Question

United Oil Company is attempting to develop a reasonably priced unleaded gasoline that will deliver higher gasoline mileages than can be achieved by its current unleaded gasolines. As part of its development process, United Oil wishes to study the effect of two independent variables—x₁, amount of gasoline additive RST (0, 1, or 2 units), and x₂, amount of gasoline additive XST (0, 1, 2, or 3 units), on gasoline mileage, y. Mileage tests are carried out using equipment that simulates driving under prescribed conditions. The combinations of x₁ and x₂ used in the experiment, along with the corresponding values of y, are given below.

RST Units (x1)	XST Units (x2)	Gas Mileage (y, mpg)
0	0	27.99
0	0	28.82
0	0	28.28
1	0	29.54
1	0	30.72
2	0	28.52
2	0	29.29
0	1	32.47
0	1	33.74
1	1	33.29
1	1	34.91
0	2	32.54
0	2	33.20
1	2	34.36
1	2	35.86
1	2	35.41
2	2	33.75
2	2	34.09
2	2	34.72
1	3	33.94
2	3	33.20
2	3	33.73

Using the model, y = β₀ + β₁x₁ + β₂x₁² + β₃x₂ + β₄x₂² + ε, calculate the point estimate. (Moreover, consider the mean mileage obtained by all gallons of the gasoline when it is made with one unit of RST and two units of XST (a combination that the data on the page margin indicates would maximize mean mileage). Do not round intermediate calculations. Round your answer to 4 decimal places.)

Answer 1

————— 2/20/2020 9:50:19 PM ————————————————————

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Probability PLOT

From the above graph data are normally distributed.

Regression Analysis: gas mil. versus x1, x2, x12, x22

Analysis of Variance

Source DF Seq SS Contribution Adj SS Adj MS F-Value P-Value

Regression 4 118.075 88.02% 118.075 29.5188 31.23 0.000

x1 1 7.563 5.64% 0.373 0.3727 0.39 0.538

x2 1 80.378 59.92% 65.798 65.7983 69.60 0.000

x12 1 5.264 3.92% 0.385 0.3855 0.41 0.532

x22 1 24.870 18.54% 24.870 24.8704 26.31 0.000

Error 17 16.070 11.98% 16.070 0.9453

Lack-of-Fit 4 9.404 7.01% 9.404 2.3511 4.59 0.016

Pure Error 13 6.666 4.97% 6.666 0.5128

Total 21 134.145 100.00%

Model Summary

S R-sq R-sq(adj) PRESS R-sq(pred)

0.972272 88.02% 85.20% 25.8655 80.72%

Point estimates

Coefficients

Term Coef SE Coef 95% CI T-Value P-Value VIF

Constant 28.894 0.496 (27.847, 29.941) 58.22 0.000

x1 0.262 0.418 (-0.619, 1.143) 0.63 0.538 2.58

x2 5.526 0.662 ( 4.129, 6.924) 8.34 0.000 10.38

x12 0.193 0.302 (-0.445, 0.831) 0.64 0.532 8.27

x22 -1.559 0.304 (-2.200, -0.918) -5.13 0.000 15.34

Regression Equation

gas mil. = 28.894 + 0.262 x1 + 5.526 x2 + 0.193 x12 - 1.559 x22

Fits and Diagnostics for All Observations

Obs gas mil. Fit SE Fit 95% CI Resid Std Resid Del Resid HI

1 27.990 28.894 0.496 (27.847, 29.941) -0.904 -1.08 -1.09 0.260579

2 28.820 28.894 0.496 (27.847, 29.941) -0.074 -0.09 -0.09 0.260579

3 28.280 28.894 0.496 (27.847, 29.941) -0.614 -0.73 -0.72 0.260579

4 29.540 29.157 0.369 (28.378, 29.935) 0.383 0.43 0.42 0.143899

5 30.720 29.157 0.369 (28.378, 29.935) 1.563 1.74 1.86 0.143899

6 28.520 29.419 0.612 (28.128, 30.710) -0.899 -1.19 -1.21 0.396313

7 29.290 29.419 0.612 (28.128, 30.710) -0.129 -0.17 -0.17 0.396313

8 32.470 32.862 0.376 (32.067, 33.656) -0.392 -0.44 -0.43 0.149942

9 33.710 32.862 0.376 (32.067, 33.656) 0.848 0.95 0.94 0.149942

10 33.290 33.317 0.346 (32.587, 34.047) -0.027 -0.03 -0.03 0.126697

11 34.910 33.317 0.346 (32.587, 34.047) 1.593 1.75 1.88 0.126697

12 32.540 33.712 0.566 (32.517, 34.906) -1.172 -1.48 -1.54 0.339116

13 33.200 33.712 0.566 (32.517, 34.906) -0.512 -0.65 -0.64 0.339116

14 34.360 34.360 0.289 (33.750, 34.971) -0.000 -0.00 -0.00 0.088517

15 35.860 34.360 0.289 (33.750, 34.971) 1.500 1.62 1.70 0.088517

16 35.410 34.360 0.289 (33.750, 34.971) 1.050 1.13 1.14 0.088517

17 33.750 35.009 0.460 (34.038, 35.980) -1.259 -1.47 -1.53 0.223997

18 34.090 35.009 0.460 (34.038, 35.980) -0.919 -1.07 -1.08 0.223997

19 34.720 35.009 0.460 (34.038, 35.980) -0.289 -0.34 -0.33 0.223997

20 33.940 34.360 0.289 (33.750, 34.971) -0.420 -0.45 -0.44 0.088517

21 33.200 33.128 0.645 (31.767, 34.489) 0.072 0.10 0.10 0.440136

22 33.730 33.128 0.645 (31.767, 34.489) 0.602 0.83 0.82 0.440136

Obs Cook’s D DFITS

1 0.08 -0.64557

2 0.00 -0.05122

3 0.04 -0.43008

4 0.01 0.17042

5 0.10 0.76224

6 0.19 -0.97689

7 0.00 -0.13423

8 0.01 -0.17908

9 0.03 0.39612

10 0.00 -0.01104

11 0.09 0.71571

12 0.23 -1.10424

13 0.04 -0.45575

14 0.00 -0.00013

15 0.05 0.53086

16 0.02 0.35548

17 0.12 -0.81999

18 0.07 -0.57915

19 0.01 -0.17640

20 0.00 -0.13776

21 0.00 0.08519

22 0.11 0.72661

Residual Plots for gas mil.

Graph shows vs fit is some pattern so model may not be appropriate.

Probability PLOT Probability Plot of gas mil. Normal - 95% CI Mean 32.38 St Dev 2.527 22 AD 1.093 P-Value 0.006 Percent 25.0 27.5 30.0 35.0 37.5 40.0 32.5 gas mil. data are normally distributed From the above graph data are normally distributed.

Residual Plots for gas mil. Residual Plots for gas mil. Normal Probability Plot Versus Fits Percent Residual 31.5 330 Fitted Value Residual Histogram Versus Order Frequency Residual . 1.5 .10 .0.5 1.0 15 4 6 18 20 22 0.0 0.5 Residual 8 10 12 14 16 Observation Order Graph shows vs fit is some pattern so model may not be appropriate.