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• The population distribution of a variable is the distribution of its values for all members of the population.
• The sample mean of an SRS of size n drawn from a large population with mean μ and standard deviation σ has a sampling distribution with mean and standard deviation
The sample mean is an unbiased estimator of the population mean μ and is less variable than a single observation. The standard deviation decreases in proportion to the square root of the sample size n. This means that to reduce the standard deviation by a factor of C, we need to increase the sample size by a factor of C2.
The central limit theorem states that, for large n, the sampling distribution of is approximately for any population with mean μ and finite standard deviation σ. This allows us to approximate probability calculations of using the Normal distribution.
Linear combinations of independent Normal random variables have Normal distributions. In particular, if the population has a Normal distribution, so does .