4.32 Cooling method for gas turbines. Refer to the Journal of Engineering for Gas Turbines and
Power (January 2005) study of a high-pressure inlet fogging method for a gas turbine engine,
Exercise 4.13 (p.188). Recall that you fit a first-order model for heat rate (y) as a function of
speed(𝑥1 ), inlet temperature(𝑥2 ), exhaust temperature(𝑥3 ), cycle pressure ratio(𝑥4 ), and air flow
rate(𝑥5 ), to data saved in the GASTURBINE file.
a.
Researchers hypothesize that the linear relationship between heat rate (y) and
temperature (both inlet and exhaust) depends on air flow rate. Write a model for heat
rate that incorporates the researchers’ theories.
b.
Use statistical software to fit the interaction model, part a, to the data in the
GASTURBINE file. Give the least squares prediction equation.
c.
Conduct a test (at α=.05) to determine whether inlet temperature and air flow rate
interact to effect heat rate.
d.
Conduct a test (at α=.05) to determine whether exhaust temperature and air flow rate
interact to effect heat rate.
e.
Practically interpret the results of the tests, parts c and d.
4.36 Cars with catalytic converters. A quadratic model was applied to motor vehicle toxic
emissions data collected over 15 recent ears in Mexico City (Environment Science and
Engineering, September 1, 2000). The following equation was used to predict the percentage (y)
of motor vehicles without catalytic converters in the Mexico City fleet for a given year (𝑥): 𝑦 =
325,790 − 321.67𝑥 + 0.794𝑥 2 .
a.
Explain why the value 𝛽0 = 325,790 has no practical interpretation.
b.
Explain why the value 𝛽1 = 321.67 should not be interpreted as a slope.
c.
Examine the value of 𝛽2 to determine the nature of the curvature (upward or downward)
in the sample data.
d.
The researchers used the model to estimate “that just after the year 2021 the fleet of cars
with catalytic converters will completely disappear.” Comment on the danger of using the model
to predict y in the year 2021.
4.40 Failure times of silicon wafer microchips. Researchers at National Semiconductor
experimented with tin-lead solder bumps used to manufacture silicon wafer integrated circuit chips
(International Wafer Level Packaging Conference, November 3-4, 2005). The failure times of the
microchips (in hours) were determined at different solder temperatures (degrees Centigrade). The
data for one experiment are given in the next table (p.208). The researchers want to predict failure
time (y) based on solder temperature (x).
a) Construct a scatterplot for the data. What type of relationship, linear or curvilinear, appears to
exist between failuretime and solder temperature?
Scatterplot of TEMP vs FAILTIME
170
160
TEMP
150
140
130
120
110
0
2000
4000
6000
8000
10000
FAILTIME
b) Fit the model (𝑦) = 𝛽0 + 𝛽1 𝑥 + 𝛽2 𝑥 2 , to the data. Give the least squares prediction equation.
S = 5.26788
R-Sq = 84.3%
R-Sq(adj) = 83.5%
Analysis of Variance
Source
Regression
Error
Total
DF
1
20
21
SS
2978.26
555.01
3533.27
MS
2978.26
27.75
F
107.32
P
0.000
Fitted Line Plot
TEMP = 158.7 - 0.004400 FAILTIME
170
S
R-Sq
R-Sq(adj)
160
5.26788
84.3%
83.5%
TEMP
150
140
130
120
110
0
2000
4000
6000
8000
10000
FAILTIME
Fitted Line: TEMP versus FAILTIME
c) Conduct a test to determine if there is upward curvature in the relationship between failuretime
and solder temperature. (Use α=.05)
The prediction is less robust to departures from normality. The distinction between variability
and uncertainty is useful in understanding the distinction between confidence intervals for the
conditional mean and prediction intervals: The confidence interval for the conditional mean
measures our degree of uncertainty in our estimate of the conditional mean; but the prediction
interval must also take into account the variability in the conditional distribution.
4.64 Cooling method for gas turbines. Refer to the Journal of Engineering for Gas Turbines and
Power (January 2005) study of a high-pressure inlet fogging method for a gas turbine engine,
Exercise 4.13 (p.188). Consider a model for heat rate (kilojoules per kilowatt per hour) of a gas
turbine as a function of cycle speed (revolutions per minute) and cycle pressure ratio. The data
are saved in the GASTURBINE file.
a) Write a complete second-order model for heat rate (y).
b) Give the null and alternative hypothesis for determining whether the curvature
terms in the complete second-order model are statistically useful for predicting heat
rate (y).
c) For the test in part b, identify the “complete” and “reduced” model.
d) Portions of the MINITAB printouts for the two models are shown below. Find the
values of 𝑆𝑆𝐸𝑅 , 𝑆𝑆𝐸𝐶 , 𝑎𝑛𝑑 𝑀𝑆𝐸𝐶 on the printouts.
e) Compute the value of the test statistics for the test of part b.
f) Find the rejection region for the test of part b using α=.10.
g) Stat the conclusion of the test in the words of the problem.