WEM05 Quantitative Techniques for Water & Environmental Management
School of Environment & Technology
Semester 1 Examinations January/February 2013
WEM05
QUANTITATIVE TECHNIQUES FOR
WATER & ENVIRONMENTAL MANAGEMENT
Instructions to Candidates:
Time allowed: TWO hours
Answer ALL questions in Section A (60) and TWO from FOUR in Section B (40)
Note that the questions in Section A do not all carry equal marks.
Special requirements: Statistical Tables, Mathematical formulae (Bird & May)
Items permitted: Any approved calculator, One A4 sheet of notes
Calculators may be used provided they are battery-operated, silent and not preprogrammed.
21st January – 1st February 2013
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WEM05 Quantitative Techniques for Water & Environmental Management
SECTION A
Answer ALL questions in this section.
Note: Questions in this section do NOT carry equal marks
Question 1 (13 marks)
In a study into the effects of change of land use on the lakes in northern
Wisconsin, measurements were recorded on the watershed area (square
kilometres) and lake area (hectares), of 53 lakes. Minitab was used to
analyse the results of the study and the output is given below:
600
Lake Area
500
400
300
200
100
0
0
1
2
3
Wshed
The regression equation is
Lake Area = - 46.7 + 176 Wshed
(i)
(ii)
Interpret the value 176 in the regression equation.
Use the following graphs to check the validity of the linear model
as a method of predicting lake area. Comment on your findings.
(2)
(5)
Residuals Versus the Fitted Values
Normal Probability Plot of the Residuals
(response is Area)
(response is Area)
200
100
1
Residual
Normal Score
2
0
0
-1
-100
-2
-100
0
100
200
Residual
(iii)
0
100
200
300
400
500
Fitted Value
Calculate estimates of the area of a lake with the following
watersheds and comment on their reliability
(a)
1.5 square kilometres
(3)
(b)
(3)
3.5 square kilometres
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WEM05 Quantitative Techniques for Water & Environmental Management
Question 2 (5 marks)
In an investigation into environmental causes of disease, the data collected
included the annual mortality rate per 100,000 males for 61 large towns in
England and Wales, and whether the towns were in the north (N) or the
south (S). An analysis of these data using Minitab produced the following
output.
Two-sample T for Mortality/100000
Region
N
S
N
35
26
Mean
1634
1377
StDev
137
140
SE Mean
23
28
Difference = mu (N) - mu (S)
Estimate for difference: 256.8
T-Test of difference = 0 (vs >): T-Value = 7.17
0.000 DF = 59
Both use Pooled StDev = 138
P-Value =
(i)
Write down H0 and H1 for this test.
(2)
(ii)
State when we should use a one-tail two-sample t-test and when
we should use a two-tail two-sample t-test.
(3)
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WEM05 Quantitative Techniques for Water & Environmental Management
Question 3 (12 marks)
A certain type of soil was determined to have a natural pH of 8.75. Ten
samples of this soil were treated with an organic fertiliser and the resulting pH
measurements were as follows:
8.8
8.7
8.3
8.4
8.1
8.2
8.6
8.5
8.3
8.7
Some analysis was carried out using Minitab and the results are shown
below.
Variable
N
pH
10
(i)
(ii)
(iii)
Mean
8.4600
StDev
0.2366
SE Mean
0.0748
Calculate the 95% confidence interval for the mean pH of the soil
after treatment and state the assumption that is being made in
calculating it.
(5)
Give an interpretation of the confidence interval and explain what
conclusion you can draw about the effect of the fertiliser.
(5)
Without doing any calculations, state, giving a reason, what the
effect would be on the confidence interval of using
(a) a larger sample size
(b) a 99% level of confidence
(2)
Question 4 (6 marks)
Researchers in the Adirondack Mountains collect data on a random sample of
streams each year. One of the variables recorded is the substrate of the
stream- the type of soil and rock over which they flow. The researchers found
that 69 of the 172 sampled streams had a substrate of shale.
Calculate a 95% confidence interval for the proportion of Adirondack streams
with a shale substrate. Explain and interpret this interval to a non-statistician.
