oops; here's the file.
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\layout Title
Statistics for Crisis Management
\layout Section
The Statistical Methodology
\layout Standard
Answering a question in a particular manner can be treated as a Bernoulli
variable, having a success (or value of 1) when answered that way, and
a failure (or value of 0) when not answered that way.
\begin_inset Formula $ p_{i}$
\end_inset
is the underlying parameter indicating the proportion of the population
that will answer in that manner, also called the frequency.
It is customary to define
\begin_inset Formula \begin{equation}
q_{i}=1-p_{i}\end{equation}
\end_inset
The fraction of respondents answering in this matter is then a sample mean,
and can be calculated as
\layout Standard
\begin_inset Formula \[
\hat{p}_{i}=\frac{x_{i}}{n}\]
\end_inset
For
\begin_inset Quotes eld
\end_inset
large
\begin_inset Quotes erd
\end_inset
samples, the Central Limit Theorem guarantees that this variable will be
asymptotically normal
\begin_inset Marginal
collapsed false
\layout Standard
jack--cite this?
\end_inset
distributed, and using the well-known variance of the Bernoulli trial,
\begin_inset Formula \begin{equation}
\hat{p}_{i}~N\left(p,\frac{pq}{n}\right)\end{equation}
\end_inset
where
\begin_inset Formula $ n$
\end_inset
is the number of responses.
\layout Standard
Generally, a large sample is taken to be thirty or more.
In the particular case of frequencies, the additional requirements are
generally made that
\begin_inset Formula \begin{equation}
np\geq 5\end{equation}
\end_inset
and
\begin_inset Formula \begin{equation}
nq\geq 5\end{equation}
\end_inset
when this is not the case, the distribution is not sufficiently normal.
\begin_inset Marginal
collapsed false
\layout Standard
cite! (Davidson?)
\end_inset
\layout Standard
For the present data, the null hypothesis for any given survey question
will be that the American response and the Guatemalan are identical.
Under this hypothesis,
\begin_inset Formula $ \hat{p}_{i,a}$
\end_inset
and
\begin_inset Formula $ \hat{p}_{i,g}$
\end_inset
are separate observations of the same underlying parameter,
\begin_inset Formula $ p_{i}$
\end_inset
.
That is, if the hypothesis is true,
\begin_inset Formula $ p_{i}$
\end_inset
is the true frequency for both American and Guatamalen firms, while
\begin_inset Formula $ \hat{p}_{i,a}$
\end_inset
and
\begin_inset Formula $ \hat{p}_{i,g}$
\end_inset
are two separate variables drawn from the distribution.
Linear combinations of normal variables are distributed normally themsleves;
in the case of straightforward addition and subtraction the combined variance
is the sum of the variances, while the combined mean is the sum or difference
of the means.
\begin_inset Marginal
collapsed false
\layout Standard
jack--cite?
\end_inset
In this case, the means are the same under the hypothesis, their difference
is hypothesized as mean zero, and is distributed
\begin_inset Formula \begin{equation}
\hat{p}_{i,a}-\hat{p}_{i,g}~N\left(0,\sigma_{ p_{i,a}}^{2}+\sigma_{
p_{i,g}}^{2}\right)\end{equation}
\end_inset
with the individual variances being of the form
\begin_inset Formula \begin{equation}
\sigma_{\hat{p}_{i,j}}^{2}=\frac{pq}{n_{i,j}}\end{equation}
\end_inset
which combine as
\begin_inset Formula \begin{equation}
\sigma_{\hat{p}_{i,a}-\hat{p}_{i,g}}^{2}=\frac{pq}{n_{i,a}}+\frac{pq}{n_{i,g}}=pq\left(\frac{1}{n_{i,a}}+\frac{1}{n_{i,g}}\right)\end{equation}
\end_inset
and therefore the quantity
\begin_inset Formula \begin{equation}
z_{p_{i}}\equiv\frac{\hat{p}_{i,a}-\hat{p}_{i,g}}{\sqrt{pq}\sqrt{\frac{1}{n_{i,a}}+\frac{1}{n_{j,b}}}}\label{eq:zdef-init}\end{equation}
\end_inset
has a standard normal distribution.
\layout Standard
Equation
\begin_inset LatexCommand \prettyref{eq:zdef-init}
\end_inset
still requires a calculation of
\begin_inset Formula $ p$
\end_inset
, which in turn yields a usable
\begin_inset Formula $ q$
\end_inset
.
