Sistemas de hipótesis

Una muestra

\[ \begin{array}{|c|c|} \hline \text{Hipótesis nula ($H_0$)} & \text{Hipótesis alternativa ($H_1$)} \\ \hline H_0: \mu_1 \geq \mu_1 \leftrightarrow \mu_1 - \mu_2 \geq 0 & H_1: \mu_1 < \mu_2 \leftrightarrow \mu_1 - \mu_2 < 0 \\ \hline H_0: \mu_1 = \mu_2 \leftrightarrow \mu_1 - \mu_2 = 0 & H_1: \mu_1 \neq \mu_2 \leftrightarrow \mu_1 - \mu_2 \neq 0 \\ \hline H_0: \mu_1 \leq \mu_2 \leftrightarrow \mu_1 - \mu_2 \leq 0 & H_1: \mu > \mu_0 \leftrightarrow \mu_1 - \mu_2 > 0 \\ \hline \end{array} \]

Estadísticos de prueba

Varianzas conocidas

Iguales

\[ \text{Si }X_1{\sim}N(\mu_1,\sigma_x^2),X_2{\sim}N(\mu_2,\sigma_x^2)\text{ entonces }\frac{\left(\overline{x}_1-\overline{x}_2\right)-\left(\mu_1-\mu_2\right)}{\sigma_x\sqrt{\frac{1}{n_1}+\frac{1}{n_2}}}{\sim}N(0,1) \]

\[ \text{Si }X_1{\sim}P\left(E_P(X_1),Var_P(X_1)\right),X_2{\sim}P\left(E_P(X_2),Var_P(X_2)\right)\text{ entonces }\frac{\left(\overline{x}_1-\overline{x}_2\right)-\left(\mu_1-\mu_2\right)}{\sigma_x\sqrt{\frac{1}{n_1}+\frac{1}{n_2}}}\stackrel{n{\rightarrow}\infty}{\sim}N(0,1) \]

Diferentes

\[ \text{Si }X_1{\sim}N(\mu_1,\sigma_1^2),X_2{\sim}N(\mu_2,\sigma_2^2)\text{ entonces }\frac{\left(\overline{x}_1-\overline{x}_2\right)-\left(\mu_1-\mu_2\right)}{\sqrt{\frac{\sigma_1^2}{n_1}+\frac{\sigma_2^2}{n_2}}}{\sim}N(0,1) \]

\[ \text{Si }X_1{\sim}P\left(E_P(X_1),Var_P(X_1)\right),X_2{\sim}P\left(E_P(X_2),Var_P(X_2)\right)\text{ entonces }\frac{\left(\overline{x}_1-\overline{x}_2\right)-\left(\mu_1-\mu_2\right)}{\sqrt{\frac{\sigma_1^2}{n_1}+\frac{\sigma_2^2}{n_2}}}\stackrel{n{\rightarrow}\infty}{\sim}N(0,1) \]

Varianzas desconocidas

Iguales

\[ \text{Si }X_1{\sim}N(\mu_1,\sigma_1^2),X_2{\sim}N(\mu_2,\sigma_2^2)\text{ entonces }\frac{\left(\overline{x}_1-\overline{x}_2\right)-\left(\mu_1-\mu_2\right)}{s_p\sqrt{\frac{1}{n_1}+\frac{1}{n_2}}}{\sim}t_{(n-1)} \]

\[ \text{Si }X_1,X_2{\sim}P\left(E_P(X),Var_P(X)\right)\text{ entonces }\frac{\left(\overline{x}_1-\overline{x}_2\right)-\left(\mu_1-\mu_2\right)}{s_p\sqrt{\frac{1}{n_1}+\frac{1}{n_2}}}\stackrel{n{\rightarrow}\infty}{\sim}N(0,1) \]

donde: \(s_p = \sqrt{\frac{(n_1 - 1)s_1^2 + (n_2 - 1)s_2^2}{n_1 + n_2 - 2}}\)

