Suppose a sample of n values is randomly selected from some population of values. These n values are then averaged. The average of the n values is itself subject to randomness, since each of the values that make up the average is random. Suppose we sample n values from the population 100 times, each time averaging the n values to obtain the sample average. What is average value of these “averages”? What is the variability of these averages? What is their distribution?
We are interested in the sampling distribution of statistics, where a statistic is any quantity whose value can be calculated from sample data. Statistics can assume random values, thus they are random variables. For example, the sample average, , is a statistic, as is , the sample variance. A random sample of size n is a collection of n independent random variables all having the same probability distribution.
We have the following results:
the sample size is large, the distribution of is approximately the
same as that of a standard
Since the sample proportion of successful items in n trials, X/n, can be thought of as a average of n random variables, each taking value 0 or 1, then the sample proportion has an approximate Normal distribution, with mean equal to the true proportion of successes in the population, p, and variability equal to p(1-p)/n. This approximation is good when if both np greater than or equal to ten and n(1-p) greater than or equal to ten.
The mean of a linear combination of random variables is equal to the linear combination of the means of the individual random variables.
The variance of a linear combination of independent random variables is equal to the linear combination of the variances of the individual random variables, with scale factors in the linear combination squared.
distribution of a linear combination of independent, Normally distributed random variables is also
Normal. For example, if X1 and X2 are independent Normal r.v.s with means m1 and m2 and
variances v1 and v2, respectively, then the distribution
of X1-X2 is