- 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.
When sampling from a normal population, such as SAT scores, the distribution of the sample means will also have a normal distribution with the same mean, but the variability in sample means will be less than the variability in individuals. This is similar to how there is less variability in sample proportions than the variability in individuals.
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.
Note:
- 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.
Now that we have found the mean and standard deviation of the sampling distribution of the sample mean, we can use it to calculate probability and make inferences about the population.