1. Chapter 6 Sampling and Sampling Distributions
Statistics for
Business and Economics 8th Edition
Chapter 6
Sampling and
Sampling Distributions
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2. Sampling Distributions of Sample Means
6.2
Sampling Distributions
Sampling Distributions of Sample
Means
Sampling Distributions of Sample Proportions
Sampling Distributions of Sample Variances
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Sample Mean
Let X1, X2, . . ., Xn represent a random sample from a population
The sample mean value of these observations is defined as
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4. Standard Error of the Mean
Different samples of the same size from the same population will yield different sample means
A measure of the variability in the mean from sample to sample is given by the Standard Error of the Mean:
Note that the standard error of the mean decreases as the sample size increases
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5. Comparing the Population with its Sampling Distribution
Sample Means Distribution
n = 2 _
P(X)
P(X) .3 .3 .2 .2 .1 .1 _
A B C D X X
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6. Developing a Sampling Distribution
(continued)
Summary Measures for the Population Distribution:
P(x)
.25 x
A B C D
Uniform Distribution
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7. Summary Measures of the Sampling Distribution:
Developing a
Sampling Distribution
(continued)
Summary Measures of the Sampling Distribution:
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8. If sample values are not independent
If the sample size n is not a small fraction of the population size N, then individual sample members are not distributed independently of one another
Thus, observations are not selected independently
A finite population correction is made to account for this: or
The term (N – n)/(N – 1) is often called a finite population correction factor
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9. If the Population is Normal
If a population is normal with mean μ and standard deviation σ, the sampling distribution of is also normally distributed with
and
If the sample size n is not large relative to the population size N, then
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10. Standard Normal Distribution for the Sample Means
Z-value for the sampling distribution of :
where: = sample mean
= population mean
= standard error of the mean
Z is a standardized normal random variable with mean of 0 and a variance of 1
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11. Sampling Distribution Properties
Normal Population Distribution
(i.e is unbiased )
Normal Sampling Distribution
(both distributions have the same mean)
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12. Sampling Distribution Properties
(continued)
Normal Population Distribution
(i.e is unbiased )
Normal Sampling Distribution
(the distribution of has a reduced standard deviation
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13. Sampling Distribution Properties
(continued)
As n increases, decreases
Larger sample size
Smaller sample size
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Central Limit Theorem
Even if the population is not normal,
…sample means from the population will be approximately normal as long as the sample size is large enough.
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Central Limit Theorem
(continued)
Let X1, X2, , Xn be a set of n independent random variables having identical distributions with mean µ, variance σ2, and X as the mean of these random variables.
As n becomes large, the central limit theorem states that the distribution of
approaches the standard normal distribution
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Central Limit Theorem
the sampling distribution becomes almost normal regardless of shape of population
As the sample size gets large enough… n↑
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17. If the Population is not Normal
(continued)
Population Distribution
Sampling distribution properties:
Central Tendency
Sampling Distribution
(becomes normal as n increases)
Variation
Larger sample size
Smaller sample size
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18. How Large is Large Enough?
For most distributions, n > 25 will give a sampling distribution that is nearly normal
For normal population distributions, the sampling distribution of the mean is always normally distributed
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Example
Suppose a large population has mean μ = 8 and standard deviation σ = 3. Suppose a random sample of size n = 36 is selected.
What is the probability that the sample mean is between 7.8 and 8.2?
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20. Copyright © 2013 Pearson Education, Inc. Publishing as Prentice Hall
Example
(continued)
Solution:
Even if the population is not normally distributed, the central limit theorem can be used (n > 25)
… so the sampling distribution of is approximately normal
… with mean = 8
…and standard deviation
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21. Example Solution (continued): (continued) Z X Population Distribution
Sampling Distribution
Standard Normal Distribution ? ? ? ? ? ? ? ? ? ?
Sample
Standardize ? ? Z X
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Acceptance Intervals
Goal: determine a range within which sample means are likely to occur, given a population mean and variance
By the Central Limit Theorem, we know that the distribution of X is approximately normal if n is large enough, with mean μ and standard deviation
Let zα/2 be the z-value that leaves area α/2 in the upper tail of the normal distribution (i.e., the interval - zα/2 to zα/2 encloses probability 1 – α)
Then
is the interval that includes X with probability 1 – α
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