Environmental Engineering Reference
In-Depth Information
The confidence interval of the ratio of predevelop-
ment to postdevelopment variance is given by Equation
(10.99) as
X
− µ
x
T
=
N
(10.101)
S
x
2
2
2
S
S
σ
σ
S
S
has a Student's
t
distribution with
N
− 1 degrees of
freedom. Consider the null hypothesis,
H
0
: The sample
is drawn from a normal population with mean
μ
x
. To
decide whether to accept or reject this hypothesis based
on sample measurements, the significance level,
α
, at
which this hypothesis is to be tested must first be speci-
fied, then, if the null hypothesis is true, there is a 1 −
α
probability that any single outcome of
T
,
t
, will be in the
range
t
∈ [
t
1−
α
/2
,
t
α
/2
]. Therefore, if
t
is found to be in the
range [
t
1−
α
/2
,
t
α
/2
], the hypothesis
H
0
(that the sample is
drawn from a normal population with mean
μ
x
) is
accepted at the
α
level of significance. using this
approach, the probability of making a Type I error is
α
.
The alternate hypothesis,
H
1
, is that the sample is not
drawn from a normally distributed population with
mean
μ
x
.
Consider the null hypothesis,
H
0
: The sample is drawn
from a normal population with mean greater than
μ
x
. In
this case, only positive deviations from the assumed
mean support this hypothesis. Therefore, at the
α
signifi-
cance level, this hypothesis would be accepted if
t
∈ [0,
t
α
] and rejected otherwise. A similar approach is taken
to assess the hypothesis that the sample is drawn from
a population with a mean less than
μ
x
. Tests in which
only one-sided deviations are considered are called
one-
tailed tests
. In cases where deviations may occur in either
a positive or negative sense, hypothesis tests are called
two-tailed tests
.
F
≤
≤
F
1
−
α
/
2
α
/
2
2
2
2
1
1
1
S
S
2
σ
σ
2
S
S
2
2
2
2
F
(
60 40
,
)
≤
≤
F
(
60 40
,
)
0 95
.
0 05
.
2
2
2
2
2
2
2
)
( .
0 752
0 683
)
σ
σ
( .
0 752
0 683
)
( .
0 629
≤
≤
1 64
.
2
2
( .
)
( .
)
σ
σ
2
2
0 762
.
≤
≤
1 98
.
2
1
σ
σ
2
0 873
.
≤
≤
1 41
.
1
Based on these results, there is 90% confidence that the
ratio of the pre- to postdevelopment standard deviation
is in the range of 0.873-1.41. Since no-change in water-
quality fluctuations would correspond to this ratio being
equal to 1.0, the measured data do not indicate that the
water-quality fluctuations have changed.
10.9 HYPOTHESIS TESTING
The objective of hypothesis testing is to determine
whether hypotheses regarding population parameters
are supported by random samples taken from the popu-
lation. In hypothesis testing, hypotheses are either
accepted or rejected based on defined acceptance crite-
ria. The hypothesis that is being proposed is called the
null hypothesis
, denoted by
H
0
, with rejection of the null
hypothesis resulting in acceptance of the
alternate
hypothesis
,
H
1
. There are two types of errors that can be
committed in hypothesis testing: Type I and Type II
errors. A Type I error occurs if a hypothesis is rejected
when it is in fact true, and a Type II error is committed
if a hypothesis is accepted when in reality it is false. The
probability of committing a Type I error is the basis of
accepting or rejecting a hypothesis, and is called the
level of significance
of the test. Hypothesis testing when
applied to sample statistics are equivalent alternatives
to using confidence intervals to bound the likely vales
of population parameters. Classical hypothesis testing
procedures are illustrated by the following commonly
encountered hypotheses.
EXAMPLE 10.17
Analysis of 31 log-concentration measurements show a
sample mean of 0.864 and a sample standard deviation
of 0.429. It is proposed that the population mean is
equal to 0.900. Would you accept this hypothesis at the
5% significance level?
Solution
From the given data:
N
= 31,
Y
= 0 86.
,
S
y
= 0.429, and
μ
y
= 0.900. Assuming that the sample is drawn from a
log-normal population, then the
t
statistic correspond-
ing to the sample outcomes is given by Equation (10.101)
as
Y
−
µ
0 864 0 900
0 429
.
−
.
y
t
=
N
=
31
= −
0 467
.
10.9.1 Mean
S
.
y
Sampling theory has shown that if
N
samples are drawn
from a normal population with mean
μ
x
, then the
random variable,
T
, defined by
If the proposed hypothesis is true at the 5% significance
level, then
t
∈ [
t
0.975
,
t
0.025
] where the
t
-values are derived
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