Transcript Chapter 7, part B

Chapter 7, continued...
IV. Introduction to Sampling
Distributions
Suppose you take a second sample of n=30 and
calculate your estimators again:
x  $52,669.70
s = $4239.07
p  .70
To see what would happen if you
repeated the sampling process 500
times, see tables 7.5 and 7.6 in the
text.
Random variables revisited
x
is a random variable: a numerical description of
an experiment.
The experiment is the process of selecting a simple
random sample. With a large population, you are
almost assured that every single time you take a
random sample, you’ll get a different value for x
Probability distributions revisited
Just like any other random variable, x has an
expected value, a variance, and a probability
distribution, which we will begin calling a
sampling distribution.
Knowledge of this sampling distribution will allow
us to make probability statements about how close
is to .
x
V. Sampling Distribution of
x
The sampling distribution of x is the probability
distribution of all possible values of the sample
mean, x .
A. Expected value of x
E(
x)=
This result is proven in the textbook. Thus for the
EAI example, the expected value of x is $51,800.
B. Standard Deviation of x
Finite Population:
Infinite Population
x 
x 
N n 
( )
N 1 n

n
This can be used if n/N.05
 x : the standard deviation of the sampling distribution
of x
: the standard deviation of the population
n: the sample size
N: the population size
EAI study
n/N= 30/2500 = .012 < .05 so we can assume this is
a “large” or infinite population.

4000
x  ( ) 
 730.30
n
30
This is referred to as the standard error of the mean.
An example
A survey of library users is taken of 321 people
leaving the Indianapolis Central Library. One
question asks “How much time did you spend in
the library?” The next slide is a histogram that
represents the frequency distribution of the
variable “SPEND”.
Population parameters:  = 41.38 minutes,  = 40.22 minutes.
Eric R. Dodge:
thanks to John Ottensman at
IUPUI.
Sampling distribution for SPEND
Now suppose you take a sample of n=8 and calculate
x . Then do this 50 times and recreate your
frequency distribution. One such result is on the
next slide.
Mean = 40.98 minutes, standard error = 14.65.
Eric R. Dodge:
thanks to John Ottensman at
IUPUI.
Take a larger sample?
When we take a sample of n=8 and do it 50 times,
we get a mean that is fairly close to the actual
mean of SPEND.
Do you think the mean of the sampling distribution
would be more accurate if we took larger (or
more) samples?
Let’s try 2500 samples, each with n=8.
Mean of the sampling distribution is 41.25, standard error
is 14.38.
Notice how the
mean gets closer
to the “true”
population mean.
This is the topic
for your next
outline.
Eric R. Dodge:
thanks to John Ottensman at
IUPUI.