Environmental Engineering Reference
In-Depth Information
(c) What percentage of the soil samples will have a concentration between 61 and
72 mg/kg?
SOLUTION: For x ¼ 72; z ¼ð7266Þ=5 ¼ 1:2; x ¼ 61; z ¼ð6166Þ=5 ¼1
ðaÞ Pðx < 72Þ¼Pðz < 1:2Þ¼Pð1 < z < 0ÞþPð0 < z < 1:2Þ
¼ 0:5þ0:3849 ðAppendix C1Þ¼0:8849ð88:49%Þ
ðbÞ Pðx > 72Þ¼Pðz > 1:2Þ¼1Pðz < 1:2Þ¼10:8849 ¼ 0:1151ð11:51%Þ
ðcÞ Pð61 < x < 72Þ¼Pð1 < z < 1:2Þ¼Pð1 < z < 0ÞþPð0 < z < 1:2Þ
¼ Pð0 < z < 1ÞþPð0 < z < 1:2Þ¼0:3413þ0:3849 ¼ 0:7262ð72:62%Þ
Student's t Distribution
The Student's t distribution was first described by William Gosset in 1908 for use
with small sample sizes (n30). It is similar to normal distribution in that it is
symmetric at about zero, and it becomes identical to normal distribution when n is
large (n!1). The t distribution is used to describe the distribution of the mean (
x)
rather than x. The mathematical equation for the t distribution is:
t ¼
xm
s= n
p
ð2
:
20Þ
p is the standard error of the mean (SEM). It is often used to calculate the
confidence interval (CI):
=
where s
s
p Þ
CI ¼
x t n1;1 2 ð
ð2
:
21Þ
where the degree of freedom (df)isn1, and 1a=2 is the one-sided confidence
level. Note that a one-sided confidence level of 95% is equivalent to 90% of the
two-sided confidence level. The t value can be obtained from Appendix C2.
EXAMPLE 2.3. If the same data shown in Example 2.2 were obtained from a sample size
of n¼8 rather than a population (i.e., x ¼ 66 mg/kg, s ¼ 5 mg/kg), what are the confidence
intervals for two-sided confidence levels of (a) 90% or (b) 99%?
SOLUTION:
ðaÞ 90%ð2-sidedÞ¼95%ð1-sidedÞ : 1a=2 ¼ 0:95; t 7;0:95 ¼ 1:895 ðAppendix C2Þ
CI ¼ 661:8955=
p ¼ 663:3mg=kg ði:e:;
62:7 69:3mg=kgÞ
(b) 99% (2-sided) ¼99.5% (1-sided): 1a/2¼0.995, t 7,0.995 ¼3.499 (Appendix C2)
CI ¼ 663:4995=
p ¼ 666:2mg=kg ði:e:;
59:8 72:2mg=kgÞ
F-distribution
Unlike the normal distribution and the t-distribution, F-distribution is skewed to the
right. It is an appropriate model for the probability distribution of the ratio of two
 
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