Transcript Relationship Between Sample Data and Population Values

Relationship Between Sample
Data and Population Values
You will encounter many situations in business
where a sample will be taken from a population,
and you will be required to analyze the sample
data. Regardless of how careful you are in using
proper sampling methods, the sample likely will
not be a perfect reflection of the population.
Sampling Distribution
A Sampling Distribution is the probability
distribution for a statistic. Its description
includes:
• all possible values that can occur for the
statistic; and
• the probability of each value or each
interval of values for a given sample.
Example
Individual
A
B
C
D
E

Annual
Income
$50,000
45,000
15,000
38,000
22,000
170,000
( Income - µX)
6
256*10
121*106
6
361*10
6
16*10
144*106
6
898*10
2
Population Parameters
• Population Mean (µX):
µX = 170,000 / 5 = $34,000
• Population Standard Deviation (X):
X = [SQRT(898*106) / 5] = $13,401.49
Draw a Random Sample of Three
• How many random samples of three can you
draw from this population?
5C3 = 10 samples of three can be drawn form
this population. Each sample has a 1 / 5C3 , or
1 / 10 chance of being selected.
• List the sample space and find sample means.
Ten Possible Samples
Sample
A, B, C
A, B, D
A, B, E
A, C, D
….
C, D, E
Income Levels Sample Mean
($1,000)
50, 45, 15
$36,666.67
50, 45, 38
44,333.33
50, 45, 22
39,000.00
50, 15, 38
34,333.33
….
….
15, 38, 22
25,000.00
The Sampling Distribution of
Sample Means ( X )
• The mean of the samples means:
µX = ( X1 + X2 + …. + Xn ) / NCn
µX = 340,000 / 10 = $34,000
• The Standard Deviation the samples means,
better known as the Standard Error of the
Mean:
X = SQRT[( Xi - µX )2 / NCn]
Standard Error of the Mean
• The standard error of the mean indicates the
spread in the distribution of all possible
sample means.
• X is also equal to the population standard
deviation divided by the SQRT of the
sample size
X = X / SQRT(n)
A Finite Population Correction
Factor (fpc)
• For n > 0.05N, the finite population
correction factor adjusts the standard error
to most accurately describe the amount of
variation.
• The fpc is SQRT[( N - n ) / ( N - 1 )]