Clinical Biochemistry of Domestic Animals (Sixth Edition) - UMK ...

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Chapter | 1 Concepts of Normality in Clinical Biochemistry
tendency of the population can be obtained from the sample,
such as the sample median and the sample mode. Also,
sample-based estimates of the measures of dispersion or
spread for the population can be obtained. The sample
variance is computed as s 2 Σ ( x − i x ) 2 /( n 1), and the
sample standard deviation, s , is obtained by taking the
square root of s 2 . The descriptive statistics are called point
estimates of the parameters and represent good approximations
of the parameters. An alternative to the point estimate
is the interval estimate , which takes into account the
underlying probability distribution of the point estimate
called the sampling distribution of the statistic.
C . Sampling Distributions
In actual practice, only a single sample is taken from a population
and, on the basis of this sample, a single point estimate
of the unknown population parameter is computed. If
time and resources would permit repeated sampling of the
population in the same manner—that is, with the same probability-based
sampling design—one point estimate would
be obtained for each sample obtained. The estimates would
not be the same because the sample would contain different
elements of the population. As the number of such repeated
sampling operations increases, a more detailed description
emerges of the distribution of possible point estimates
that could be obtained by sampling the population.
This is the sampling distribution of the statistic.
Some fundamental facts relating to the sampling distribution
of the sample mean follow: (1) The center of the
sampling distribution of x − is equal to μ , the center of the
underlying distribution of elements in the population. (2)
The spread of the sampling distribution of x − is smaller than
σ 2 , the spread of the underlying distribution of elements
in the population. Specifically, the variance of the sampling
distribution of x − (denoted σ 2 x
− ) equals σ 2 /n , where n
is the sample size. So increasing the sample size serves to
increase the likelihood of obtaining an x − close to the center
of the distribution because the spread of the sampling
distribution is being reduced. (3) The central limit theorem
(Daniel, 2005 ; Schork and Remington, 2000 ; Zar, 1999 )
states that regardless of the underlying distribution of the
population of elements from which the sample mean is
based, if the sample size is reasonably large ( n 30), the
sampling distribution of x − is approximated well by the
Gaussian distribution. So x − drawn from any distribution has
a sampling distribution that is approximately N (μ , σ 2 /n ) for
n 30. If the distribution of the underlying population of
elements is Gaussian or approximated well by a Gaussian
distribution, the sampling distribution of x − will be approximated
well by the Gaussian distribution regardless of the
sample size on which x − is based.
Probabilities of the sampling distribution of x − , N ( μ ,
σ 2 / n ), can be evaluated using the method described in
Section II.B.
Example 3
Suppose the underlying population of elements is N (4,16)
and a sample of size n 9 is drawn from this population
using SRS. It is desired to find the probability of observing
a sample mean less than 3.1 or greater than 6.2. In solving
this problem, the relevant sampling distribution is specified:
x − is N (4,16/9). Note that the sampling distribution of x −
is Gaussian because the problem stated that the underlying
population was Gaussian. (Otherwise the stated sample size
would have needed to be 30 or larger to invoke the central
limit theorem.) The probability of observing x − 3.1 in the
distribution of x − is equivalent to the probability of observing
z (3.1 4)/(4/3) 0.675 in the standard Gaussian distribution.
Going to Table 1-2 , z 0.675 is approximately the
75th percentile of the standard Gaussian distribution, and by
symmetry z 0.675 is approximately the 25th percentile.
Thus, the probability of observing a z value less than or equal
to 0.675 is approximately 0.25. The probability of observing
a sample mean greater than 6.2 is equivalent to the probability
of observing z (6.2 4)/(4/3) 1.65. Table 1-2 gives
the probability of observing a z 1.65 as approximately 0.95
so the probability of observing a z 1.65 equals 1 0.95 or
0.05. The desired probability of observing a sample mean
less than 3.1 or greater than 6.2 is the sum of 0.25 and 0.05,
which is 0.3 or 3 chances in 10.
D . Constructing an Interval Estimate of
the Population Mean, μ
This brief exposure to sampling distributions and their standardized
forms provides the framework for generating an
interval estimate for μ . Consider the probability statement
Prob( 2 z 2) 0.9544. Because z ( x − μ )/(σ /√ n ),
this probability statement is equivalent to the statement
Prob( 2 ( x − μ )/(σ /√ n ) 2) 0.9544. Some standard
algebraic manipulation of the inequality within the
parentheses gives Prob(x − ( 2σ /√ n ) μ x − (2 σ /
√ n )) 0.9544. This is the form of the confidence statement
about the unknown parameter μ . With repeated
sampling of the underlying population, 95.44% of the
intervals constructed by adding and subtracting 2 σ / √ n to
and from the sample mean would be expected to cover the
true unknown value of μ . The quantities of x − 2 σ /√ n
and x − 2 σ /√ n are called the lower and upper confidence
limits , respectively, and the interval bounded below by
x− 2 σ /√ n and above by x − 2 σ /√ n —that is, ( x − 2 σ /√ n ,
x− 2 σ /√ n ) is the 95.44% confidence interval for μ . In
practice, only one sample is taken from the population, and
thus there is a 95.44% chance that the one interval estimate
obtained will cover the true value of μ . Note that 95.44% or
0.9544 is called the confidence level .
The degree of confidence that is to be had is determined
by the amount of error that is to be tolerated in the estimation
procedure. For a 0.9544 level of confidence, the error rate is
1 0.9544 0.0456. The error rate is designated by alpha, α .