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Chapter 7Chapter 7

Created by Bethany Stubbe and Stephan Kogitz

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Chapter Seven

Sampling Methods and the Sampling Methods and the Central Limit TheoremCentral Limit Theorem

ONEExplain why a sample is often the only feasible way to learn something 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.

GOALSWhen you have completed this chapter, you will be able to:

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FIVE Use the Central Limit Theorem to find probabilities of selecting possible sample means from a specified population.

Chapter Seven continued

Sampling Methods and the Sampling Methods and the Central Limit TheoremCentral Limit TheoremGOALSWhen you have completed this chapter, you will be able to:

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Why Sample the Population?

To contact the whole population would often be time consuming.

The cost of studying all the items in a population is often prohibitive.

The adequacy of sample results.

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Why Sample the Population

The destructive nature of certain tests.

The physical impossibility of checking all items in the population.

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Probability Sampling

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

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Methods of Probability Sampling

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

Systematic Random Sampling: A random starting point is selected and then every kth member of the population is selected.

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Methods of Probability Sampling

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

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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Methods of Probability Sampling

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

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The Sampling Distribution of the Sample Mean

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

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Sampling Distribution of the Sample Mean

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

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EXAMPLE 1The law firm of Tybo and Associates has five partners. At

their weekly partners meeting each reported the number of hours they billed clients for their services last week.

Partner Hours

1. Dunn 22

2. Hardy 26

3. Kiers 30

4. Malinowski 26

5. Tillman 22

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Example 1

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

There are 10 different samples. This is the combination of 5 objects taken 2 at a time.

10)!25(!2

!525

C

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Example 1 continued

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

Organize the sample means into a sampling distribution.SampleMean

Frequency RelativeFrequencyprobability

22 1 1/10

24 4 4/10

26 3 3/10

28 2 2/10

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EXAMPLE 1 continued

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

The mean of the sample means is 25.2 hours.

2.2510

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

X

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Example 1 continued

The population mean is also 25.2 hours.

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

If all samples of a specified size are selected from any population, the sampling distribution of the sample mean is approximately a normal distribution. This approximation improves with larger samples.

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Mean of the Sample Means

The mean of the distribution of the sample mean will be exactly equal to the population mean if we are able to select all possible samples of a particular size from the a given population.

X

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Standard Error of the Mean

If the standard deviation of the population is σ, the standard deviation of the distribution of the sample mean is

nx /

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Sampling from a Normal Population

If a population follows the normal distribution, the sampling distribution of the sample mean will also follow the normal distribution.

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Finding the z Value of When σ is known

To determine the probability a sample mean falls within a particular region, use:

n

Xz

If the population is normally distributed

Assume the standard deviation, σ, is known.

X

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Finding the z Value of When σ is Unknown

If the population is not normally distributed, but the sample is at least 30 observations, the sampling distribution of the sample mean is approximately normal.

Assume the population standard deviation is not known, use the sample standard deviation.

X

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Finding the z Value of When σ is Unknown continued

To determine the probability a sample mean falls within a particular region, use:

ns

Xz

If the population standard deviation, σ, is unknown:

X

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Example 2

The mean selling price of 100 ml. Tube of toothpaste is $1.30. The distribution is positively skewed, with a standard deviation of $0.28. What is the probability of selecting a sample of 35 stores and finding the sample mean within $.08?

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Example 2 continued

The first step is to find the z-values corresponding to $1.24 and $1.36. These are the two points within $0.08 of the population mean.

69.13528.0$

30.1$38.1$

ns

Xz

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Example 2 continued

69.13528.0$

30.1$22.1$

ns

Xz

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Example 2 continued

Next we determine the probability of a z-value between -1.69 and 1.69. It is:

9090.)4545(.2)69.169.1( zP

We would expect about 91 percent of the sample means to be within $0.08 of the population mean.