Chapter 7 Sampling and Sampling Distributions I Introduction

Chapter 7: Sampling and Sampling Distributions

I. Introduction Virtually anything we know in life is based on some kind of sample. We live life as one big sample and make generalizations based on our experiences which are necessarily limited and biased. Recall “statistical inference” means that we obtain information about a population from information contained in a sample.

A basic question must be asked. Because our sample information isn’t perfect, everything relies upon probability to guide us in determining how much confidence we can place in our estimates. Basic question: “What are the limits within which the findings of a particular sample can probably be relied on as a basis for generalizing to a larger, even infinite population? ”

An Example A running example used in the text book will allow us to develop several sampling concepts. EAI is a company that wishes to compile a profile of 2500 managers. We can consider this the population. The mean salary ( ) is $51, 800 and the standard deviation ( ) is $4000. Furthermore, 1500/2500 (p=. 60) of the managers have completed a training program.

Parameters =$4000 =$51, 800 p=. 60 These are population parameters, or characteristics. • Can a personnel director get an accurate profile of the population from a sample of n=30? • •

II. Simple Random Sampling One of the most common, and accurate methods for taking a sample is to do so randomly. How would you take a random sample of: • College students? • Voters? • Internet users? • Sports fans? • Shoppers at a mall?

A. Finite Population A sample of size n from a finite population of size N is a sample selected such that each possible sample of size n has the same probability of being selected.

How? Maybe the personnel manager puts 2500 names in a hat and randomly draws one. Mixes up the names and draws again until n=30 have been selected. This is sampling, without replacement. Or maybe they could use a table of random numbers or a computer that generates 30 random numbers.

B. Infinite Population A sample from an infinite population is a sample selected such that the following conditions are satisfied: 1. Each element is selected from the same population. 2. Each element is selected independently.

Lots of possibilities! How many different simple random samples of size 30 can be selected from a finite population of size 2500? Use the combination rule!

How? Suppose a fast food restaurant studies the customer service time between 11: 30 a. m. and 1: 30 p. m. . They don’t know N, the # of customers that will arrive during those hours. But if they time every 10 th customer until they have a large sample size, this method should satisfy the earlier conditions. Can you think of another way to sample for this problem?

III. Point Estimation Use data from the sample to compute sample statistics to serve as estimates of the population parameters. One sample yields: s = $3347. 72 These are point estimators of , , and p. Remember this is just one sample!