STAT350 Chap 7

patterns

1. If the underlying population is normal, the distribution of the sample mean appears to be normal, regardless of the sample size. 2. Even if the underlying population is not normal, the distribution of the sample mean becomes more normal as n increases. 3. The sampling distribution of the mean is centered at the mean of the underlying population. 4. As the sample size, n, increases, the variance of the distribution of the sample mean decreases.

parameter

A numerical descriptive measure of a population.

unbiased estimator

A statistic is an unbiased estimator of a population parameter if the mean of the statistic is equal to the population parameter.

statistic

Any quantity computed from values in a sample.

Central Limit Theorem

As the sample size n increases, the sampling distribution of ̄ will increasingly approximate a normal distribution, with mean µ and variance σ2/n, regardless of the shape of the underlying population distribution.

sampling distribution

The probability distribution of a statistic.

We can use either of two techniques to obtain, or find, a sampling distribution.

a. Recall that to approximate the distribution of a population, we construct a histogram (or stem-and-leaf plot) using values from the population. If the sample is representative, then the histogram should be similar in shape, center, and spread to the population distribution. Similarly, to approximate the distribution of a statistic, we obtain (many) values of the statistic and construct a histogram. The resulting graph approximates the sampling distribution of the statistic. For example, to approximate the sampling distribution of the mean of a sample of size n = 10 from a population: (i) obtain several samples of size 10 from the population; (ii) compute the sample mean for each sample; (iii) construct a histogram using all the sample means. The histogram approximates the sampling distribution of the sample mean. b. In some cases, the exact sampling distribution of a statistic can be obtained. If the statistic is a discrete random variable, the sampling distribution includes all the values the statistic assumes and the associated probabilities. If the statistic is a continuous random variable, the sampling distribution consists of a probability density curve.