MASSACHUSETTS INSTITUTE OF TECHNOLOGY
Department of Civil and Environmental Engineering
1.017 Computing and Data Analysis for Environmental Applications
Practice Quiz 3
December 5, 2001
Please answer all questions on a separate piece(s) of paper with your name clearly identified:
Problem 1 (15 points)
Provide a brief answer to each of the following questions:
a) Suppose that you have a choice of collecting a random sample of 20, 40, 80, or 160
measurements of a random variable
y
.
You will use the sample to compute the sample
mean
m
y
, which you will use as an estimate of
E
[
y
].
Draw a sketch of the probability
density of
m
y
for
n
= 20 measurements and superimpose on it a sketch of the same density
for
n
= 40 measurements.
b) Plot the standard deviation of the sample mean vs. the number of measurements in the
sample.
How does the standard deviation at
n
= 160 compare to the standard deviation at
n
= 20?
c) Is the sample mean a consistent estimate of
E
[
y
]?
Why?
Solution:
a) Plot should look normal. Standard deviation for
n
= 40 should be 0.70 standard deviation
for
n
= 20, peak for
n
= 40 should be 1.41 peak for
n
= 20.
b) Standard deviation of sample mean should decrease as
n
1/2
.
c) Yes, because
Lim
0
]
[
=
∞
→
y
n
m
Var
Problem
2 (25 points)
You are studying the cadmium concentration (
y
) in trout tissue in a large lake.
You catch ten trout
and measure the following cadmium concentrations (mg/L):
0.106 0.040
0.128 0.225 0.167 0.024 0.317 0.072 0.099 0.029
a) Considering the possible values of the data, what would be a reasonable assumption for
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 Spring '05
 GeorgeKocur
 Normal Distribution, Standard Deviation, Environmental Engineering, Massachusetts Institute of Technology, Cumulative distribution function, Department of Civil and Environmental Engineering

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