The central limit theorem plays an important role in the sampling distribution. This theorem has important implication for constructing confidence intervals and testing hypotheses.
Central Limit Theorem: For a population described by any probability distribution having mean µ and finite variance σ2, the sampling distribution of the sample mean X ̅ computed from samples of size n from the population will be approximately normal when the sample size is large (n≥30). This sampling distribution will have mean µ and variance σ2/n.
It makes it easy to make inference about the population mean by using the sample mean irrespective of the distribution of the population as long as it has a finite variance. It is because the sample mean follows an approximately normal distribution for large sample sizes.
As long as our sample size is large (n≥30), then according to the central limit theorem:
- The distribution of the sample mean X ̅ will be approximately normal.
- The mean of the distribution of X ̅ will be equal to the mean of the population (µ) from which the samples are drawn
The variance of the distribution of X will be equal to the variance of the population divided by the sample size (σ2/n).
Previous LOS: Time-series and cross-sectional data
Next LOS: Standard error of the sample mean