Returning to the hypothesis that both groups are the same, the best estimate
of the true frequency will come from the underlying frequency
\begin_inset Formula $ p_{i}$
\end_inset
can be best estimated by using the degrees of freedom for each observation
to form a weighted average,
\begin_inset Marginal
collapsed false
\layout Standard
jack--cite? I can show that it's BLUE if we want
\end_inset
\begin_inset Formula \begin{equation}
\hat{p}_{i}=\frac{n_{i,a}\hat{p}_{i,a}+n_{i,g}\hat{p}_{i,g}}{n_{i,a}+n_{i,g}}\end{equation}
\end_inset
which substituted innto yields the final
\begin_inset Formula $ z$
\end_inset
distributed test statistic of
\begin_inset Formula \begin{eqnarray}
z_{i} & = &
\frac{\hat{p}_{i,a}-\hat{p}_{i,g}}{\sqrt{\hat{p}_{i}\hat{q}_{i}}\sqrt{\frac{1}{n_{i,a}}+\frac{1}{n_{j,b}}}}\nonumber
\\
& = &
\frac{\hat{p}_{i,a}-\hat{p}_{i,g}}{\sqrt{\frac{\left(n_{i,a}\hat{p}_{i,a}+n_{i,g}\hat{p}_{i,g}\right)\left(1-\left(n_{i,a}\hat{p}_{i,a}+n_{i,g}\hat{p}_{i,g}\right)\right)}{\left(n_{i,a}+n_{i,g}\right)^{2}}}\sqrt{\frac{1}{n_{i,a}}+\frac{1}{n_{j,b}}}}\nonumber
\\
& = &
\frac{\left(\hat{p}_{i,a}-\hat{p}_{i,g}\right)\left(n_{i,a}+n_{i,g}\right)}{\sqrt{\left(n_{i,a}\hat{p}_{i,a}+n_{i,g}\hat{p}_{i,g}\right)\left[1-\left(n_{i,a}\hat{p}_{i,a}+n_{i,g}\hat{p}_{i,g}\right)\right]}\sqrt{\frac{1}{n_{i,a}}+\frac{1}{n_{j,b}}}}
\end{eqnarray}
\end_inset
which can easily be calculated partwise on a spreadsheet.
\layout Section
Results
\layout Standard
In many areas, the data shows conclusively that American and Guatamalen
attitudes and experiences with crisis management are different.
Table **
\begin_inset Marginal
collapsed true
\layout Standard
x-ref
\end_inset
shows the z values for all questions for which the methods of Section ***
\begin_inset Marginal
collapsed true
\layout Standard
xreftt
\end_inset
can be calculated.
Any value with a magnitude greater than
\begin_inset Formula $ 1.96$
\end_inset
allows the hypothesis that the response is the same for both countries to
be rejected at the
\begin_inset Formula $ 95\% $
\end_inset
level.
Similarly, values with a magnitude of
\begin_inset Formula $ 2.576$
\end_inset
or greater can be rejected at the
\begin_inset Formula $ 99\% $
\end_inset
level.
Values beyond
\begin_inset Formula $ 3$
\end_inset
can be rejected at any reasonable level.
\layout Subsection
The OP1 category.
\layout Standard
Nearly all of the respondents answered with extemal values; either they
are very concerned or very oncerned with unconcerned with OP1.
\begin_inset Marginal
collapsed true
\layout Standard
are the 2's and 3's even valid responses????
\end_inset
The positive value for level one indicates that americans answered in this
manner far more often.
That is, americans are far more concerned with the issue.
OPOCCUR1 suggests a reason for this: OP1 happens far more often in the
American experience than the Guatamalan.
\layout Standard
*****
\layout Standard
Jack, we need to go over what these mean.
Also, are the intermediate responses valid? If not, we need to do some
paranoia adjustments.
\layout Section
Future Research
\layout Standard
These results come from a moderately sized sample and a simple statistical
analysis.
At this level, it can be seen that significant differences exist between
both the expectations and the experiences of businesses in the two countries.
\layout Standard
A larger data set, drawn from a larger crosssection of both countries, would
strengthen the findings; the conclusions here justify such an effort.
Additionally, further statistical analysis of the present data set is possible.
The only tests done so far are upon individual questions.
A Logit or other regression model may lead to further insights.
\the_end