Diferentes

\[ \text{Si }X_1{\sim}N(\mu_1,s_1^2),X_2{\sim}N(\mu_2,s_2^2)\text{ entonces }\frac{\left(\overline{x}_1-\overline{x}_2\right)-\left(\mu_1-\mu_2\right)}{\sqrt{\frac{s_1^2}{n_1}+\frac{s_2^2}{n_2}}}{\sim}N(0,1) \]

\[ \text{Si }X_1{\sim}P\left(E_P(X_1),Var_P(X_1)\right),X_2{\sim}P\left(E_P(X_2),Var_P(X_2)\right)\text{ entonces }\frac{\left(\overline{x}_1-\overline{x}_2\right)-\left(\mu_1-\mu_2\right)}{\sqrt{\frac{s_1^2}{n_1}+\frac{s_2^2}{n_2}}}\stackrel{n{\rightarrow}\infty}{\sim}N(0,1) \]

Regiones de rechazo de la hipótesis nula (\(H_0\))

Varianzas conocidas

Iguales

\[ \begin{array}{|c|c|c|} \hline \text{Hipótesis nula ($H_0$)} & \text{Hipótesis alternativa ($H_1$)} & \text{Región de rechazo ($H_0$)} \\ \hline H_0: \mu \geq \mu_0 \leftrightarrow \mu_1 - \mu_2 \geq 0 & H_1: \mu < \mu_0 \leftrightarrow \mu_1 - \mu_2 < 0 & \left( -\infty, \left( \overline{x}_1 - \overline{x}_2 \right) - Z_{1-\alpha}\sigma_x\sqrt{\frac{1}{n_1}+\frac{1}{n_2}} \right) \\ \hline H_0: \mu = \mu_0 \leftrightarrow \mu_1 - \mu_2 = 0 & H_1: \mu \neq \mu_0 \leftrightarrow \mu_1 - \mu_2 \neq 0 & \left( -\infty, \left( \overline{x}_1 - \overline{x}_2 \right) - Z_{1-\frac{\alpha}{2}}\sigma_x\sqrt{\frac{1}{n_1}+\frac{1}{n_2}} \right) \cup \left( \left( \overline{x}_1 - \overline{x}_2 \right) + Z_{1-\frac{\alpha}{2}}\sigma_x\sqrt{\frac{1}{n_1}+\frac{1}{n_2}}, +\infty \right) \\ \hline H_0: \mu \leq \mu_0 \leftrightarrow \mu_1 - \mu_2 \leq 0 & H_1: \mu > \mu_0 \leftrightarrow \mu_1 - \mu_2 > 0 & \left( \left( \overline{x}_1 - \overline{x}_2 \right) + Z_{1-\alpha}\sigma_x\sqrt{\frac{1}{n_1}+\frac{1}{n_2}}, +\infty \right) \\ \hline \end{array} \]

Diferentes

\[ \begin{array}{|c|c|c|} \hline \text{Hipótesis nula ($H_0$)} & \text{Hipótesis alternativa ($H_1$)} & \text{Región de rechazo ($H_0$)} \\ \hline H_0: \mu \geq \mu_0 \leftrightarrow \mu_1 - \mu_2 \geq 0 & H_1: \mu < \mu_0 \leftrightarrow \mu_1 - \mu_2 < 0 & \left( -\infty, \left( \overline{x}_1 - \overline{x}_2 \right) - Z_{1-\alpha}\sqrt{\frac{\sigma_1^2}{n_1}+\frac{\sigma_2^2}{n_2}} \right) \\ \hline H_0: \mu = \mu_0 \leftrightarrow \mu_1 - \mu_2 = 0 & H_1: \mu \neq \mu_0 \leftrightarrow \mu_1 - \mu_2 \neq 0 & \left( -\infty, \left( \overline{x}_1 - \overline{x}_2 \right) - Z_{1-\frac{\alpha}{2}}\sqrt{\frac{\sigma_1^2}{n_1}+\frac{\sigma_2^2}{n_2}} \right) \cup \left( \left( \overline{x}_1 - \overline{x}_2 \right) + Z_{1-\frac{\alpha}{2}}\sqrt{\frac{\sigma_1^2}{n_1}+\frac{\sigma_2^2}{n_2}}, +\infty \right) \\ \hline H_0: \mu \leq \mu_0 \leftrightarrow \mu_1 - \mu_2 \leq 0 & H_1: \mu > \mu_0 \leftrightarrow \mu_1 - \mu_2 > 0 & \left( \left( \overline{x}_1 - \overline{x}_2 \right) + Z_{1-\alpha}\sqrt{\frac{\sigma_1^2}{n_1}+\frac{\sigma_2^2}{n_2}}, +\infty \right) \\ \hline \end{array} \]

