**APPENDIX D**

**MORE ABOUT STATISTICS**

**APPENDIX D**

**MORE ABOUT STATISTICS**

In this Appendix we will discuss, in additional detail, some statistical
topics including error, sample size, the Student\'s \"t\" distribution,
the \"f\" distribution, and confidence intervals.

**[STATISTICAL INFERENCE]{.underline}**

In statistical inference we make generalizations based on samples, and,
traditionally, such inferences have been divided into problems of
estimation and hypothesis testing. In estimation we assign a numerical
value to a population parameter on the basis of sample data. In any CER,
we are attempting to predict the population behavior(s) from a sample.
We need to ask ourselves the question, \"How good is our estimation?\"
When we test a hypothesis, we accept or reject assumptions concerning
the parameters or the form of a population. When we use a CER, we need
to be confident of the CERs ability to predict the future - without this
confidence, the CER should not be used.

**Z-scores**

In general, if X is a measurement belonging to a set of data having the
mean $\overline{x}$ (or $m$ for a population) and the standard deviation
s (or $s$ for a population), then its value in standard units, denoted
by Z, is

$Z = \frac{x - \overline{x}}{S}$ or, $Z = \frac{C - m}{s}$

depending on whether the data constitute a sample or a population. In
these units, Z tells us how many standard deviations a value lies about
or below the mean of the set of data to which it belongs.

**Error**

An estimate is generally called a point estimate, since it consists of a
single number, or a single point on the real number scale. Although this
is the most common way in which estimates are expressed, it leaves room
for many questions. For instance, it does not tell us on how much
information the estimate is based, and it does not tell us anything
about the possible size of the error. And, of course, we must expect an
error. An estimate\'s reliability depends upon two things - the size of
the sample and the size of the population standard deviation, $s$. Any
statistics textbook will show that the error term is,

$E = Z(\alpha/2) \ast \frac{s}{\sqrt{n}}$ where,

Z($\alpha$/2) represents the number of standard deviations from the mean
that we are willing to allow our estimate to be \"off\" either way by
probability of $1 - \alpha$. This result applies when n is large and the
population is infinite. The two values which are most commonly used for
$1 - \alpha$ are 0.95 and 0.99, with corresponding Z scores, Z ( 0.025)
= 1.96 (standard deviations) for $1 - \alpha$ = 0.95 and Z (0.005) =
2.575 for $1 - \alpha$= 0.99, respectively.

There is one complication with this result. To be able to judge the size
of the error we might make when we use $\overline{x}$ as an estimate of
$m$, we must know the value of the population standard deviation, $s$.
Since this is not the case in most practical situations, we have no
choice but to replace $s$ with an estimate, usually the sample standard
deviation, s. In general, this is considered to be reasonable provided
the sample is sufficiently large (n ≥ 30).

**Sample Size**

The formula for E can also be used to determine the sample size that is
needed to attain a desired degree of precision. Suppose that we want to
use the mean of a large random sample to estimate the mean of a
population, and we want to be able to assert with probability
$1 - \alpha$ that the error of this estimate will be less than some
prescribed quantity E. Solving the the previous equation for n, we get,

$$n = \left\lbrack \frac{Z(\frac{\alpha}{2}) \ast s}{E} \right\rbrack^{2}$$

**Confidence Intervals**

For large random samples from infinite populations, the sampling
distribution of the mean is approximately normal with the mean $m$ and
the standard deviation $\frac{s}{\sqrt{n}}$ , namely, that,

$$Z = \frac{\overline{x} - m}{s/\sqrt{n}}$$

is a random variable having approximately the standard normal
distribution. Since the probability is $1 - \alpha$ that a random
variable having the standard normal distribution will take on a value
between -Z($\frac{\alpha}{2}$) and Z($\frac{\alpha}{2}$), namely, that
$- Z(\frac{\alpha}{2})$ \< Z \< $Z(\frac{\alpha}{2})$, we can substitute
into this inequality the foregoing expression for z and it yeilds,

$$- Z(\frac{\alpha}{2}) < \frac{\overline{x} - m}{\frac{s}{\sqrt{n}}} < Z(\frac{\alpha}{2})$$

Using some algebraic manipulation, we get

$$\overline{x} - Z(\frac{\alpha}{2}\frac{) \ast s}{\sqrt{n}} < m < \overline{x} + z(\frac{\alpha}{2}\frac{) \ast s}{\sqrt{n}}$$

and we can assert with probability $1 - \alpha$ that it will be
satisfied for any given sample. In other words, we can assert with
($1 - \alpha$)% confidence that the interval, above, determined on the
basis of a large random sample, contains the population mean we are
trying to estimate. When σ is unknown and n is at least 30, we replace σ
by the sample standard deviation, s.

