Section 7.2
Sampling Distribution
of the Sample Mean
For a finite population, the sampling
distribution of the sample mean is the
For a finite population, the sampling
distribution of the sample mean is the set
of means from all possible samples of a
specific size, n.
Simulated Sampling Distribution
4 steps to construct:
Simulated Sampling Distribution
4 steps to construct:
1) Take random sample from population
Simulated Sampling Distribution
4 steps to construct:
1) Take random sample from population
2) Compute summary statistic for sample
Simulated Sampling Distribution
4 steps to construct:
1) Take random sample from population
2) Compute summary statistic for sample
3) Repeat steps 1 and 2 many times
Simulated Sampling Distribution
4 steps to construct:
1) Take random sample from population
2) Compute summary statistic for sample
3) Repeat steps 1 and 2 many times
4) Display distribution of summary statistic
Find mean and standard deviation
using Freq. Table of Population
(not a sample distribution)
Page 428
Find mean and standard deviation
using Freq. Table of Population
(not a sample distribution)
L1
L2
Find mean and standard deviation
using Freq. Table of Population
(not a sample distribution)
L1
L2
STAT
CALC
1-Var Stats L1, L2
Find mean and standard deviation
using Freq. Table of Population
(not a sample distribution)
Find mean and standard deviation
using a sample distribution
(not of population)
Suppose we make 5 sampling distributions
of the sample mean for samples of size:
• n=1
• n=4
• n = 10
• n = 20
• n = 40
Find mean and standard deviation
using a sample distribution, n=1
(from a population)
Note: this column is a Sample
Distribution using a sample size n=1
(looks just like the Population Frequency Graph)
Note: this column is of a
Population
Using Population Freq. Table, generate Sampling
Distributions of different sample sizes (next slides)
n=4
n = 10
n = 20
n = 40
4 Steps not shown…see
Page 428 & 429 in
textbook for calculations
when n = 4
Note: this graph is of a Population
To clarify Step 1 in textbook, page 428,
we can use a random table of digits OR a calculator to
assign digits to possible outcomes
Using random
001- 524
table of digits:
525 - 725
726 - 904
905 - 974
975 – 999, 1000
Or using
Calculator:
MATH
PRB
randInt (1, 1000)
Results
After repeating
Steps 1-4
many times
(i.e. Fathom)
using different
size samples,
the results
are:
As n increases,
what happens to:
•Shape
•Center
•Spread
As n increases, what
happens to:
•Shape: more normal
•Center
•Spread
As n increases, what
happens to:
•Shape: more normal
•Center: stays same
•Spread
As n increases, what
happens to:
•Shape: more normal
•Center: stays same
•Spread: decreases
Common Symbols
Page 430
Note: a calculator cannot tell if a list (L1 or L2) is
from a population or sample distribution…you
have to know which is which.
Properties of Sampling
Distribution of the Sample Mean
These properties depend on using
________ samples.
Properties of Sampling Distribution
of the Sample Mean
These properties depend on using random
samples.
Properties of Sampling Distribution
of the Sample Mean
These properties depend on using random
samples.
These properties will cover center, spread,
and shape.
Properties of Sampling Distribution
of the Sample Mean
If a random sample of size n is selected
from a population with mean  and
standard deviation  , then:
Properties of Sampling Distribution
of the Sample Mean
Center
The mean,  x, of the sampling distribution of
x equals the mean of the population,  :
 x =
Properties of Sampling Distribution
of the Sample Mean
Center
The mean,  x, of the sampling distribution of
x equals the mean of the population,  :
 x =
In other words, the means of random
samples are centered at the population
mean.
Properties of Sampling Distribution
of the Sample Mean
Spread
The standard deviation,  x, of the sampling
distribution, sometimes called the standard
error of the mean, equals the standard
deviation of the population,  , divided by
the square root of the sample size n.

x
=

n
Properties of Sampling Distribution
of the Sample Mean
Spread
The standard deviation,  x, of the sampling
distribution, sometimes called the standard
error of the mean, equals the standard
deviation of the population,  , divided by
the square root of the sample size n.
 X= 
n
When sample size increases, spread
________.
Properties of Sampling Distribution
of the Sample Mean
Spread
The standard deviation,  x, of the sampling
distribution, sometimes called the standard
error of the mean, equals the standard
deviation of the population,  , divided by
the square root of the sample size n.
 X= 
n
When sample size increases, spread
decreases
Properties of Sampling Distribution
of the Sample Mean
Shape
The shape of the sampling distribution will
be approximately normal if the population
is approximately normal.
Properties of Sampling Distribution
of the Sample Mean
Shape
The shape of the sampling distribution will
be approximately normal if the population
is approximately normal.
For other populations, the sampling
distribution becomes more normal as n
increases (Central Limit Theorem).
Using These Properties
1. When can you use property that mean of
sampling distribution of the mean is equal to
the mean of the population,  x =  ?
Using These Properties
1. When can you use property that mean of
sampling distribution of the mean is equal to
the mean of the population,  x =  ?
Anytime you use random sampling.
Shape of pop., size of sample, how large
pop. is, or whether sample with or
without replacement do not matter.
Using These Properties
2. When can we use property that standard
error of sampling distribution of the mean,