Page 4 of 16
(6)
WEM05 Quantitative Techniques for Water & Environmental Management
Question 5 (8 marks)
The water louse, Asellus, can live in polluted oxygen-poor water. By pumping
water over the gills, gill movements can be counted. The number of gill
movements per minute of Asellus specimens in stagnant water was compared
with that for specimens living in an aquarium in oxygen-rich water, to assess if
they differed in the two types of water.
Minitab was used to analyse the results and the output is given below:
Mann-Whitney Confidence Interval and Test
Stagnant
N=7
Median = 49.00
Oxygen-rich
N=10
Median = 43.5
Point estimate for ETA1 - ETA2 is 5.00
95.5% CI for ETA1-ETA2 is (0.001,10.002)
W= 85.5
Test of ETA1= ETA2 vs ETA1 not= ETA2 is significant at 0.0318
(i)
Write down H0 and H1 for this test.
(2)
(ii)
State, giving your reasoning, your decision about H0. State the
conclusions that can be reached from the analysis.
(6)
Page 5 of 16
WEM05 Quantitative Techniques for Water & Environmental Management
Question 6 (10 marks)
Chemical and manufacturing plants often discharge toxic waste materials into
nearby rivers and streams. These toxicants have a detrimental effect on the
plant and animal life inhabiting the river and the river bank.
Measurements are taken of a particular contaminant level, L, (parts per
million) along a stream at varying distances x (metres) downstream from a
fixed point
0  x  20 .
It is found that L and x are related by the equation:
L  30 
(i)
(ii)

1
x 3  50x 2  600x
100

Find an expression for the rate of change of level of contaminant
with respect to the distance x.
(6)
Find the level of contaminant and the rate of change of level in the
stream, with respect to the distance x=1.
(4)
Question 7 (6 marks)
An accident at a chemical plant results in a spillage of chemicals into a river.
The level of contamination depends on the distance x (metres) downstream.
A clean-up operation commences and the rate of change of contamination
(with respect to distance) is given by:
4
x
r  cos  
5
5
for 0  x  20 .
The level of contaminant at position x = 0 is 5 ppm.
Find an expression relating the levels of contamination and distance x.
Page 6 of 16
(6)
WEM05 Quantitative Techniques for Water & Environmental Management
SECTION B
Answer TWO questions from this section.
Note: each question in this section carries 20 marks
Question 8
In a study into the levels of the groundwater contaminant methyl tert-butyl
ether (MTBE) in the water supplies of New Hampshire, data were collected
from a sample of 223 wells. Each well was classified according to ownership
(private or public) and detectable levels of MTBE (below limit or detectable)
and the results are shown in the table below.
Well ownership
Levels of MTBE
Private
Public
Below limit
81
72
Detectable
22
48
(i)
State the type of data that is to be analysed.
(1)
(ii)
Carry out a suitable hypothesis test to decide if the data provide
evidence of an association between the well ownership and the
levels of MTBE in the supply. State clearly the hypotheses and
the conclusions.
(9)
(iii)
What are the constraints of the test used in part (ii)
(2)
(iv)
Calculate a 95% confidence interval for the proportion of private
wells with detectable levels of MTBE and explain its meaning.
(5)
The proportion of public wells with detectable levels of MTBE has
a 95% confidence interval of (31.2%, 48.8%). Discuss how it
relates to parts (i) and (ii).
(3)
(v)
Page 7 of 16
WEM05 Quantitative Techniques for Water & Environmental Management
Question 9
An investigation was carried out to see whether soap kills bacteria. Two
solutions were prepared, one containing soap and the other a control solution
of sterile water. Each solution was placed on ten petri dishes and E. coli
bacteria were added. The dishes were incubated for 24 hours and the number
of bacteria colonies on each dish was counted. Minitab output from analysing
the data is given below:
Boxplot of Soap, Control
85
80
Data
75
70
65
60
55
Soap
Control
Two-Sample T-Test and CI: Soap, Control
Two-sample T for Soap vs Control
N
10
10
Soap
Control
Mean
66.00
74.00
StDev
6.06
8.37
SE Mean
1.9
2.6
Difference = mu (Soap) - mu (Control)
Estimate for difference: -8.00000
95% CI for difference: (-14.86158, -1.13842)
T-Test of difference = 0 (vs not =): T-Value = -2.45
Both use Pooled StDev = 7.3030
(i)
P-Value = 0.025
DF = 18
Making use of the boxplots only, compare the bacterial count for
the control and the soap.
