Sampling Distributions of Sample Means

Sample Mean
Let the random variables X1, X2,……….,Xn  denote a random sample from a population. The sample mean value of these random variables is defined as

Consider the sampling distribution of the random variable . At this point we cannot determine the shape of the sampling distribution, but we can determine the mean and variance of the sampling distribution. We know that the expectation of a linear combination of random variables is the linear combination of the expectations:

Thus, the mean of the sampling distribution of the sample means is the population mean. If samples of n random and independent observations are repeatedly and independently drawn from a population, then as the number of samples becomes very large, the mean of the sample means approaches the true population mean.
The variance of the sample mean is denoted by  , is given by and the corresponding standard deviation, called the standard error of

Standard Normal Distribution for the Sample Means

Whenever the sampling distribution of the sample means is a normal distribution, we can compute a standardized normal random variable, Z, that has mean 0 and variance 1:

Example: Suppose that the annual percentage salary increases for the chief executive officers of all midsize corporations are normally distributed with mean 12.2% and standard deviation 3.6%. A random sample of nine observations is obtained from this population and the sample mean computed. What is the probability that the sample mean will be less than 10%?

Example: A spark plug manufacturer claims that the lives of its plugs are normally distributed with mean 36,000 miles and standard deviation 4,000 miles. A random sample of 16 plugs had an average life of 34,500 miles. If the manufacture’s claim is correct, what is the probability of finding a sample mean of 34,500 or less?