1. Basic Business Statistics (8th Edition)
Chapter 7
Sampling Distributions
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2. Why Study Sampling Distributions
Sample statistics are used to estimate population parameters
e.g.: estimates the population mean
Problems: Different samples provide different estimates
Large samples give better estimates; large sample costs more
How good is the estimate?
Approach to solution: Theoretical basis is sampling distribution
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3. Sampling Distribution
Theoretical probability distribution of a sample statistic
Sample statistic is a random variable
Sample mean, sample proportion
Results from taking all possible samples of the same size
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4. Developing Sampling Distributions
Assume there is a population …
Population size N=4
Random variable, X, is age of individuals
Values of X: 18, 20, 22, 24 measured in years C B D A
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5. Developing Sampling Distributions
(continued)
Summary Measures for the Population Distribution
P(X) .3 .2 .1 X
A B C D
(18) (20) (22) (24)
Uniform Distribution
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6. All Possible Samples of Size n=2
Developing Sampling Distributions
(continued)
All Possible Samples of Size n=2
16 Sample Means
16 Samples Taken with Replacement
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7. Sampling Distribution of All Sample Means
Developing Sampling Distributions
(continued)
Sampling Distribution of All Sample Means
Sample Means Distribution
16 Sample Means
P(X) .3 .2 .1 _ X
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8. Summary Measures of Sampling Distribution
Developing Sampling Distributions
(continued)
Summary Measures of Sampling Distribution
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9. Comparing the Population with its Sampling Distribution
Sample Means Distribution
n = 2
Population
N = 4
P(X)
P(X) .3 .3 .2 .2 .1 .1 _ X
A B C D
(18) (20) (22) (24) X
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10. Properties of Summary Measures
e.g.: Is unbiased
Standard error (standard deviation) of the sampling distribution is less than the standard error of other unbiased estimators
For sampling with replacement:
As n increases, decreases
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11. When the Population is Normal
Population Distribution
Central Tendency
Variation
Sampling Distributions
Sampling with Replacement
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12. When the Population is Not Normal
Population Distribution
Central Tendency
Variation
Sampling Distributions
Sampling with Replacement
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13. Central Limit Theorem
Sampling Distribution Becomes Almost Normal Regardless of Shape of Population
As Sample Size Gets Large Enough
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14. How Large is Large Enough?
For most distributions, n>30
For fairly symmetric distributions, n>15
For normal distribution, the sampling distribution of the mean is always normally distributed
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15. Standardized Normal Distribution
Example:
Standardized Normal Distribution
Sampling Distribution
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16. Population Proportions
Categorical variable
e.g.: Gender, voted for bush, college degree
Proportion of population that has a characteristic
Sample proportion provides an estimate
If two outcomes, X has a binomial distribution
Possess or do not possess characteristic
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17. Sampling Distribution of Sample Proportion
Approximated by normal distribution
Mean:
Standard error:
Sampling Distribution
P(ps) .3 .2 .1 ps
p = population proportion
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18. Standardizing Sampling Distribution of Proportion
Standardized Normal Distribution
Sampling Distribution
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19. Standardized Normal Distribution
Example:
Standardized Normal Distribution
Sampling Distribution
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20. Sampling from Finite Sample
Modify standard error if sample size (n) is large relative to population size (N )
Use finite population correction factor (FPC)
Standard error with FPC
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