An interval such as this is called a confidence interval, its endpoints
are called confidence limits, and the probability $1 - \alpha$ is called
the degree of confidence. Again, the values most commonly used for
$1 - \alpha$ are 0.95 and 0.99, the corresponding values of
$Z(\frac{\alpha}{2})$ are 1.96 and 2.575, and the resulting confidence
intervals are referred to as 95% and 99% confidence intervals for μ.

**Confidence Intervals for Means (Small Samples)**

To develop corresponding theory which applies also to small samples, it
will be necessary to assume that the population we are sampling has
roughly the shape of a normal distribution. We can then base our methods
on the statistic $t = \frac{\overline{x} - m}{\frac{s}{\sqrt{n}}}$,
whose sampling distribution is a continuous distribution called the t
distribution. This distribution is symetrical and bell-shaped with zero
mean. The exact shape of the t distribution depends as a parameter
called the number of degrees of freedom, given by n-1, the sample size
less one. For the t distribution we define t(α/2) in the same way in
which Z(α/2) was defined. However, t(α/2) depends on n-1 (degrees of
freedom) and its value must be looked up in a table of values. In the
same way as before we can arrive at the following small sample
confidence interval for μ:

$$\overline{x} - t(\frac{\alpha}{2}\frac{) \ast s}{\sqrt{n}} < m < \overline{x} + t(\frac{\alpha}{2}\frac{) \ast s}{\sqrt{n}}$$

The degree of confidence is $1 - \alpha$ and the only difference between
this formula and the large sample formula is the t(α/2) takes the place
of Z(α/2).

**Analysis of Variance and the F Statistic**

The F statistic is a statistic for a test concerning the differences
among means. It is defined as:

F = [estimate of σ^2^ based on the variation among the]{.underline}
$\overline{x}$[\'s]{.underline}

estimate of σ^2^ based on the variation within the samples

and is called a variance ratio. The F distribution is a theoretical
distribution which depends on two parameters called the numerator and
denominator degrees of freedom. When the F statistic is used to compare
the means of k samples of size n, the numerator and denominator degrees
of freedom are, respectively, k-1 and k(n-1).

This is a simple form of an analysis of variance. The basic idea of an
analysis of variance is to express a measure of the total variation of a
set of data as a sum of terms, which can be attributed to specific
sources, or causes of variation. Two such sources of variation could
be 1) actual differences, and, 2) chance differences. As a measure of
the total variation of an observation consisting of k samples of size n,
we use the total sum of squares,

$\text{SST} = \sum_{i = 1}^{k}{}\sum_{j = 1}^{n}{(X_{\text{ij}} - \overline{x}\text{.}\text{.}\text{.})^{2}}$
,

where X~ij~ is the jth observation of the ith sample, i = 1, 2, \... ,
k, and j = 1, 2, \... , n, and $\overline{x}$, the mean of all the k
measurements or observations is called the grand mean. If we divide the
total sum of squares by kn-1, we get the variance of the combined data.

Letting ${\overline{X}}_{i}$ devote the mean of the ith sample, i = 1,
2, \..., k, we can write the following identity:

$$\text{SST} = n \ast \sum_{i = 1}^{k}{({\overline{X}}_{i} - \overline{X}\text{.}\text{.}\text{.})^{2} + \sum_{i = 1}^{k}\sum_{j = 1}^{n}(}X_{\text{ij}} - \overline{X}i)^{2}$$

Looking closely at the two terms into which the total sum of squares SST
has been partitioned, we find that the first term is a measure of the
variation among the means. Similarly, the second term is a measure of
the variation within the individual samples, or chance variation.
Dividing the first term by k-1 and the second by k (n-1), we get the
numerator and the denominator of the F Statistic as defined, above. The
first term is often referred to as the treatment sum of squares, SST and
the second term as the error sum of squares, SSE, experimental error, or
chance.

Refer to a statistics textbook for a further explanation of these and
other statistical subjects.