10.9: Lista de fórmulas
- Page ID
- 151185
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- Listado de todas las fórmulas utilizadas a lo largo del capítulo.
\[SS_{xx}=\sum x^2-\frac{1}{n}\left ( \sum x \right )^2\; \; SS_{xy}=\sum xy-\frac{1}{n}\left ( \sum x \right )\left ( \sum y \right )\; \; SS_{yy}=\sum y^2-\frac{1}{n}\left ( \sum y \right )^2\]
Coeficiente de correlación:
\[r=\frac{SS_{xy}}{\sqrt{SS_{xx}SS_{yy}}}\]
Ecuación de regresión de mínimos cuadrados (ecuación de la línea de regresión de mínimos cuadrados):
\[\hat{y}=\hat{\beta _1}x+\hat{\beta _0}\; \; \text{where}\; \; \hat{\beta _1}=\frac{SS_{xy}}{SS_{xx}}\; \; \text{and}\; \; \hat{\beta _0}=\bar{y}-\hat{\beta _1}\bar{x}\]
Suma de los errores al cuadrado para la línea de regresión de mínimos cuadrados:
\[SSE=SS_{yy}-\hat{\beta _1}SS_{xy}\]
Desviación estándar de la muestra de errores:
\[S_\varepsilon =\sqrt{\frac{SSE}{n-2}}\]
\(100(1-\alpha )\%\)intervalo de confianza para\(\beta _1\):
\[\hat{\beta _1}\pm t_{\alpha /2}\frac{S_\varepsilon }{\sqrt{SS_{xx}}}\; \; \; (df=n-2)\]
Estadística de prueba estandarizada para pruebas de hipótesis relativas a\(\beta _1\):
\[T=\frac{\hat{\beta _1}-B_0}{S_\varepsilon /\sqrt{SS_{xx}}}\; \; \; (df=n-2)\]
Coeficiente de determinación:
\[r^2=\frac{SS_{yy}-SSE}{SS_{yy}}=\frac{SS_{xy}^{2}}{SS_{xx}SS_{yy}}=\hat{\beta _1}\frac{SS_{xy}}{SS_{yy}}\]
\(100(1-\alpha )\%\)intervalo de confianza para el valor medio de\(y\) at\(x=x_p\):
\[\hat{y_p}\pm t_{\alpha /2}S_\varepsilon \sqrt{\frac{1}{n}+\frac{(x_p-\bar{x})^2}{SS_{xx}}} \; \; \; (df=n-2)\]
\(100(1-\alpha )\%\)intervalo de predicción para un nuevo valor individual de\(y\) at\(x=x_p\):
\[\hat{y_p}\pm t_{\alpha /2}S_\varepsilon \sqrt{1+\frac{1}{n}+\frac{(x_p-\bar{x})^2}{SS_{xx}}} \; \; \; (df=n-2)\]
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