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Estimating the Population Mean Income of Lexus OwnersPowerPoint Presentation

Estimating the Population Mean Income of Lexus Owners

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Estimating the Population Mean Income of Lexus Owners

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- 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.

- 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

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 Means

Near Normal!

- 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:

- 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.

- 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)

Drawing Conclusions about a Pop.

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

Draw Conclusion about

Population Mean, m

- 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.

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

- 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

Can Use t-Table Provided Distribution

of Sample Mean is Normal

- 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???

- 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?

- Why Must We Estimate Population Mean?
- Why Would You Want to Reduce MOE?
- How Can MOE Be Reduced Without Lowering Confidence Level?