Chapter 4 (cont.) The Sampling Distribution

What is the sampling distribution? The sampling distribution is a theoretical probability distribution that allows us to determine the probabilities of possible values associated with a sample statistic such as a sample mean or a sample proportion. Every sample statistic has its own sampling distribution. Works on the same principles as the normal probability distribution.

Logic And Terminology Problem: The populations we wish to study are almost always so large that we are unable to gather information from every case. Solution: We choose a random sample -- a carefully chosen subset of the population – and use information gathered from the cases in the sample to generalize to the population.

Remember: Statistics are mathematical characteristics of samples. Parameters are mathematical characteristics of populations. Statistics are used to estimate parameters. The Sampling Distribution is the link between a statistic and a parameter. PARAMETER STATISTIC

The Sampling Distribution We can use the sampling distribution to calculate our population parameter based on our sample statistic. The single most important concept in inferential statistics. It is the distribution of a statistic for all possible samples of a given size (N). The sampling distribution is a theoretical concept based on the principles of the normal curve.

The Sampling Distribution Every application of inferential statistics involves 3 different distributions. Information from the sample is linked to the population via the sampling distribution. Population Sampling Distribution Sample

The Sampling Distribution of 1. Normal in shape. 2. Has a mean equal to the population mean. 3. Has a standard deviation (standard error) equal to the population standard deviation divided by the square root of N.

Sampling Distribution of a Proportion Mean of the distribution: S.E.: Note that as N increases, S.E. decreases

Central Limit Theorem For any trait or variable, even those that are not normally distributed in the population, as sample size grows larger, the sampling distribution of sample means will become normal in shape. The importance of the Central Limit Theorem is that it removes the constraint of normality in the population. When n>30, the distribution is approximately normal.

The Sampling Distribution Summary The Sampling Distribution is always normal so we can use Table A (z-table) or to find areas. We do not know the value of the population mean (μ) but the mean of the Sampling Distribution is the same value as μ (pop. mean). We do not know the value of the pop. standard deviation (σ) but the Stnd. Dev. of the S.D. is equal to σ divided by the square root of N. We can substitute s for σ

Sampling Distribution Three Distributions Shape Central Tendency Dispersion Sample Varies _ X s Sampling Distribution Normal μy=μ σy= σ/√N Population μ σ