Download BUSN429Chap7

SAMPLING AND SAMPLE DISTRIBUTIONS
To find out about a population we can:
1. sample the entire population or
2. select a random sample
- a sample is quicker and cheaper
statistical inference
- use a sample to draw conclusions about a population
POINT ESTIMATE
- calculate a statistic from a sample
- use it as an estimate of the population parameter
used to estimate  (population mean)
s (sample s.d.)
used to estimate  (population s.d.)
p (sample proportion)used to estimate p (pop. proportion)
x (sample mean)
- point estimate as opposed to an interval estimate
(e.g. estimate that x is between two numbers)
SIMPLE RANDOM SAMPLING
simple random sampling - probability of any combination of data
points being selected is equal
- selection is independent
Finite Populations
i.e. 1, 2, 3, 4, 5
- if a sample is drawn with n = 2, then
N!
5!

there are n!{N  n)! 2 ! 3!  10 different samples.
- if a sample that is drawn can only be drawn once, then
the number of possible samples decreases and
the probability of choosing any given sample is greater
- therefore, to be random (and therefore independent)
the sample must be repeatable (replaced).
Infinite Populations
(or for very large populations)
- little or no change in probabilities even if no replacement
THE SAMPLING DISTRIBUTION OF MEANS
- when we select a sample from a population:
not all the values are the same and
not all the values are equal to the sample mean
- the sample values follow some distribution that has variation (and a
mean)
- when take means of several samples from the same pop.
not all the sample means are the same and
not all the sample means are equal to the pop. mean
- the sample means follow some distribution that has variation (and a
mean)
Eg: A group of 4 student’s weekly study times are:
Student
Hrs/wk
A
4
B
6
C
8
D
10
To create a sampling distribution of the means for samples: list all
possible samples (let’s make sample size, n, = 2), the mean of each
sample, and the probability of each mean.
x
Sample Means p( x )
Students
Hours
AB
4,6
4+6
2
5
1/6
AC
4,8
4+8
2
6
1/6
AD
4,10
4+10
2
6+8
2
7
n
2/6
BC
6,8
7
BD
6,10
6+10
2
8
1/6
CD
8,10
8+10
2
9
1/6
Distribution of the
population values:
Distribution of the
sample means:
0.3
0.3
0.2
0.2
p(x)
p(x)
0.1
0.1
0
0
4
6
8
10
5
x
Mean of the pop. values:
 = 4+6+8+10 = 7
4
6
7
8
9
x
Mean of the sample means:
x =
5+6+7+7+8+9 = 7
6
mean of the sampling distribution of means = the pop. mean
x = 
Standard Deviation of Sampling Distribution of Means
S.D. of the pop. values:
  x  

S.D. of the sample means:
2
x





2
x 
N
= 5 = 2.234
# of samples
= 5/3 = 1.291
x 

n
- called standard error of the mean
x 

n
= 5 / 2 = 1.5811
Why does it not match?
Because the sample size is a large proportion of the pop.
When the sample size is more than 5% of the population (n/N > .05),
must adjust the standard error using the finite population correction
factor.
x 
x 

n

n
Nn
N 1
Nn
N  1 = (5 / 2) (2/3) = 1.291
Type of Distribution
If the population is normally distributed, then the sampling
distribution of x is normally distributed.
If the population is not normally distributed,
but the sample size is large,
then the sampling distribution of x is normally distributed
Central Limit Theorem
If we draw a random sample of size n from a population,
the distribution of the sample mean x
can be approximated by a normal probability distribution
as the sample size becomes large (n  30).
If the population is not normally distributed,
but the sample size is small,
then the sampling distribution of x is not necessarily normally
distributed
Sample Distribution of the Sample Proportion
- when interested in the proportion of items in a population that have
a particular characteristic:
Registrar wants to know the proportion of female students. The
characteristic is “female”.
A manufacturer of computer chips wants to know the proportion of
nondefective chips in a production run.
The characteristic of interest is “nondefective” chips.
What prop. of the people will vote for party A in an election?
Proportion
Proportion = # of items that have the characteristic = x
total # of items
n
p = population proportion
p = sample proportion
 p = mean of the sample proportions = p
p = p
p 
pq
n
Note: q = (1-p)
For Finite Populations:
p 
pq N  n
n N 1
If
n
N  0.05
Type of Distribution
Central Limit Theorem
If we draw a random sample of size n from a population,
the distribution of the sample proportion p
can be approximated by a normal probability distribution
as the sample size becomes large (np  5, nq  5).