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\begin{document}

\title{Problemas sobre pron\selectlanguage{ngerman}ósticos~~~~\selectlanguage{spanish}}



\author[1]{Yamirlethy Rodríguez-Esquivel }%
\affil[1]{Tecnológico Nacional de México - Campus Zacatecas Occidente}%


\vspace{-1em}



  
  \date{May 04, 2024}


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\selectlanguage{ngerman}\section*{Fórmulas:}

{\label{748564}}

\[X_t=a+bt\]

\[a=\overline{X}-b\overline{t}\]

\(\)\(% b=\frac{\left(n\sum_{i=1}^nX_it_i-\sum_{i=1}^nX_i\sum_{i=1}^nt_i\right)}{n\sum_{i=1}^nt_i^2-[\sum_{i=1}^{n}t_i]^2}\)\[b=\frac{\left(n\sum_{i=1}^nX_it_i-\sum_{i=1}^nX_i\sum_{i=1}^nt_i\right)}{n\sum_{i=1}^nt_i^2-\left[\sum_{i=1}^nt_i\right]^2}\]

\section*{Ejercicio:}

{\label{453709}}

Para~ la~ economía~ española,~ disponemos~ de~ los~ datos~ anuales~
redondeados~ sobre~ consumo~ final~ de los~ hogares~ a~ precios~
corrientes~ (Y)~ y~ renta~ nacional~ disponible~ neta~ (X),~ tomados~
de~ la Contabilidad~ Nacional~ de~ España~ base~ 1995~ del~ INE~ ,~
para~ el~ período~ 1995-2002,~ ambos expresados en miles de millones de
euros:\selectlanguage{spanish}
\begin{figure}[H]
\begin{center}
\includegraphics[width=0.70\columnwidth]{figures/tab/tab}
\caption{{Tabla de datos
{\label{782032}}%
}}
\end{center}
\end{figure}

Considerando~ que~ el~ consumo~ se~ puede~ expresar~ como~ funci\selectlanguage{ngerman}ón~
lineal~ de~ la~ renta~~\[X_t=a+bt\] ,determine:

a) Los parámetros a y b de la recta de regresión.

b) La predicción del valor que tomará el consumo para una renta de
650.000 millones de euros.

\subsection*{Solución: ~}

{\label{413947}}\par\null\selectlanguage{spanish}
\begin{longtable}[]{@{}ll@{}}
\toprule
t & x\tabularnewline
381'7 & 258'6\tabularnewline
402'2 & 273'6\tabularnewline
426'5 & 289'7\tabularnewline
454'3 & 308'9\tabularnewline
486'5 & 331'0\tabularnewline
520'2 & 355'0\tabularnewline
553'3 & 377'1\tabularnewline
590'0 & 400'4\tabularnewline
\bottomrule
\end{longtable}

\subsubsection*{Primero debemos calcular
b:}

{\label{281162}}

Para conocer el valor de b es necesario calcular los datos que nos pide
la f\selectlanguage{ngerman}órmula:

\[b=\frac{\left(n\sum_{i=1}^nX_it_i-\sum_{i=1}^nX_i\sum_{i=1}^nt_i\right)}{n\sum_{i=1}^nt_i^2-\left[\sum_{i=1}^nt_i\right]^2}\]

\(\sum_{i=1}^nX_it_i=\left(381.7\cdot258.6\right)+\left(402.2\cdot273.6\right)+\left(426.5\cdot289.7\right)+\left(454.3\cdot308.9\right)+\left(486.5\cdot331.0\right)+\left(520.2\cdot355.0\right)+\left(553.3\cdot377.1\right)+\left(590.0\cdot400.4\right)\)

\(\sum_{i=1}^nX_it_i=1263227.79\)

\par\null

\(\sum_{i=1}^nX_i=258.6+273.6+289.7+308.9+331+355+377.1+400.4\)

\(\sum_{i=1}^nX_i=2594.3\)

\par\null

\(\sum_{i=1}^nt_i=381.7+402.2+426.5+454.3+486.5+520.2+553.3+590\)

\(\sum_{i=1}^nt_i=3814.7\)

\par\null

\(\sum_{i=1}^nt_i^2=145694.89+161764.84+181902.25+206388.49+236682.25+270608.04+306140.89+348100\)

\(\sum_{i=1}^nt_i^2=1857281.65\)

\par\null

\(\left[\sum_{i=1}^nt_i\right]^2=\left(3814.7\right)^2\)

\(\left[\sum_{i=1}^nt_i\right]^2=14551936.09\)

\par\null

\textbf{Sustituimos los valores en nuestra fórmula:}

\(b=\frac{\left[8\left(1263227.79\right)-\left(2594.3\right)\left(3814.7\right)\right]}{\left[8\left(1857281.65\right)-14551936.1\right]}\)

\(b=0.6834\)

\par\null

\subsubsection*{Ahora calculamos el valor de
a:}

{\label{259326}}

Para conocer el valor de a es necesario calcular los datos que nos pide
la fórmula y como ya conocemos cuanto vale b solo debemos calcular
\(\overline{X\ }y\ \overline{t}\): ~

\[a=\overline{X}-b\overline{t}\]

\(\overline{X}=\frac{\left(258.6+273.6+289.7+308.9+331+355+377.1+400.4\right)}{8}\)

\(\overline{X}=324.2875\)

\par\null

\(\overline{t}=\frac{\left(381.7+402.2+426.5+454.3+486.5+520.2+553.3+590\right)}{8}\)

\(\overline{t}=476.8375\)

\par\null

\textbf{Sustituimos los valores en nuestra fórmula:}

\textbf{\(a=324.2875-\left(0.6834\cdot476.8375\right)\)}

\textbf{\(a=-1.59
\)}

\par\null

\textbf{Ahora que conocemos los datos a y b podemos sustituir los
valores en nuestra formula para realizar el inciso b):}

\[X_t=a+bt\]

\subsubsection*{Sustituimos:}

{\label{937588}}

\(X_t=-1.59+0.6834t\)~ ~~

Como en este caso queremos saber la predicción del valor que tomará el
consumo para una renta de 650.000 millones de euros. ~

\(t=650.00\)

\(X_t=-1.59+0.6834\left(650.00\right)\)

\(X_{650}=442.62\)

Así podemos determinar que el pronóstico del valor que tomará el consumo
para una renta de 650.000 millones de euros es de 442.62~ millones de
euros.

\section*{Gráfica:}

{\label{266316}}\selectlanguage{spanish}
\begin{figure}[H]
\begin{center}
\includegraphics[width=0.70\columnwidth]{figures/graf/graf}
\caption{\selectlanguage{ngerman}{Pronóstico
{\label{297843}}%
}}
\end{center}
\end{figure}\selectlanguage{ngerman}

\selectlanguage{spanish}
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