Stat 328 Exam 1
Summer 2004
Prof. Vardeman
The book Statistical Thinking for Managers by John, Whitaker, and Johnson describes a case in which a
company was considering modifying its large trucks by the installation of (somewhat expensive) airfoils
intended to improve fuel efficiency. A single truck from the company’s fleet made a daily run (of
between 300 and 400 km) non = 86 times with an airfoil attached and noff = 82 times without the airfoil
over a period of about 5 months. (Some “off” runs were made, followed by some “on” runs, followed by
some “off” runs, etc. so that weather effects, wear on the truck, etc. may be assumed to have equally
affected mileages under the two configurations.) Attached to this exam are two JMP reports providing
summary statistics for daily values of y = km traveled per liter of gas consumed in this study.
a) JMP uses different conventions for quartiles than does the textbook. Find the 3rd quartile of the “On”
data set according the text’s convention. (Note that the raw data are all available in stem and leaf form.)
Q3 = _____________________
b) Make below a boxplot for the “On” data using the JMP quartiles and the scale provided. (Write values
of the “5 numbers” above appropriate features of the plot.)
c) 1 km/l is 2.53 mpg . If the values read into JMP had been in miles per gallon instead of kilometers per
liter, what would JMP have reported for the sample mean and sample standard deviation of airfoil “On”
fuel efficiencies?
mean = ____________ mpg
standard deviation = ____________ mpg
The stem-and-leaf plot, histogram, and boxplot for the “On” data perhaps suggest that a normal model
might be used for km traveled per liter on days the airfoil is attached. In parts d) and e) on the next
page, suppose that one models daily efficiencies for this truck with airfoil attached as normal with
µ = 2.00 km/l and σ = .15 km/l .
1
d) On about what fraction of days will this truck have a fuel efficiency below 1.90 km/l ? (Show your
work.)
e) If the truck makes 5 runs in a given work week, what is the probability that the weekly average of
efficiencies is below 1.90 km/l ? (Show your work.)
Now drop the µ = 2.00 and σ = .15 assumptions and return to data analysis on the basis of the JMP
reports.
f) Give limits that you are “90% sure” would contain 1 more daily fuel efficiency for this truck with the
airfoil “On.” (Plug in, but you need not simplify.)
g) Use the text’s default method and give limits that you are “90% sure” contain the difference in mean
daily fuel efficiencies with the airfoil “On” and with the airfoil “Off” for this truck (say µon − µoff ).
(Again, plug in, but you need not simplify.)
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h) Is there a statistically detectable increase in efficiency of this truck provided by the airfoil? (Produce
and interpret an appropriate p-value using the book’s default method.)
i) There is a fairly serious flaw in the whole plan of the fuel efficiency study, at least if it was meant to
provide information on effect of airfoil installation across the company’s entire large fleet. This is in spite
of the fact that apparently a very large number of data values were collected. Discuss this matter in 100
carefully chosen words or less. You might, for example, want to think about the concept of “pairing,” the
matter of how big the “sample size” is in terms number of trucks studied, and what the difference
µon − µoff estimated in g) and tested in part h) represents in practical terms.
3
Distributions
On
Normal(1.89895,0.14719)
1.4 1.5 1.6 1.7 1.8 1.9 2 2.1 2.2 2.3 2.4
Off
1.3 1.41.5 1.61.7 1.81.9 2 2.1 2.22.3 2.42.5
Normal(1.8778,0.15483)
Quantiles
100.0% maximum
99.5%
97.5%
90.0%
quartile
75.0%
median
50.0%
quartile
25.0%
10.0%
2.5%
0.5%
minimum
0.0%
Quantiles
100.0% maximum
99.5%
97.5%
90.0%
quartile
75.0%
median
50.0%
quartile
25.0%
10.0%
2.5%
0.5%
minimum
0.0%
2.3000
2.3000
2.2578
2.0830
1.9625
1.9000
1.8100
1.7170
1.5558
1.4800
1.4800
2.4600
2.4600
2.1185
2.0570
1.9700
1.9000
1.7575
1.7060
1.5190
1.3100
1.3100
Moments
Mean
Std Dev
Std Err Mean
upper 95% Mean
lower 95% Mean
N
Moments
Mean
Std Dev
Std Err Mean
upper 95% Mean
lower 95% Mean
N
1.8989535
0.1471936
0.0158723
1.9305119
1.8673951
86
1.8778049
0.1548318
0.0170983
1.9118252
1.8437846
82
Leaf
0
07
0588
0113457789
0001112223333344555566667799
00112444555566677888999
12356677889
33669
4
8
Stem and Leaf
Stem
23
22
21
20
19
18
17
16
15
14
14|8 represents 1.48
02
0112234567788
00002222233334555666777789
112344445567889
0022334445555566889
3589
1
Leaf
6
1
2
13
26
15
19
4
1
Count
1
Count
1
2
4
10
28
23
11
5
1
1
1
Stem and Leaf
Stem
24
23
22
21
20
19
18
17
16
15
14
13
13|1 represents 1.31