(3)
(ii)
State the null and alternative hypotheses that have been tested.
(2)
(iii)
State the assumptions made in the hypothesis test. Show how
one of these assumptions can be justified from the data.
(4)
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WEM05 Quantitative Techniques for Water & Environmental Management
Question 9 continues overleaf…
Page 9 of 16
WEM05 Quantitative Techniques for Water & Environmental Management
Question 9 continued
(iv)
(v)
(vi)
Give a reasoned decision about the hypotheses and state your
conclusions clearly.
(4)
Write down and interpret fully the confidence interval that is
given in the Minitab output.
(5)
Would the 99% confidence interval be wider or narrower than the
interval given? Justify your answer.
(2)
Question 10
The pH of soil specimens taken from each of four sites in an area is given
below
Site
pH
1
7.00 7.00 7.10 7.10 7.05 7.50 7.40
2
3
4
8.30 8.45 8.00 8.00 8.20 7.85 8.05
7.85 8.00 8.15 8.15 8.00 7.65 7.20
7.05 7.70 7.30 7.75 7.80 7.70 7.65
Minitab was used to analyse pH for the four sites.
Boxplot of Site1, Site2, Site3, Site4
8.6
8.4
8.2
Data
8.0
7.8
7.6
7.4
7.2
7.0
Site1
Site2
Site3
Site4
Question 10 continues overleaf ...
Page 10 of 16
WEM05 Quantitative Techniques for Water & Environmental Management
Question 10 continued...
Results for: ONEWAYANOVA.MTW
Grouping Information Using Tukey Method
Site2
Site3
Site4
Site1
N
7
7
7
7
Mean
8.1214
7.8571
7.5643
7.1643
Grouping
A
A B
B
C
Means that do not share a letter are significantly different.
Tukey 95% Simultaneous Confidence Intervals
All Pairwise Comparisons
Individual confidence level = 98.90%
Site1 subtracted from:
Lower Center Upper
Site2 0.5702 0.9571 1.3441
Site3 0.3059 0.6929 1.0798
Site4 0.0131 0.4000 0.7869
------+---------+---------+---------+--(-----*-----)
(------*-----)
(------*-----)
------+---------+---------+---------+---0.60
0.00
0.60
1.20
Site2 subtracted from:
Lower
Site3 -0.6512
Site4 -0.9441
Center
Upper ------+---------+---------+---------+---0.2643 0.1226
(------*-----)
-0.5571 -0.1702 (------*-----)
------+---------+---------+---------+---0.60
0.00
0.60
1.20
Site3 subtracted from:
Lower
Site4 -0.6798
Center Upper
-0.2929 0.0941
------+---------+---------+---------+--(-----*------)
------+---------+---------+---------+---0.60
0.00
0.60
1.20
Question 10 continues overleaf...
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WEM05 Quantitative Techniques for Water & Environmental Management
Question 10 continued...
(i)
S
(3)
tate the assumptions made in this analysis.
(ii)
W
ithout referring to any hypothesis test, describe what the box-plots tell
you about the data.
(3)
C
(iii)
opy and complete the following ANOVA table in your answer book.
(5)
Source
Factor
Error
Total
DF
3
?
27
SS
3.5388
1.6536
5.1924
MS
?
?
F
?
P
?
(iv)
S
tate H0 and H1 for this study. Using the completed ANOVA table to draw
conclusion about H0.
(5)
U
(v)
sing the Tukey test results given overleaf carry out pair wise
comparisons for the pH values in the five sites.
(4)
Page 12 of 16
WEM05 Quantitative Techniques for Water & Environmental Management
Question 11
Meteorological conditions have been shown to have some effect on levels of
air pollution. To investigate this relationship, the maximum daily levels of a
particular oxidant (a photochemical pollutant) were measured for 30 days
during one summer. In addition, the morning averages of four meteorological
variables were measured: wind speed, temperature, humidity and insolation (a
measure of the amount of sunlight).
