United Oil Company is attempting to develop a reasonably priced unleaded gasolin
ID: 3157871 • Letter: U
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.
Using the model, y = 0 + 1x1 + 2x12 + 3x2 + 4x22 + , 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.)
RST
Units
(x1) XST
Units
(x2) Gas Mileage
(y, mpg) 0 0 27.19 0 0 28.24 0 0 28.78 1 0 29.62 1 0 30.27 2 0 28.40 2 0 29.69 0 1 32.52 0 1 33.31 1 1 33.36 1 1 34.33 0 2 32.21 0 2 33.94 1 2 34.84 1 2 35.98 1 2 35.17 2 2 33.34 2 2 34.03 2 2 34.27 1 3 33.70 2 3 33.29 2 3 33.35
Explanation / Answer
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.
Regression Analysis
R²
0.941
Adjusted R²
0.927
n
22
R
0.970
k
4
Std. Error
0.689
Dep. Var.
y
ANOVA table
Source
SS
df
MS
F
p-value
Regression
129.4502
4
32.3626
68.14
3.08E-10
Residual
8.0739
17
0.4749
Total
137.5242
21
Regression output
confidence interval
variables
coefficients
std. error
t (df=17)
p-value
95% lower
95% upper
Intercept
28.2730
0.3172
89.147
3.80E-24
27.6039
28.9421
x1
3.0537
0.6444
4.739
.0002
1.6940
4.4133
x1^2
-1.3570
0.3077
-4.410
.0004
-2.0062
-0.7077
x2
5.1891
0.4513
11.499
1.93E-09
4.2370
6.1412
x2^2
-1.2856
0.1649
-7.797
5.17E-07
-1.6334
-0.9377
Predicted values for: y
95% Confidence Interval
95% Prediction Interval
x1
x1^2
x2
x2^2
Predicted
lower
upper
lower
upper
Leverage
1
1
2
4
35.20568
34.63040
35.78097
33.64202
36.76935
0.157
The regression line is
mean mileage =28.2730+3.0537*RST-1.3570*RST2+ 5.1891*XST -1.2856*XST2
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.)
when RST =1 and XST=2,
mean mileage =28.2730+3.0537*1-1.3570*12+ 5.1891*2 -1.2856*22
= 35.2057
Point estimate=35.2057
Regression Analysis
R²
0.941
Adjusted R²
0.927
n
22
R
0.970
k
4
Std. Error
0.689
Dep. Var.
y
ANOVA table
Source
SS
df
MS
F
p-value
Regression
129.4502
4
32.3626
68.14
3.08E-10
Residual
8.0739
17
0.4749
Total
137.5242
21
Regression output
confidence interval
variables
coefficients
std. error
t (df=17)
p-value
95% lower
95% upper
Intercept
28.2730
0.3172
89.147
3.80E-24
27.6039
28.9421
x1
3.0537
0.6444
4.739
.0002
1.6940
4.4133
x1^2
-1.3570
0.3077
-4.410
.0004
-2.0062
-0.7077
x2
5.1891
0.4513
11.499
1.93E-09
4.2370
6.1412
x2^2
-1.2856
0.1649
-7.797
5.17E-07
-1.6334
-0.9377
Predicted values for: y
95% Confidence Interval
95% Prediction Interval
x1
x1^2
x2
x2^2
Predicted
lower
upper
lower
upper
Leverage
1
1
2
4
35.20568
34.63040
35.78097
33.64202
36.76935
0.157
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