7 25.8 26.46 −0.86 10.0 6.0 39.2 10.77 −0.77 16.0 7.0 39.3 14.57 1.43 16.8 7.0 38.4 16.88 −0.08 20.7 7.4 34.0 23.38 −2.68 38.8 8.3 33.7 25.07 13.73 16.9 6.4 28.0 16.23 0.67 27.0 8.1 33.0 24.31 2.69 16.0 4.6 26.4 15.06 0.94 24.9 9.8 37.9 28.64 −3.74 7.3 2.8 34.5 15.08 −7.78 12.8 1.9 36.3 8.15 4.65 (a) Use the given information to calculate SSResid, SSTo, and SSRegr. (Round your answers to four decimal places.) SSTo=SSResid=SSRegr= (b) Calculate R2 for this regression model. (Round your answer to three decimal places.) R2 = How would you interpret this value? The value R2 gives the percentage of observed variation in water content that can be explained by the fitted model.The value R2 gives the percentage of observed variation in shear strength of sandy soil that can be explained by the fitted model. The value R2 gives the percentage of water content values in the sample that are equal to the values predicted by the model.The value R2 gives the percentage of shear strength of sandy soil values in the sample that are equal to the values predicted by the model. (c) Use the value of R2 from part (b) and a 0.05 level of significance to carry out a model utility F test. Calculate the test statistic. (Round your answer to two decimal places.) F = Use technology to calculate the P-value. (Round your answer to four decimal places.) P-value = What can you conclude? Fail to reject H0. We have convincing evidence that the multiple regression model is useful.Reject H0. We do not have convincing evidence that the multiple regression model is useful. Reject H0. We have convincing evidence that the multiple regression model is useful.Fail to reject H0. We do not have convincing evidence that the multiple regression model is useful.

Question

The article "The Undrained Strength of Some Thawed Permafrost Soils"† contained the accompanying data on the following.

y = shear strength of sandy soil (kPa)
x₁ = depth (m)
x₂ = water content (%)

The predicted values and residuals were computed using the estimated regression equation

ŷ = −145.41 − 14.24x₁ + 12.70x₂ + 0.079x₃ − 0.236x₄ + 0.441x₅

where x₃ = x₁², x₄ = x₂², and x₅ = x₁x₂.

y	x₁	x₂	Predicted y	Residual
14.7	9.0	31.6	23.83	−9.13
48.0	36.5	27.1	47.07	0.93
25.6	36.7	25.8	26.46	−0.86
10.0	6.0	39.2	10.77	−0.77
16.0	7.0	39.3	14.57	1.43
16.8	7.0	38.4	16.88	−0.08
20.7	7.4	34.0	23.38	−2.68
38.8	8.3	33.7	25.07	13.73
16.9	6.4	28.0	16.23	0.67
27.0	8.1	33.0	24.31	2.69
16.0	4.6	26.4	15.06	0.94
24.9	9.8	37.9	28.64	−3.74
7.3	2.8	34.5	15.08	−7.78
12.8	1.9	36.3	8.15	4.65

(a)

Use the given information to calculate SSResid, SSTo, and SSRegr. (Round your answers to four decimal places.)

SSTo=SSResid=SSRegr=

(b)

Calculate R² for this regression model. (Round your answer to three decimal places.)

R² =

How would you interpret this value?

The value R² gives the percentage of observed variation in water content that can be explained by the fitted model.The value R² gives the percentage of observed variation in shear strength of sandy soil that can be explained by the fitted model. The value R² gives the percentage of water content values in the sample that are equal to the values predicted by the model.The value R² gives the percentage of shear strength of sandy soil values in the sample that are equal to the values predicted by the model.

(c)

Use the value of R² from part (b) and a 0.05 level of significance to carry out a model utility F test.

Calculate the test statistic. (Round your answer to two decimal places.)

F =

Use technology to calculate the P-value. (Round your answer to four decimal places.)

P-value =

What can you conclude?

Fail to reject H₀. We have convincing evidence that the multiple regression model is useful.Reject H₀. We do not have convincing evidence that the multiple regression model is useful. Reject H₀. We have convincing evidence that the multiple regression model is useful.Fail to reject H₀. We do not have convincing evidence that the multiple regression model is useful.