Varianzas desconocidas

Iguales

\[ \begin{array}{|c|c|c|} \hline \text{Hipótesis nula ($H_0$)} & \text{Hipótesis alternativa ($H_1$)} & \text{Región de rechazo ($H_0$)} \\ \hline H_0: \mu \geq \mu_0 \leftrightarrow \mu_1 - \mu_2 \geq 0 & H_1: \mu < \mu_0 \leftrightarrow \mu_1 - \mu_2 < 0 & \left( -\infty, \left( \overline{x}_1 - \overline{x}_2 \right) - t_{\left(n-1,1-\alpha\right)}s_p\sqrt{\frac{1}{n_1}+\frac{1}{n_2}} \right) \\ \hline H_0: \mu = \mu_0 \leftrightarrow \mu_1 - \mu_2 = 0 & H_1: \mu \neq \mu_0 \leftrightarrow \mu_1 - \mu_2 \neq 0 & \left( -\infty, \left( \overline{x}_1 - \overline{x}_2 \right) - t_{\left(n-1,1-\frac{\alpha}{2}\right)}s_p\sqrt{\frac{1}{n_1}+\frac{1}{n_2}} \right) \cup \left( \left( \overline{x}_1 - \overline{x}_2 \right) + t_{\left(n-1,1-\frac{\alpha}{2}\right)}s_p\sqrt{\frac{1}{n_1}+\frac{1}{n_2}}, +\infty \right) \\ \hline H_0: \mu \leq \mu_0 \leftrightarrow \mu_1 - \mu_2 \leq 0 & H_1: \mu > \mu_0 \leftrightarrow \mu_1 - \mu_2 > 0 & \left( \left( \overline{x}_1 - \overline{x}_2 \right) + t_{\left(n-1,1-\alpha\right)}s_p\sqrt{\frac{1}{n_1}+\frac{1}{n_2}}, +\infty \right) \\ \hline \end{array} \]

donde: \(s_p = \sqrt{\frac{(n_1 - 1)s_1^2 + (n_2 - 1)s_2^2}{n_1 + n_2 - 2}}\)

Diferentes

\[ \begin{array}{|c|c|c|} \hline \text{Hipótesis nula ($H_0$)} & \text{Hipótesis alternativa ($H_1$)} & \text{Región de rechazo ($H_0$)} \\ \hline H_0: \mu \geq \mu_0 \leftrightarrow \mu_1 - \mu_2 \geq 0 & H_1: \mu < \mu_0 \leftrightarrow \mu_1 - \mu_2 < 0 & \left( -\infty, \left( \overline{x}_1 - \overline{x}_2 \right) - t_{\left(n-1,1-\alpha\right)}\sqrt{\frac{s_1^2}{n_1}+\frac{s_2^2}{n_2}} \right) \\ \hline H_0: \mu = \mu_0 \leftrightarrow \mu_1 - \mu_2 = 0 & H_1: \mu \neq \mu_0 \leftrightarrow \mu_1 - \mu_2 \neq 0 & \left( -\infty, \left( \overline{x}_1 - \overline{x}_2 \right) - t_{\left(n-1,1-\frac{\alpha}{2}\right)}\sqrt{\frac{s_1^2}{n_1}+\frac{s_2^2}{n_2}} \right) \cup \left( \left( \overline{x}_1 - \overline{x}_2 \right) + t_{\left(n-1,1-\frac{\alpha}{2}\right)}\sqrt{\frac{s_1^2}{n_1}+\frac{s_2^2}{n_2}}, +\infty \right) \\ \hline H_0: \mu \leq \mu_0 \leftrightarrow \mu_1 - \mu_2 \leq 0 & H_1: \mu > \mu_0 \leftrightarrow \mu_1 - \mu_2 > 0 & \left( \left( \overline{x}_1 - \overline{x}_2 \right) + t_{\left(n-1,1-\alpha\right)}\sqrt{\frac{s_1^2}{n_1}+\frac{s_2^2}{n_2}}, +\infty \right) \\ \hline \end{array} \]