 x, is equal to
?
n
Using These Properties
With a population of any shape and with any
sample size as long as you randomly
sample with replacement
or
Using These Properties
With a population of any shape and with any
sample size as long as you randomly
sample with replacement
or
you randomly sample without replacement
and the sample size is less than 10% of
population size.
Using These Properties
3. When can we assume the sampling
distribution is approximately normal?
Using These Properties
3. When can we assume the sampling
distribution is approximately normal?
If we are told the population is
approximately normally distributed (or bellshaped or mound-shaped), you can
assume sampling distribution is
approximately normal regardless of
sample size or _______.
Using These Properties
3. When can we assume the sampling
distribution is approximately normal?
If we are told the population is
approximately normally distributed (or bellshaped or mound-shaped), you can
assume sampling distribution is
approximately normal regardless of
sample size or if we are told sample size
is very large.
Using These Properties
3. When can we assume the sampling distribution
is approximately normal?
If we are told the population is approximately
normally distributed (or bell-shaped or moundshaped), you can assume sampling distribution
is approximately normal regardless of sample
size or if we are told sample size is very large
(think Central Limit Theorem).
Using These Properties
4. Isn’t the size of the population really
important?
Using These Properties
4. Isn’t the size of the population really
important?
As long as the sample was randomly
selected and as long as the population is
much larger than the sample, it does not
matter how large the population is.
Example: Average Number of Children
What is the probability that a random sample
of 20 families in the United States will
have an average of 1.5 children or fewer?
Example: Average Number of Children
What is the probability that a random sample
of 20 families in the United States will
have an average of 1.5 children or fewer?
Could add up heights of
all bars ≤ 1.5
Example: Average Number of Children
What is the probability that a random sample
of 20 families in the United States will
have an average of 1.5 children or fewer?
Could add up heights of
all bars ≤ 1.5
What shape is this?
Example: Average Number of Children
What is the probability that a random sample
of 20 families in the United States will
have an average of 1.5 children or fewer?
How do we find the
area under the
normal curve?
Average of 1.5 children or fewer?
normalcdf(lower bound, upper bound, mean,
standard deviation)
Example: Average Number of Children
What is the probability that a random sample
of 20 families in the United States will
have an average of 1.5 children or fewer?
Recall, for the population μ = 0.9 and σ = 1.1
What are the sampling distribution mean
and standard deviation (standard error)?
Example: Average Number of Children
What is the probability that a random sample
of 20 families in the United States will
have an average of 1.5 children or fewer?
Recall,  = 0.9 and  = 1.1
What is the probability that a random sample
of 20 families in the United States will
have an average of 1.5 children or fewer?
normalcdf (-1E99, 1.5, .9, .25) =
0.9918024711
Reasonably Likely Averages
What average numbers of children are
reasonably likely in a random sample of
20 families?
Reasonably Likely Averages
What average numbers of children are
reasonably likely in a random sample of 20
families?
Reasonably likely values are those in the
middle 95% of the sampling
distribution.
Reasonably Likely Averages
What average numbers of children are
reasonably likely in a random sample of 20
families?
Reasonably Likely Averages
What average numbers of children are
reasonably likely in a random sample of 20
families?
For approx. normal distributions,
reasonably likely outcomes are those
within 1.96 standard errors of the mean.
 x  1.96( SE )
Reasonably Likely Averages
What average numbers of children are
reasonably likely in a random sample of
20 families?
0.9  1.96(0.25)
which gives an average number of children
between 0.41 and 1.39
Page 437, P7
Page 437, P7
Page 437, P7
population
n = 10
n=4
Page 437, P7
population
n = 10
mean ≈ 1.7 for each
n=4
Page 437, P7
population
n = 10
mean ≈ 1.7 for each
SD ≈ 1
n=4
Page 437, P7
population
n = 10
mean ≈ 1.7 for each
SD ≈ 1
SE ≈ 0.3
n=4
Page 437, P7
population
n = 10
n=4
mean ≈ 1.7 for each
SD ≈ 1
SE ≈ 0.3
SE ≈ 0.5
Page 437, P7
a. Plot I is the population, starting out with only
five different values and a slight skew.
Plot III is for a sample size of 4, having more
possible values, a smaller spread, and a more
nearly normal shape.
Plot II is for a sample size of 10, which has even
more possible values, an even smaller spread, and
a shape that is closest to normal of all the
distributions.
Page 437, P7
b. Theoretically, the mean will be the same
1.7 for each distribution, which is
consistent with the estimates.
Page 437, P7
Page 437, P7
d. The population distribution is roughly
mound-shaped with a slight skew. The two
sampling distributions; however, are
approximately normal with the distributions
becoming more nearly normal as the sample
size increases. This is consistent with the
Central Limit Theorem.
Page 438, P8
Page 438, P8
Page 438, P10
Page 438, P10
Decrease in
price means < 0
Page 438, P10
Page 438, P10
Page 438, P10
Page 443, E21
Page 443, E21
Page 443, E21
normalcdf (510, 1E99, 500, 50)
Page 443, E21
normalcdf (510, 1E99, 500, 20)