Page 27 - Statistics for Environmental Engineers

L1592_Frame_C02 Page 18 Tuesday, December 18, 2001 1:40 PM

Example 2.8

y
The average for the n = 27 nitrate measurements is = 7.51 and the sample standard deviation
is s = 1.38. The estimated standard error of the mean is:

---------- =
s y = 1.38 0.27 mg/L
27
If the parent distribution is normal, the sampling distribution of will be normal. If the parent distribution
y
y
is nonnormal, the distribution of will be more nearly normal than the parent distribution. As the number
y
of observations n used in the average increases, the distribution of becomes increasingly more normal.
This fortunate property is the central limit effect. This means that we can use the normal distribution
2
with mean η and variance σ /n as the reference distribution to make probability statements about y
(e.g., that the probability that is less than or greater than a particular value, or that it lies in the interval
y
between two particular values).
2
Usually the population variance, σ , is not known and we cannot use the normal distribution as the
and use the t distribution.
reference distribution for the sample average. Instead, we substitute s y for σ y
2
If the parent distribution is normal and the population variance is estimated by s , the quantity:
y η–
t = ------------
s/ n
which is known as the standardized mean or as the t statistic, will have a t distribution with ν = n − 1
degrees of freedom. If the parent population is not normal but the sampling is random, the t statistic
will tend toward the t distribution (just as the distribution of tends toward being normal).
y
2
If the parent population is N(η, σ ), and assuming once again that the observations are random and
2
2
independent, the sample variance s has especially attractive properties. For these conditions, s is
2
distributed independently of y in a scaled χ (Chi-square) distribution. The scaled quantity is:
s
χ = ν ------ 2 .
2
σ 2
2
This distribution is skewed to the right. The exact form of the χ distribution depends on the number of
2
degrees of freedom, ν, on which s is based. The spread of the distribution increases as ν increases. The
2 2 2
tail area under the Chi-square distribution is the probability of a value of χ = νs σ exceeding a given
value.
2
Figure 2.11 illustrates these properties of the sampling distributions of , s , and t for a random
y
sample of size n = 4.
Example 2.9
y
For the nitrate data, the sample mean concentration of = 7.51 mg/L lies a considerable distance
below the true value of 8.00 mg/L (Figure 2.12). If the true mean of the sample is 8.0 mg/L and
the laboratory is measuring accurately, an estimated mean as low as 7.51 would occur by chance
only about four times in 100. This is established as follows. The value of the t statistic is:

–
------------ =
t = y η– 7.51 8.00 – 1.842
--------------------------- =
s / n 1.38/ 27
with ν = 26 degrees of freedom. Find the probability of such a value of t occurring by referring
to the tabulated tail areas of the t distribution in Appendix A. Because of symmetry, this table
© 2002 By CRC Press LLC