Varianzas desconocidas \(n{\rightarrow}\infty\)

Iguales

\[ \begin{array}{|c|c|c|} \hline \text{Hipótesis nula ($H_0$)} & \text{Hipótesis alternativa ($H_1$)} & \text{Región de rechazo ($H_0$)} \\ \hline H_0: \mu \geq \mu_0 \leftrightarrow \mu_1 - \mu_2 \geq 0 & H_1: \mu < \mu_0 \leftrightarrow \mu_1 - \mu_2 < 0 & \left( -\infty, \left( \overline{x}_1 - \overline{x}_2 \right) - Z_{1-\alpha}s_p\sqrt{\frac{1}{n_1}+\frac{1}{n_2}} \right) \\ \hline H_0: \mu = \mu_0 \leftrightarrow \mu_1 - \mu_2 = 0 & H_1: \mu \neq \mu_0 \leftrightarrow \mu_1 - \mu_2 \neq 0 & \left( -\infty, \left( \overline{x}_1 - \overline{x}_2 \right) - Z_{1-\frac{\alpha}{2}}s_p\sqrt{\frac{1}{n_1}+\frac{1}{n_2}} \right) \cup \left( \left( \overline{x}_1 - \overline{x}_2 \right) + Z_{1-\frac{\alpha}{2}}s_p\sqrt{\frac{1}{n_1}+\frac{1}{n_2}}, +\infty \right) \\ \hline H_0: \mu \leq \mu_0 \leftrightarrow \mu_1 - \mu_2 \leq 0 & H_1: \mu > \mu_0 \leftrightarrow \mu_1 - \mu_2 > 0 & \left( \left( \overline{x}_1 - \overline{x}_2 \right) + Z_{1-\alpha}s_p\sqrt{\frac{1}{n_1}+\frac{1}{n_2}}, +\infty \right) \\ \hline \end{array} \]

donde: \(s_p = \sqrt{\frac{(n_1 - 1)s_1^2 + (n_2 - 1)s_2^2}{n_1 + n_2 - 2}}\)

Diferentes

\[ \begin{array}{|c|c|c|} \hline \text{Hipótesis nula ($H_0$)} & \text{Hipótesis alternativa ($H_1$)} & \text{Región de rechazo ($H_0$)} \\ \hline H_0: \mu \geq \mu_0 \leftrightarrow \mu_1 - \mu_2 \geq 0 & H_1: \mu < \mu_0 \leftrightarrow \mu_1 - \mu_2 < 0 & \left( -\infty, \left( \overline{x}_1 - \overline{x}_2 \right) - Z_{1-\alpha}\sqrt{\frac{s_1^2}{n_1}+\frac{s_2^2}{n_2}} \right) \\ \hline H_0: \mu = \mu_0 \leftrightarrow \mu_1 - \mu_2 = 0 & H_1: \mu \neq \mu_0 \leftrightarrow \mu_1 - \mu_2 \neq 0 & \left( -\infty, \left( \overline{x}_1 - \overline{x}_2 \right) - Z_{1-\frac{\alpha}{2}}\sqrt{\frac{s_1^2}{n_1}+\frac{s_2^2}{n_2}} \right) \cup \left( \left( \overline{x}_1 - \overline{x}_2 \right) + Z_{1-\frac{\alpha}{2}}\sqrt{\frac{s_1^2}{n_1}+\frac{s_2^2}{n_2}}, +\infty \right) \\ \hline H_0: \mu \leq \mu_0 \leftrightarrow \mu_1 - \mu_2 \leq 0 & H_1: \mu > \mu_0 \leftrightarrow \mu_1 - \mu_2 > 0 & \left( \left( \overline{x}_1 - \overline{x}_2 \right) + Z_{1-\alpha}\sqrt{\frac{s_1^2}{n_1}+\frac{s_2^2}{n_2}}, +\infty \right) \\ \hline \end{array} \]

