Download ISE 362 HOMEWORK FIVE Due Date: Thursday 11/22/2016 1

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Due Date: Thursday 11/22/2016
1. Suppose that in a certain chemical process the reaction time y (hour) is related to the
temperature (oF) in the chamber in which the reaction takes place according to the simple linear
regression model with equation y = 5.00 – 0.01(x) and σ = 0.50. Suppose three observations are
made independently on reaction time, each one for a temperature of 250OF. What is the
probability that all three times are between 2.28 and 2.72 hours?
Ans:
2. Engineers frequently use models in problem formulation and solution. Sometimes these
models are based on our physical, chemical, or engineering science knowledge of the
phenomenon, and in such cases we call these models deterministic models. However, there are
“many” situations in which two or more variables of interest are related, and the deterministic
model relating these variables is unknown. In these cases it is necessary to build a nondeterministic model relating the variables based on observed data. Consider the situation where
y is the salt concentration (mg/l) found in surface streams in an upstate watershed and x is the
percentage of the watershed area consisting of paved roads. There is no obvious physical
mechanism that relates the salt concentration to the roadway area. Suppose that the relationship
between salt concentration and roadway area can be described by the simple linear regression
model with the true regression line y = 3 + 15(x) and with σ2 = 4.0. What is the probability that
salt concentration exceeds 32.56 when roadway area is 2.0?
Ans:
3. A scatter plot of x = rainfall volume (m3) & y = runoff volume (m3) for a particular location
are shown below. The accompanying values were read from the plot. Calculate point estimates
of the slope and intercept of the true regression line. Calculate a point estimate of the true
average runoff volume when rainfall volume is 50. Calculate a point estimate of the standard
deviation σ. Also, find the proportion of the observed variation in runoff volume can be
attributed to the simple linear regression relationship between runoff and rainfall.
x
y
5
4
x
y
55
38
12
10
14
13
67
46
17
15
72
53
23
15
81
70
30
25
96
82
112
99
Runoff volume
100
90
80
70
60
50
40
30
20
10
0
0
25
50
75
100
Rainfall volume
Point Estimates of β1 and β0:
Point Estimate of μy when x = 50:
Point Estimate of σ:
Proportion:
125
150
40
27
127
100
47
46
4. The accompanying data was read from the graph below that appeared in a recent study. The
independent variable is SO2 deposition rate (mg/m2/day) and the dependent variable is steel
weight loss (g/m2). Calculate the equation of the estimated regression line. Also; what
percentage of observed variation in steel weight loss can be attributed to the model relationship
in combination with variation in deposition rate?
x
13
19
40
44
46
114
y
280
350
470
500
560
1200
100
120
1200
1000
y
800
600
400
200
0
20
40
60
80
x
Ans:
5. The following summary statistics were obtained from a study that used regression analysis to
investigate the relationship between pavement deflection and surface temperature of the
pavement at various locations on a state highway. Here x = temperature (oF) and y = deflection
adjustment factor (y ≥ 0):
n = 15 Σ xi = 1,425 Σ yi = 10.68 Σ xi2 = 139,037.25 Σ xiyi = 987.645
Compute β1, β2 & the equation of the estimated regression line. Also, suppose temperature were
measured in oC rather than in oF. What would be the new estimated regression line?
(Hint: oF = (9/5) oC + 32; now substitute for the “old x” in terms of the new x.”)
Ans:
6. Infestation of crops by insects has long been of great concern to farmers and agricultural
scientists. A study reports data on x = age of a cotton plant (days) and y = percentage of
damaged squares. Consider the accompanying n = 12 observations:
x
9
12
12
15
18
18
y
11
12
23
30
29
52
x
21
21
27
30
30
33
y
41
65
60
72
84
93
The summary statistics are: Σ xi = 246, Σ xi2 = 5,742, Σ yi = 572, Σ yi2 = 35,634, Σ xiyi =14,022.
Why is the relationship between x and y not deterministic? Also, determine the equation of
the least squares line.
ANS:
7. The Turbine Oil Oxidation Test (TOST) and the Rotating Bomb Oxidation Test (RBOT) are
two different procedures for evaluating the oxidation stability of steam turbine oils. The
accompanying observations on x = TOST time (hr) & y = RBOT time (min) for 12 oil
specimens have been reported:
TOST
4200
3600
3750
3675
4050
2770
RBOT
370
340
375
310
350
200
TOST
4870
4500
3450
2700
3750
3300
RBOT
400
375
285
225
345
285
Summary Values: n = 12, Σ xi =44,615, Σ xi2 = 170,355,425, Σ yi = 3,860, Σ yi2 = 1,284,450
Σ xiyi =14,755,500.
Calculate value of the sample correlation coefficient r. Would the value of r be affected if
you let x = RBOT time and y = TOST time? (Why)?
Ans:
8. Regression analysis can be used to build a model to predict the purity of oxygen produced
in a chemical distillation process for a given percentage of hydrocarbons that are present in the
main condenser of the distillation unit. This regression model can be used for process
optimization, such as finding the percentage of hydrocarbons that maximizes purity, or for
process control purposes. The table below represents the data for 20 observations from this
type of chemical distillation process. Fit a simple linear regression model to the oxygen purity
data. Provide an estimate for the variance of the error term є.
Observation #
Hydrocarbon Level(%)
Purity(%)
1
0.99
90.00
2
1.02
89.00
3
1.15
91.40
4
1.19
94.00
5
1.30
96.00
6
1.25
95.00
7
0.83
87.00
8
1.20
92.00
9
1.50
99.40
10
1.30
93.60
11
1.20
93.40
12
1.10
92.40
13
0.96
90.60
14
1.00
89.50
15
1.10
89.90
16
1.20
89.10
17
1.10
93.60
18
1.30
92.00
19
1.40
94.80
20
0.91
87.30
Ans:
THE END