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Chapter

Eight

McGraw-Hill/Irwin

© 2005 The McGraw-Hill Companies, Inc., All Rights Reserved.

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Chapter EightSampling Methods and the Central Sampling Methods and the Central Limit TheoremLimit TheoremGOALSWhen you have completed this chapter, you will be able to:ONEExplain why a sample is the only feasible way to learn about a population.

TWO Describe methods to select a sample.

THREEDefine and construct a sampling distribution of the sample mean.

FOURExplain the central limit theorem. Goals

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Chapter Eight continuedSampling Methods and the Sampling Methods and the Central Limit TheoremCentral Limit Theorem

GOALSWhen you have completed this chapter, you will be able to:FIVE Use the Central Limit Theorem to find probabilities of selecting possible sample means from a specified population.

Goals

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Why Sample the Population?

Why sample?

The destructive nature of certain tests.

The physical impossibility of checking all items in the population.

The cost of studying all the items in a

population.

The adequacy of sample results in most cases.

The time-consuming aspect of contacting the whole population.

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Probability Sampling/Methods

Systematic Random Sampling The items or individuals of the population are arranged in some order. A random starting point is selected and then every kth member of the population is selected for the sample.

Simple Random Sample A sample formulated so that each item or person in the population has the same chance of being included.

A probability sample is a sample selected such that each item or person in the population being studied has a known likelihood of being included in the sample.

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Methods of Probability Sampling

Stratified Random Sampling: A population is first divided into subgroups, called strata, and a sample is selected from each stratum.

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Cluster Sampling

Cluster Sampling: A population is first divided into primary units then samples are selected from the primary units.

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Methods of Probability Sampling

The sampling error is the difference between a sample statistic and its corresponding population parameter.

In nonprobability sample inclusion in the sample is based on the judgment of the person selecting the sample.

The sampling distribution of the sample mean is a probability distribution consisting of all possible sample means of a given sample size selected from a population.

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Example 1

Partner Hours

Dunn 22

Hardy 26

Kiers 30

Malory 26

Tillman 22

The law firm of Hoya and Associates has five partners. At their weekly partners meeting each reported the number of hours they billed clients for their services last week.

If two partners are selected randomly, how many different samples are possible?

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Example 1

10)!25(!2

!525

C

5 objects taken 2 at a time.

A total of 10 different samples

Partners Total Mean 1,2 48 24 1,3 52 26 1,4 48 24 1,5 44 22 2,3 56 28 2,4 52 26 2,5 48 24 3,4 56 28 3,5 52 26 4,5 48 24

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Example 1 continued

Sample Mean Frequency Relative Frequency probability

22 1 1/10

24 4 4/10

26 3 3/10

28 2 2/10

As a sampling distribution

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Example 1 continued

2.2510

)2(28)3(26)2(24)1(22

X

Compute the mean of the sample means. Compare it with the population mean.

The mean of the sample means

The population mean

2.255

2226302622

Notice that the mean of the sample means is exactly equal to the population mean.

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Central Limit Theorem x =

n

For a population with a mean and a variance 2 the sampling distribution of the means of all possible samples of size n generated from the population will be approximately normally distributed.

This approximation improves with larger samples.The mean of the sampling distribution equal to m and the variance equal to 2/n.

The standard error of the mean is the standard deviation of the standard deviation of the sample means given as:

Central Limit TheoremCentral Limit Theorem

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Sample Means

the sample size is large enough even when the underlying population

may be nonnormal

Sample means follow the normal probability distribution under two conditions:

the underlying population follows the normal

distribution

OR

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ns

Xz

Use in place of s if the population standard deviation is known.

Sample Means

To determine the probability that a sample

mean falls within a particular region, use

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Example 2

Suppose the mean selling price of a gallon of gasoline in the United States is $1.30. Further, assume the distribution is positively skewed, with a standard deviation of $0.28. What is the probability of selecting a sample of 35 gasoline stations and finding the sample mean within $.08?

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Example 2 continued

69.13528.0$

30.1$38.1$

ns

Xz

69.13528.0$

30.1$22.1$

ns

Xz

Step One : Find the z-values corresponding to $1.24 and $1.36. These are the two points within $0.08 of the population mean.

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Example 2 continued

9090.)4545(.2)69.169.1( zP

Step Two: determine the probability of a z-value between -1.69 and 1.69.

We would expect about 91 percent of the sample

means to be within $0.08 of the population mean.