Ejercicios y ejemplos

Supongamos una población de trabajadores con un salario mínimo de $1.350.000 pesos, donde se desea comparar los salarios de hombres y mujeres. Se asume que el salario de los hombres tiene una varianza de $30.000, mientras que el de las mujeres tiene una varianza de $20.000.

Simularemos los salarios de ambos grupos, asumiendo que los hombres tienen un salario medio de $1.400.000 y las mujeres $1.300.000. Tomaremos muestras de cada grupo y realizaremos una prueba de hipótesis para determinar si hay evidencia suficiente para rechazar la hipótesis nula de que las varianzas de los salarios de hombres y mujeres son iguales.

Carga de librerías

library(tidyverse)
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library(mosaic)
## Registered S3 method overwritten by 'mosaic':
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## 
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Simulación de la población

# Fijar los parámetros poblacionales
mu_hombres <- 1400000
sigma2_hombres <- 30000
mu_mujeres <- 1300000
sigma2_mujeres <- 20000

# Simular la población
set.seed(123)  # Para reproducibilidad
salarios_hombres <- rnorm(n=125000, mean=mu_hombres, sd=sqrt(sigma2_hombres))
salarios_mujeres <- rnorm(n=125000, mean=mu_mujeres, sd=sqrt(sigma2_mujeres))

# Crear un data frame
INGRESOS <- data.frame(
  tipo = rep(c("Hombres", "Mujeres"), each = 125000),
  salarios = c(salarios_hombres, salarios_mujeres)
)

Distribución de la población

INGRESOS %>% 
  ggplot(mapping=aes(x=tipo, y=salarios)) +
  geom_boxplot(colour="purple", fill="blue") +
  labs(title="Boxplot de Salarios", y="Salarios", x="")

Vamos a seleccionar una muestra aleatoria

muestra <- INGRESOS %>% 
  sample_frac(
    size=0.3
  )

Vamos a probar hipótesis

\[ \begin{array}{|c|c|c|} \hline \text{Hipótesis nula ($H_0$)} & \text{Hipótesis alternativa ($H_1$)} & \text{Región de rechazo ($H_0$)} \\ \hline H_0: \sigma_1^2 = \sigma_2^2 \leftrightarrow \frac{\sigma_1^2}{\sigma_2^2} = 1 & H_1: \sigma_1^2 \neq \sigma_2^2 \leftrightarrow \frac{\sigma_1^2}{\sigma_2^2} \neq 1 & \left( 0, F_{\alpha/2, n_1-1, n_2-1} \right) \cup \left( F_{1-\alpha/2, n_1-1, n_2-1}, +\infty \right) \\ \hline \end{array} \]

# Realizar la prueba de hipótesis (Prueba F)
prueba_f <- var.test(
  x=muestra[muestra$tipo=="Hombres", "salarios"],
  y=muestra[muestra$tipo=="Mujeres","salarios"],
  ratio = 1,
  alternative = "two.sided",
  conf.level = 0.95
)

# Resultados
print(prueba_f)
## 
##  F test to compare two variances
## 
## data:  muestra[muestra$tipo == "Hombres", "salarios"] and muestra[muestra$tipo == "Mujeres", "salarios"]
## F = 1.4862, num df = 37441, denom df = 37557, p-value < 2.2e-16
## alternative hypothesis: true ratio of variances is not equal to 1
## 95 percent confidence interval:
##  1.456464 1.516640
## sample estimates:
## ratio of variances 
##           1.486247