Teorema del Límite Central
El Teorema del Límite Central (CLT) es un principio fundamental en statistics that describes how the distribution of sample means becomes approximately normal, regardless of the original distribution of the population, as the sample size increases. This theorem is crucial for making inferences about populations based on sample data.
Specifically, the CLT states that if you take sufficiently large random samples from a population, the means of those samples will form a distribución normal, even if the population itself is not normally distributed. The larger the sample size, the closer the sample means will be to a normal distribution. Typically, a sample size of 30 or more is considered sufficient for the CLT to hold.
Matemáticamente, el teorema puede expresarse de la siguiente manera: Si X es una variable aleatoria con media µ y desviación estándar σ, entonces la distribución muestral de la media muestral (denotada como X̄) se acercará a una distribución normal con media µ y desviación estándar σ/√n, donde n es el tamaño de la muestra, a medida que n se acerca a infinito.
Esta propiedad del Teorema del Límite Central es particularmente útil en prueba de hipótesis and intervalo de confianza estimation, as it allows statisticians to use the normal distribution as an approximation for various métodos estadísticos, even when we are working with skewed or non-normal population distributions.
En resumen, el Teorema del Límite Central proporciona una base para estadística inferencial, enabling researchers to make predictions and decisions based on sample data with a high degree of accuracy.