We are interested in determining what type of model best describes the relationship between two variables x and y.

a.For a given data set, an estimated regression equation relating x and y of the form
 = b0 + b1x was developed using Excel. The results are shown below. Comment on the adequacy of this equation for predicting y. Let ? = .01.
?
SUMMARY OUTPUT

Regression Statistics
Multiple R0.5095
R Square0.2596
Adjusted R Square0.1362
Standard Error2.0745
Observations8
?
ANOVA     ?
df
SS
MS
F
Significance F
Regression1
9.0536
9.0536
2.1037
0.1971
Residual6
25.8214
4.3036
  Total7
34.875
         ?
Coefficients
Standard Error
t Stat
P-value
 Intercept2.7857
1.6164
1.7234
0.1356
 x0.4643
0.3201
1.4504
0.1971
 ?
b.An estimated regression equation for the same data set (as in part a) of the form
 = b0 + b1x + b2x2 was developed. The Excel output is shown below. Comment on the adequacy of this equation for predicting y. Let ? = .01.
?
SUMMARY OUTPUT
           Regression Statistics
    Multiple R0.9680
    R Square0.9370
    Adjusted R Square0.9118
    Standard Error0.6628
    Observations8
          ANOVA     ?
df
SS
MS
F
Significance F
Regression2
32.6786
16.3392
37.1951
0.0010
Residual5
2.1964
0.4393
  Total7
34.875
         ?
Coefficients
Standard Error
t Stat
P-value
 Intercept-2.8393
0.9247
-3.0706
0.0278
 x3.8393
0.4714
8.1437
0.0005
 x-squared-0.375
0.0511
-7.3335
0.0007
 ?
c.Use the results of Part b and predict y when x = 4.

What will be an ideal response?

a.

 = 2.7857 + .4643x

r2 = .2596; adjusted coefficient of determination is .1362; only 13.62% of the variation is explained.
p-value = .1971; no evidence of any significant relationship. The model is not adequate for predicting y.

  
b.

 = -2.8393 + 3.8393x - .375x2
R2 = 93.7%;  = 91.18%, which means 91.18% of the variation in y is explained by both x and x2.

Both x and x2 are significant; both p-values < .01.

The p-value for the analysis of variance is .001, which is less than .01. Therefore, the model is adequate for predicting y.

  
c.6.518



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