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})^-1mathbf{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