Minitab has been used to analyse the data and fit a multiple regression model
to the data. Use the output provided in the following pages to answer the
questions.
(i)
Use the matrix plot to comment on the applicability of a multiple
regression model for these data.
MatrixPlot 'Wind Speed'-'Oxidant';
57.5
Wind Speed
42.5
85.75
Temperature
73.25
66.5
Humidity
45.5
66.5
Insolation
37.5
19.75
Oxidant
9.25
.5
.5
4 2 57
.25 .7 5
7 3 85
.5
.5
45 6 6
.5
.5
3 7 66
5
5
9 .2 19 .7
Question 11 continues overleaf...
Page 13 of 16
(4)
WEM05 Quantitative Techniques for Water & Environmental Management
Question 11 continued...
(ii)
In regression model 1, all four meteorological variables have been
included in the regression model. Minitab indicates that there are two
unusual observations.
What do the R and X beside these observations indicate? What
implications might these unusual observations have for the regression
model?
(5)
Regression Analysis: Oxidant versus Wind Speed, Temperature, ...
The regression equation is
Oxidant = - 15.5 - 0.443 Wind Speed + 0.569 Temperature + 0.0929 Humidity
+ 0.0228 Insolation
Unusual Observations
Obs
Wind Spe
Oxidant
11
47.0
11.000
23
65.0
4.000
Fit
17.586
0.425
SE Fit
0.671
2.170
Residual
-6.586
3.575
St Resid
-2.32R
1.83 X
Residuals Versus the Fitted Values
(response is Oxidant)
2.0
Standardized Residual
1.5
1.0
0.5
0.0
-0.5
-1.0
-1.5
-2.0
-2.5
0
5
10
15
20
25
Fitted Value
Question 11 continues overleaf/...
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WEM05 Quantitative Techniques for Water & Environmental Management
Question 11 continued...
(iii) The stepwise regression procedure in Minitab has been used to decide
which variables to include in the multiple regression model. Write out
the fitted regression model that you think best describes the data, and
use the Minitab output to justify your model choice.
(8)
Stepwise Regression: Oxidant versus Wind Speed, Temperature, ...
Alpha-to-Enter: 0.15
Response is Oxidant
Alpha-to-Remove: 0.15
on
4 predictors, with N =
Step
Constant
1
45.317
2
-5.203
3
-16.607
Wind Spe
T-Value
P-Value
-0.633
-6.30
0.000
-0.427
-4.94
0.000
-0.446
-5.24
0.000
0.52
4.81
0.000
0.60
5.12
0.000
Temperat
T-Value
P-Value
Humidity
T-Value
P-Value
30
0.098
1.56
0.131
S
R-Sq
R-Sq(adj)
C-p
3.95
58.63
57.15
25.2
2.95
77.73
76.08
3.6
2.87
79.64
77.29
3.2
(iv) As a final check, the standardized residuals for the model in part (iii)
were plotted against day. What does the graph indicate? Provide a brief
explanation for this pattern.
(3)
Residuals Versus Day
(response is Oxidant)
2.0
Standardized Residual
1.5
1.0
0.5
0.0
-0.5
-1.0
-1.5
-2.0
-2.5
0
10
20
Day
30
Page 15 of 16
WEM05 Quantitative Techniques for Water & Environmental Management
Formulae
1. A 95% confidence interval for population mean is given by:
x  t.
s
s
   x  t.
n
n
where t is from a t-distribution.
2. For tests of association, use the test statistic:
2

(O – E) 2
E
where O is the observed frequency and E is the expected frequency.
3. Useful formulae for integration and differentiation:
d
dx (constant) = 0
d
n-1
n
dx (x ) = nx
d
ax
ax
dx (e ) = a e
d
1
dx (ln ax) = x
d
dx (sin ax) = a cos ax
d
dx (cos ax) = –a sin ax
n
 t dt

=
t n 1
+ c n ≠ –1
n 1
1
sin(at) dt = – cos(at) + c
a
1
 t dt
= ln |t| + c
e
at
dt =
1 at
e + c
a
(at  b) n1
 (at  b) dt = a(n  1) + c
n

cos(at) dt =
n ≠ –1
1
sin(at) + c
a
Page 16 of 16