SciRuby/statsample

\part{Equations}
\section{Convention}
\begin{align*}
n &= \text{sample size}\\
N &= \text{population size}\\
p &= \text{proportion inside a sample}\\
P &= \text{proportion inside a population}
\end{align*}
\section{Ruby::Regression::Multiple}

To compute the standard error of coefficients, you obtain the estimated variance-covariance matrix of error.

Let \mathbf{X} be matrix of predictors data, including a constant column; \mathbf{MSE} as mean square error; SSE as Sum of squares of errors; n the number of cases; p as number of predictors

\begin{equation}
\mathbf{MSE}=\frac{SSE}{n-p-1}
\end{equation}

\begin{equation}
\mathbf{E}=(\mathbf{X'}\mathbf{X})^-1\mathbf{MSE}
\end{equation}

The root squares of diagonal should be standard errors


\section{Ruby::SRS}
Finite Poblation correction is used on standard error calculation on poblation below 10.000. Function 
\begin{verbatim}
fpc_var(sam,pop)
\end{verbatim}
calculate FPC for variance with
\begin{equation}
fpc_{var} = \frac{N-n} {N-1}
\end{equation}

with n  as sam and N as pop

Function 
\begin{verbatim}
fpc = fpc(sam,pop)
\end{verbatim}

calculate FPC for standard deviation with 
\begin{equation}
fpc_{sd} = \sqrt{\frac{N-n} {N-1}}
\label{fpc}
\end{equation}
with n  as sample size and N as population size.

\subsection{Sample Size estimation for proportions}

On infinite poblations, you should use method
\begin{verbatim}
estimation_n0(d,prop,margin=0.95)
\end{verbatim}
which uses
\begin{equation}
n = \frac{t^2(pq)}{d^2}
\label{n_i}
\end{equation}
where
\begin{align*}
t &= \text{t value for given level of confidence ( 1.96 for 95\% )}\\
d &= \text{margin of error}
\end{align*}

On finite poblations, you should use
\begin{verbatim}
estimation_n(d,prop,n_pobl, margin=0.95)
\end{verbatim}
which uses
\begin{equation}
n = \frac{n_i}{1+(\frac{n_i-1}{N})}
\end{equation}

Where $n_i$ is n on \ref{n_i} and N is population size