1. Sampling Distributions
Chapter 6
Sampling Distributions
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Random Sampling
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Definitions
A population: Consist of the totality of observations with which we are concerned
A sample is a subset of a population
Let X1 , X2 , … , Xn be n independent random variables, each having the same probability distribution f(x). We then define X1 , X2 , … , Xn to be random sample of size n from the population f(x).
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Random Samples
The rv’s X1,…,Xn are said to form a (simple
random sample of size n if
The Xi’s are independent rv’s.
Every Xi has the same probability distribution.
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5. Some Important Statistics
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Statistic
A statistic is any quantity whose value can be calculated from sample data. Prior to obtaining data, there is uncertainty as to what value of any particular statistic will result. or
Any function of the random variables constituting of random sample called a statistic.
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If X1 , X2 , … , Xn represent a random sample of size n, then the sample mean is defined by the statistic
If X1 , X2 , … , Xn represent a random sample of size n, then the sample variance is defined by the statistic
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8. Sampling Distribution of the Mean
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Using the Sample Mean
Let X1,…, Xn be a random sample from a distribution with mean value and standard deviation Then
In addition, with To = X1 +…+ Xn,
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Normal Population Distribution
Let X1,…, Xn be a random sample from a normal distribution with mean value and standard deviation Then for any n, is normally distributed with mean and standard deviation
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The Central Limit Theorem
Let X1,…, Xn be a random sample from a distribution with mean value and variance Then if n sufficiently large, has approximately a normal distribution with
and To also has
approximately a normal distribution with
The larger the value of
n, the better the approximation.
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The Central Limit Theorem
small to moderate n
large n
Population distribution
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Rule of Thumb
If n > 30, the Central Limit Theorem can be used.
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14. Developing the Distribution
Of the Sample Mean
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15. Developing a Sampling Distribution
Assume there is a population …
Population size N=4
Random variable, x, is age of individuals
Values of x: 18, 20, 22, 24 (years) D A C B
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16. Developing a Sampling Distribution
(continued)
Summary Measures for the Population Distribution:
P(x) .3 .2 .1 x
A B C D
Uniform Distribution
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17. Now consider all possible samples of size n=2
Developing a Sampling Distribution
(continued)
Now consider all possible samples of size n=2
16 Sample Means
16 possible samples (sampling with replacement)
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18. Sampling Distribution of All Sample Means
Developing a
Sampling Distribution
(continued)
Sampling Distribution of All Sample Means
Sample Means Distribution
16 Sample Means
P(x) .3 .2 .1 _ x
(no longer uniform)
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19. Summary Measures of this Sampling Distribution:
Developing a
Sampling Distribution
(continued)
Summary Measures of this Sampling Distribution:
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20. Comparing the Population with its Sampling Distribution
Sample Means Distribution
n = 2
P(x)
P(x) .3 .3 .2 .2 .1 .1 _ x
A B C D x
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21. 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
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22. z-value for Sampling Distribution of x
Z value for the sampling distribution of
Where
the sample mean
the population mean
population standard deviation
n sample size
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23. Sampling Distribution Properties
Normal Population Distribution
(i.e is unbiased )
Normal Sampling Distribution
(has the same mean)
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24. Sampling Distribution Properties
(continued)
For sampling with replacement
Decreases
As n increases
Smaller sample size
Larger sample size
Business Statistics: A Decision-Making Approach, 6e © 2005 Prentice-Hall, Inc.

25. If the Population is not Normal
We can apply the 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
…and the sampling distribution will have
Business Statistics: A Decision-Making Approach, 6e © 2005 Prentice-Hall, Inc.

26. Central Limit Theorem
the sampling distribution becomes almost normal regardless of shape of population
As the sample size gets large enough… n↑
Business Statistics: A Decision-Making Approach, 6e © 2005 Prentice-Hall, Inc.

27. 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
(Sampling with replacement)
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28. Example
Suppose a 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?
Business Statistics: A Decision-Making Approach, 6e © 2005 Prentice-Hall, Inc.

29. Example
(continued)
Solution:
Even if the population is not normally distributed, the central limit theorem can be used (n > 30)
… so the sampling distribution of is approximately normal
… with mean = 8
…and standard deviation
Business Statistics: A Decision-Making Approach, 6e © 2005 Prentice-Hall, Inc.

30. Example Solution (continued): (continued) z Population Distribution
Sampling Distribution
Standard Normal Distribution ? ? ? ? ? ? ? ? ? ?
Sample
Standardize ? ? z
Business Statistics: A Decision-Making Approach, 6e © 2005 Prentice-Hall, Inc.

31. Population Proportions, P
P = the proportion of population having
some characteristic
Sample proportion ( ) provides an estimate of P is:
Business Statistics: A Decision-Making Approach, 6e © 2005 Prentice-Hall, Inc.

32. Sampling Distribution of
Approximated by a normal distribution if:
Sampling Distribution
P( ) .3 .2 .1
where
and
(where p = population proportion)
Business Statistics: A Decision-Making Approach, 6e © 2005 Prentice-Hall, Inc.

33. z-Value for Proportions
Standardize to a z value with the formula:
Business Statistics: A Decision-Making Approach, 6e © 2005 Prentice-Hall, Inc.

34. Example
If the true proportion of voters who support Proposition A is p = .4, what is the probability that a sample of size 200 yields a sample proportion between .40 and .45?
i.e.: if p = .4 and n = 200, what is
P(.40 ≤ ≤ .45) ?
Business Statistics: A Decision-Making Approach, 6e © 2005 Prentice-Hall, Inc.

35. Example if p = .4 and n = 200, what is P(.40 ≤ ≤ .45) ? Find :
(continued)
if p = .4 and n = 200, what is
P(.40 ≤ ≤ .45) ?
Find :
Convert to standard normal:
Business Statistics: A Decision-Making Approach, 6e © 2005 Prentice-Hall, Inc.

36. Standardized Normal Distribution
Example
(continued)
if p = .4 and n = 200, what is
P(.40 ≤ ≤ .45) ?
Use standard normal table: P(0 ≤ z ≤ 1.44) = .4251
Standardized Normal Distribution
Sampling Distribution
.4251
Standardize
.40
.45
1.44 z
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37. The Distribution of a Linear Combination
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Linear Combination
Given a collection of n random variables X1,…, Xn and n numerical constants a1,…,an, the rv
is called a linear combination of the Xi’s.
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Expected Value of a Linear Combination
Let X1,…, Xn have mean values and variances of respectively
Whether or not the Xi’s are independent,
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Variance of a Linear Combination
If X1,…, Xn are independent,
and
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Difference Between Two Random Variables
and, if X1 and X2 are independent,
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Difference Between Normal Random Variables
If X1, X2,…Xn are independent, normally distributed rv’s, then any linear combination of the Xi’s also has a normal distribution. The difference X1 – X2 between two independent, normally distributed variables is itself normally distributed.
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