Business Economis

This chapter continues the study of probability distributions by examining continuous probability distributions. We learn about three continuous probability distributions: the uniform probability distribution, the normal probability distribution, and the exponential probability distribution.

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Learning Objectives

8-1 Explain why populations are sampled and describe four methods to sample a population

8-2 Define sampling error

8-3 Demonstrate the construction of a sampling distribution of the sample mean

8-4 Recite the central limit theorem and define the mean and standard error of the sampling distribution of the sample mean

8-5 Apply the central limit theorem to calculate probabilities

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Reasons for Sampling a Population

The results of a sample may adequately estimate the value of the population parameter, saving time and money

It may be too time-consuming to contact all members of the population

It may be impossible to check or locate all the members of the population

The cost of studying all the items in the population may be prohibitive

Often testing destroys the sampled item and it cannot be returned to the population

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There are many practical reasons why we prefer to select portions or samples of a population to observe and measure when studying characteristics of a population.

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Probability Sampling Methods

In a simple random sample, all members of the population have the same chance of being selected for the sample

In a systematic sample, a random starting point is selected, and then every kth item thereafter is selected for the sample

In a stratified sample, the population is divided into several groups, called strata, and then a random sample is selected from each stratum

In clustered sampling, the population is divided into primary units, then samples are drawn from the primary units

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These are not the only methods of sampling available to a researcher. If you become involved in a major research project, consult books devoted to sample theory and sample design.

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Simple Random Sampling (1 of 2)

The most widely used method of sampling is a simple random sample

SIMPLE RANDOM SAMPLE A sample selected so that each item or person in the population has the same chance of being selected.

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In an unbiased sample, all members of the population have a chance of being selected for the sample. If the population is relatively small, one could place all the items of interest, say names, in a box and select the sample. For large samples like the one in this example, this is very time-consuming.

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Simple Random Sampling (2 of 2)

Example

There were 750 Major League Baseball players at the end of the 2016 season. A committee of 10 players is to be formed to study the issue of concussions. To make sure every player has an equal chance of being selected, write each name on a piece of paper, place the names in a box and mix them up, then draw 10 names.

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Using a Table of Random Numbers (1 of 2)

Suppose the population of interest is the 750 Major League Baseball players on the active rosters of the 30 teams at the end of the 2016 season. A committee of 10 players is to be formed to study the issue of concussions. To make sure every player has an equal chance of being selected, use a table of random numbers.

Prepare of list of all the players and number them 1 through 750

Randomly pick a starting place in the random number table

Select 10 three-digit numbers between 1 and 750

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Other ways to select a sample are to use a table of random numbers like the one in Appendix B.4 or to use a statistical software program or Excel to select a simple random sample. This example shows how to select random numbers using a portion of a random number table. To choose a starting point, you could close your eyes and simply point at a number in the table. Another way is to randomly pick a column and a row. Suppose the time is 3:04; go to column three and down to row four. Here, the number is 03759, so we use the first three digits in the five digit number. To continue selecting, you could move in any direction, in this example, we move right. The next number is 44723, so we choose player 447, the next number is 961 and is too high so we skip it and we skip the next number, 784 because it is also too high. Then the next number is 18910, so we select player 189 and so on until we have a list of 10 players. The instructions on how to use Excel to select a random sample are in Appendix C.

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Using a Table of Random Numbers (2 of 2)

In column 3: 037 (Starting point)

In column 4: 447 (Second player)

In column 7: 189 (Third player)

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Systematic Random Sampling (1 of 2)

If you do not have a list of the entire population to begin with, you can use the systematic random sample

SYSTEMATIC RANDOM SAMPLE A random starting point is selected, and then every kth member of the population is selected.

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Caution: If the population is in some order already, like invoices arranged in increasing dollar amounts, the systematic procedure should not be used.

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Systematic Random Sampling (2 of 2)

Example

Stood’s Grocery Store wants to study the length of time customers spend in their store

Randomly select the days of the week, the times, and the starting point of the study, then systematically select the customers and measure the time each spends in the store

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Stratified Random Sampling (1 of 2)

When the population can be divided into groups based on some characteristic, use stratified random sampling

STRATIFIED RANDOM SAMPLE A population is divided into subgroups, called strata, and a sample is randomly selected from each stratum.

Example

A study of 50 of the 352 largest US firms’ ad spending

Begin by identifying the strata, then use random sampling within each group based on relative frequencies to collect the sample

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This example of stratified random sampling groups the firms by profitability, measured by the percent return on equity. The advantage of stratified random sampling is that this method may more accurately reflect the population characteristic since at least one firm will be selected from each strata, even those with low numbers; which might not happen when using random sampling or systematic random sampling.

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Stratified Random Sampling (2 of 2)

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Cluster Sampling (1 of 2)

Cluster sampling is a common type of sampling, used to reduce the cost of sampling over large geographic areas

CLUSTER SAMPLING A population is divided into clusters using naturally occurring geographic or other boundaries. Then clusters are randomly selected and a sample is collected by randomly selecting from each cluster.

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In this example, we wish to determine the views of residents in the greater Chicago area about state and federal environmental protection policies. Selecting a random sample of residents in the region and personally contacting each one would be time-consuming and expensive. Instead, let the counties serve as the primary unit and select a random sample of the counties and then each of the selected county’s residents. This is a combination of cluster sampling and simple random sampling.

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Cluster Sampling (2 of 2)

Example

Suppose we wish to sample residents of the 12 counties in the greater Chicago area about government policy. Randomly select 3 counties and then select a random sample of the residents in each of the 3 counties.