Estimating the Population Mean Income of Lexus Owners

1
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.

2
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?

3
Assume Small Population of Lexus Owners Incomes
(N 200)
4
Distribution of N 200 Incomes
30
75 125 175 225 275 325
5
Constructing a Distribution of
Samples of Size 5 from N 200 Owners
6
Distribution of Sample Mean Incomes (Column 7)
Distribution of Sample Means Near Normal!
7
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

8
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 Relatively Far
  Above or Below Population Mean.
• A Few Sample Means will be Either Very Far Above
  or Below Population Mean.

9
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)

10
Drawing Conclusions about a Pop. Mean Using a
Sample Mean
Select Simple Random Sample
Compute Sample Mean and Std. Dev. For n lt 10,
Sample Bell-Shaped? For n gt10 CLT Ensures Dist of
Normal
Draw Conclusion about Population Mean, m
11
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.

12
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
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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.

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Estimated Standard Error Expression
15
Confidence Factor for MOE Appendix 5
Can Use t-Table Provided Distribution of
Sample Mean is Normal
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95 Confidence Interval
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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???

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Would Tract Be Eligible?

• Situation A 7,700 150
• Situation B 8,250 150
• Situation C 8,050 150

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Width versus Meaningfulness of Two-Sided
Confidence Intervals
Ideal _________ Level of Confidence and
_________ Confidence Interval . How Obtain?
20
Chapter Summary

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