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Estimating the Population Mean Income of Lexus Owners. Sample Mean + Margin of Error Called a Confidence Interval To Compute Margin of Error, One of Two Conditions Must Be True: The Distribution of the Population of Incomes Must Be Normal, or

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Estimating the Population Mean Income of Lexus Owners

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### Estimating the Population Mean Income of Lexus Owners

• Sample Mean + Margin of Error

• Called a Confidence Interval

• To Compute Margin of Error, One of Two Conditions Must Be True:

• The Distribution of the Population of Incomes Must Be Normal, or

• The Distribution of Sample Means Must Be Normal.

### A Side-Trip Before Constructing Confidence Intervals

• What is a Population Distribution?

• What is a Distribution of the Sample Mean?

• How Does Distribution of Sample Mean Differ From a Population Distribution?

• What is the Central Limit Theorem?

Assume Small Population of

Lexus Owners’ Incomes (N = 200)

Mean

### Distribution of N = 200 Incomes

30

75 125 175 225 275 325

Mean

Obs 1

Obs 2

Obs 3

Obs 4

Obs 5

Constructing a Distribution of

Samples of Size 5 from N = 200 Owners

Estimated Std. Error

### Distribution of Sample Mean Incomes (Column #7)

Distribution of Sample Means

Near Normal!

### Central Limit Theorem

• Even if Distribution of Population is Not Normal, Distribution of Sample Mean Will Be Near Normal Provided You Select Sample of Five or Ten or Greater From the Population.

• For a Sample Sizes of 30 or More, Dist. of the Sample Mean Will Be Normal, with

• mean of sample means = population mean, and

• standard error = [population deviation] / [sqrt(n)]

• Thus Can Use Expression:

### Why Does Central Limit Theorem Work?

• As Sample Size Increases:

• Most Sample Means will be Close to

Population Mean,

• Some Sample Means will be Either RelativelyFar Above or Below Population Mean.

• A Few Sample Means will be Either Very Far Above or Below Population Mean.

### Impact of Side-Trip on MOE

• Determine Confidence, or Reliability, Factor.

• Distribution of Sample Mean Normal from Central Limit Theorem.

• Use a “Normal-Like Table” to Obtain Confidence Factor.

• Determine Spread in Sample Means (Without Taking Repeated Samples)

Mean Using a Sample Mean

Select Simple Random Sample

Compute Sample Mean and

Std. Dev. For n < 10, Sample Bell-Shaped?

For n >10 CLT Ensures Dist of Normal

Population Mean, m

### Federal Aid Problem

• Suppose a census tract with 5000 families is eligible for aid under program HR-247 if average income of families of 4 is between \$7500 and \$8500 (those lower than 7500 are eligible in a different program). A random sample of 12 families yields data on the next page.

### Federal Aid Study Calculations

Representative Sample

7,300 7,700 8,100 8,400

7,800 8,300 8,500 7,600

7,400 7,800 8,300 8,600

### Estimated Standard Error

• Measures Variation Among the Sample Means If We Took Repeated Samples.

• But We Only Have One Sample! How Can We Compute Estimated Standard Error?

• Based on Constructing Distribution of Sample Mean Slide, Will Estimated Standard Error Be Smaller or Larger Than Sample Standard Deviation (s)?

• Estimated Std. Error ______ than s.

For Federal

Aid Study

### Confidence Factor for MOE: Appendix 5

Can Use t-Table Provided Distribution

of Sample Mean is Normal

### Interpretation of Confidence Interval

• 95% Confident that Interval \$7,983 + \$280 Contains Unknown Population (Not Sample) Mean Income.

• If We Selected 1,000 Samples of Size 12 and Constructed 1,000 Confidence Intervals, about 950 Would Contain Unknown Population Mean and 50 Would Not.

• So Is Tract Eligible for Aid???

### Would Tract Be Eligible?

• Situation A: 7,700+ 150

• Situation B: 8,250+ 150

• Situation C: 8,050+ 150

Width versus Meaningfulness of

Two-Sided Confidence Intervals

Ideal: _________ Level of Confidence and

_________ Confidence Interval .

How Obtain?

### Chapter Summary

• Why Must We Estimate Population Mean?

• Why Would You Want to Reduce MOE?

• How Can MOE Be Reduced Without Lowering Confidence Level?