STATISTICS 401
Sample Examination 2 (100 points)
NOTE: Please show all work to obtain full credit.
1. Trace metals in drinking water affect the flavor and an unusually high concentration can pose a health
hazard. Ten pairs of data were taken measuring zinc concentration in bottom water and surface water
from various locations of a stream. Do the data suggest that the true average concentration of zinc in the
bottom water exceeds that of surface water? The data are given below aggregated:
Location
1
2
3
4
5
6
7
8
9
10
Bottom
.430
.266
.567
.531
.707
.716
.651
.589
.469
.723
Surface
.415
.238
.390
.610
.605
.663
.632
.523
.411
.612
Difference d
.015
.028
.177
-.079
.102
.053
.019
.066
.058
.111
P
d = .55
P 2
d = .072194
Let µd = µbottom − µsurf ace be the mean difference in zinc concentration of the two populations and
assume that the differences di have an approximately Normal distribution.
(a) (10) Compute a t-statistic to test the research hypothesis that the mean zinc concentration in
the bottom water exceeds that of surface water. State the null and alternative hypothesis
in terms of µd . Compute an approximate p-value and make the decision using α = 0.05.
1
(b) (10) Construct a 90% confidence interval for µd . Use this interval to test the hypothesis in part
(a), explaining how you arrived at your conclusion. What is the α-level of this test?
2. Many people purchase sports utility vehicles (SUV’s) because they think they are sturdier and hence safer
than regular cars. However, data have indicated that the costs of repairs for SUVs are higher than for
midsize cars when both vehicles are in an accident. To verify this observation, a random sample of 9 new
SUVs and 8 midsize cars are tested for front impact resistance. The amounts of damage (in hundreds of
dollars) to the vehicles when crashed at 20 mph head-on into a stationary barrier are recorded below:
Vehicle
SUV
Midsize
14.25
11.95
23.47
15.42
19.15
14.27
Damage (in $100’s)
16.17 27.38 17.37
11.42 18.12 10.36
-1.28
-0.67
24.64
13.65
0.0
13.33
20.25
0.67
21.54
1.28
25
Repair Cost
Midsiz
20
0.8
0.9
0.6
0.7
0.4
0.5
0.2
SUV
0.3
10Midsize
0.1
15
Vehicle
Normal Quantile
Means and Std Deviations
Level
Midsize
SUV
Number
8
9
Mean
14.4300
19.7000
Std Dev
3.39912
4.86642
Let µ1 and µ2 represent mean cost of repair for the SUVs and midsize vehicles, respectively.
(a) (10) State all evidence you can find to support or reject the assumption of equal population
variances by examining the output above. Perform a statistical test using α = .05 to verify
this assumption. State the null and alternative hypotheses and the rejection region clearly.
2
(b) (10) State and test the appropriate hypotheses to determine if the mean repair cost for SUVs
was higher. Use α = .05 to make your decision.
(c) (10) Construct a 90% confidence interval for µ1 − µ2 . Test the research hypothesis that the
repair cost for SUVs is larger than the repair cost for midsize cars by $500 on the average,
using this confidence interval.
3. The following data are obtained from a chemical process where the yield (y, gm.) of the process is
thought to be related to the reaction temperature (x, ◦ C):
x
y
x(cont.)
y(cont.)
50
122
76
171
53
118
79
175
54
128
80
182
55
122
82
180
56
125
85
183
59
136
87
188
3
62
144
90
200
65
142
93
190
67
149
94
206
71
161
95
207
72
167
97
210
74
169
100
219
75
162
Some summary statistics are:
n = 25
Σ x2i = 145705
Σ xi = 1871
Σ yi2 = 713582
Σ yi = 4156
Σ xi yi = 322273
(a) (8) Fit a linear regression y = β0 + β1 x + of the yield in this chemical process (y) on
temperature (x) using the above data. (Do the intermediate computations accurately.)
Report the estimates of β0 and β1 . According to this model, what is the average increase
in yield (in gms.) if the temperature is increased by 10 ◦ C?
(b) (7) Compute an analysis of variance table for the regression. What is the estimate of the error
variance σ2 ? Use the F-statistic from the analysis of variance table to test H0 : β1 = 0 vs.
Ha : β1 6= 0. Use α = .05 and show work.
Source
d.f.
SS
Regression
Error
Total
29
4
MS
F
(c) (8) Compute the estimated standard error of β̂1 . Use it to compute a 95% confidence interval
for β1 . Use this interval to test whether the slope is positive.
(d) (7) Use the numbers in the analysis of variance table to calculate a statistic that measures how
well your model will predict the yield based on the temperature. Explain what this statistic
says to the experimenter.
(e) (8) Estimate the mean yield (in gms.), µ70 , for similar chemical reactions where the temperature
is maintained at 70 ◦ C. Construct a 95% confidence interval for this mean.
(f) (6) A new process produces a mean yield of 150 gms. at 70 ◦ C of the same chemical. Use the
above interval to perform a test of appropriate hypotheses to determine if the mean yield
for reactions 70 ◦ C using the current process, is better than the new process (µ70 exceeds
150 gms.), stating the α-level of the test.
5
(g) (6) This question concerns the 3 plots attached (see next page). Identify the plot (or plots,
use letters A,B,C) you would use to answer each of the questions about model assumptions
stated below. Give your answer to each question justifying your answer using the plot or
plot(s). An yes/no answer alone will not earn any points.
Questions
Do the errors(’s) have a
normal distribution?
Plot(s)
Your answer
————————————————————————————————————
Is the variance (σ2 )
constant for all x’s?
————————————————————————————————————
Is the straight line model
adequate?
6
Diagnostics Plots
Plot A:
Residual by Predicted Plot
Yield Residual
5
0
-5
-10
120
100
180
160
140
200
Yield Predicted
Plot B:
Residual Normal Quantile Plot
Yield Residual
5
0
-5
Normal Quantile
Plot C:
Residual by Temperature Plot
0.9
0.8
0.7
0.6
0.5
0.4
0.3
0.2
0.1
-10