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. 94 94 94 94 94 93 93 93 93 89

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? 94 94 94

3. Assume Small Population of Lexus Owners’ Incomes (N = 200)95 95 95 95 95 90 95 95 95 95

4. Distribution of N = 200 Incomes
Mean 30
1. Range is from 76K to 371K
2. Mean is about 225K (balance point)
3. Distribution of the population is definitely NOT NORMAL or BELL-SHAPED. Slightly over 30 observations in each class

5. Constructing a Distribution of
Samples of Size 5 from N = 200 Owners
Obs 1
Obs 2
Obs 3
Obs 4
Obs 5
Mean
Suppose we took 40 samples of size five from the population. Only 13 samples shown above, 27 not shown.
How many are within 25K of the population mean of 225K? 7/13
For a single sample of five, the incomes range widely. Do for first five samples.
Range for the sample means shown is from 161.2K to 292K = 131
Standard deviation of sample mean. If you took repeated samples of size five here is how you would compute it 97 97 97 97 97 97 97 97 92 97

6. Distribution of Sample Mean Incomes (Column #7)
Estimated Std. Error
Distribution of Sample Means
Near Normal! 98 98 98

7. Central Limit Theorem Thus Can Use Expression:
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: 99 99 99 99 99 99 99 99 99 94

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.
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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)
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10. 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
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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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13. 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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14. Estimated Standard Error Expression
For Federal
Aid Study
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15. Confidence Factor for MOE: Appendix 5
Can Use t-Table Provided Distribution
of Sample Mean is Normal
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16. 95% Confidence Interval
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17. Interpretation of Confidence Interval
95% Confident that Interval $7, $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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18. Would Tract Be Eligible?
Situation A: 7,
Situation B: 8,
Situation C: 8,
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19. Width versus Meaningfulness of Two-Sided Confidence Intervals
Ideal: _________ Level of Confidence and
_________ Confidence Interval .
How Obtain?
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20. Why Must We Estimate Population Mean?
Chapter Summary
Why Must We Estimate Population Mean?
Why Would You Want to Reduce MOE?
How Can MOE Be Reduced Without Lowering Confidence Level?
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