- Create a sampling distribution given [latex]\mu[/latex] and [latex]n[/latex].
- Know and check the conditions of the Central Limit Theorem.
- Use the normal approximation to compute probabilities involving sample means when appropriate.
mean and standard deviation of the sampling distribution
The mathematical formulas to find the mean and standard deviation of the sampling distribution of the sample mean for samples of size [latex]n[/latex] are:
- Mean of the sampling distribution of the sample mean [latex]=\mu[/latex]
- Standard deviation of the sampling distribution of the sample mean [latex]=\frac{\sigma}{\sqrt{n}}[/latex]
where [latex]\mu[/latex] and [latex]\sigma[/latex] represent the mean and standard deviation of the original population, respectively.
You should notice that the mean and standard deviation calculated above is nearly the same as the mean and standard deviation found using the simulation of [latex]1,000[/latex] random samples. You just witnessed the Central Limit Theorem at work for sample means. The Central Limit Theorem states that, as the sample size gets larger, the distribution of the sample mean will become closer to a normal distribution.
Central Limit Theorem
If the population distribution is normal, the distribution of the sample means will also follow a normal distribution, for any sample size.
If the population distribution is not normal, the Central Limit Theorem states that the distribution of the sample means still follows an approximate normal distribution as long as the sample size is large (e.g., [latex]n \ge 30[/latex]) and the population distribution is not strongly skewed.