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Chapter 7
Review
Sampling Distribution
Sampling distribution: distribution of
summary statistics you get from taking
repeated random samples of same fixed
size, n
Reasonably Likely vs Rare Events
What are these?
Reasonably Likely vs Rare Events
Reasonably likely events: values that lie in
the middle 95% of sampling distribution
If normal distribution: mean ± 1.96(standard
deviation)
Reasonably Likely vs Rare Events
Reasonably likely events: values that lie in
the middle 95% of sampling distribution
If normal distribution: mean ± 1.96(standard
deviation)
Rare events: values that lie in outer 5% of
sampling distribution
If normal distribution: outside the interval
mean ± 1.96(standard deviation)
A parameter is __________________.
A parameter is a summary number
describing a population.
A statistic is ____________________.
A statistic is a summary number calculated
from a sample taken from a population.
A sampling distribution is ______________.
A sampling distribution is the distribution of a
sample statistic in repeated random
samples of the same fixed size, n.
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).
Properties of the Sampling Distribution
of the Sum of a Sample
Three properties based on the following
premise: If a random sample of size n is
selected from a distribution with mean 
and standard deviation  , then:
Properties of the Sampling
Distribution of the Sum of a Sample
Three properties based on the following
premise: If a random sample of size n is
selected from a distribution with mean 
and standard deviation  , then:
(1) The mean of the sampling distribution of
the sum is  sum = n 
Properties of the Sampling
Distribution of the Sum of a Sample
Three properties based on the following
premise: If a random sample of size n is
selected from a distribution with mean 
and standard deviation  , then:
(2) The standard error of the sampling
distribution of the sum is
 sum = n  
Note: SE does not decrease as n increases
Properties of the Sampling
Distribution of the Sum of a Sample
Three properties based on the following premise:
If a random sample of size n is selected from a
distribution with mean  and standard
deviation  , then:
(3) The shape of the sampling distribution
will be approximately normal if the
population is approximately normally
distributed. For other populations the
sampling distribution will become more
normal as n increases.
Sampling distributions of the mean
Which distribution represents the population,
n = 4, and n = 10? How do you know?
Page 437, P7
population
n = 10
n=4
Properties of the Sampling Distribution
of the Number of Successes
If a random sample of size n is selected
from a population with proportion of
successes p, then the sampling
distribution of the number of successes X:
Properties of the Sampling Distribution
of the Number of Successes
If a random sample of size n is selected
from a population with proportion of
successes p, then the sampling
distribution of the number of successes X:
• has mean x = np
Properties of the Sampling Distribution
of the Number of Successes
If a random sample of size n is selected
from a population with proportion of
successes p, then the sampling
distribution of the number of successes X:
• has mean x = np
• has standard error  x  np1  p 
Properties of the Sampling Distribution
of the Number of Successes
If a random sample of size n is selected
from a population with proportion of
successes p, then the sampling
distribution of the number of successes X:
• has mean x = np
• has standard error  x  np1  p 
• will be approximately normal as long as n
is large enough
Properties of the Sampling Distribution
of the Sample Proportion
If a random sample of size n is selected
from a population with proportion of
successes p, then the sampling
distribution of p has these properties:
Properties of the Sampling Distribution
of the Sample Proportion
If a random sample of size n is selected
from a population with proportion of
successes p, then the sampling
distribution of p has these properties:
• Mean of the sampling distribution is equal
to the mean of the population, or
 p
p
Properties of the Sampling Distribution
of the Sample Proportion
If a random sample of size n is selected
from a population with proportion of
successes p:
• Standard error of the sampling
distribution is equal to the standard
deviation of the population divided by the
square root of the sample size:
p
(
1

p
)
p
n
Properties of the Sampling Distribution
of the Sample Proportion
If a random sample of size n is selected
from a population with proportion of
successes p, then the sampling
distribution of p has these properties:
Properties of the Sampling Distribution
of the Sample Proportion
If a random sample of size n is selected
from a population with proportion of
successes p, then the sampling
distribution of p has these properties:
As the sample size gets larger, the shape
of the sampling distribution becomes more
normal and will be approximately
normal if n is large enough (both np and
n(1 – p) are at least 10).
Questions?
You may use both sides of one 4x6 note
card for the test—not two note cards
stapled or taped together
Properties of the Sampling Distribution
of the Sum
Suppose two values are taken randomly
from two populations with means μ1 and
μ2 respectively and variances σ12 and σ22.
Properties of the Sampling Distribution
of the Sum
Suppose two values are taken randomly
from two populations with means μ1 and μ2
respectively and variances σ12 and σ22.
Then the sampling distribution of the sum of
the two values has mean:
μsum = μ1 + μ2
Properties of the Sampling Distribution
of the Sum
Suppose two values are taken randomly
from two populations with means μ1 and μ2
respectively and variances σ12 and σ22.
If the two values were selected
independently, the variance of the sum
is:
σ2sum = σ12 + σ22
Properties of the Sampling Distribution
of the Difference
Suppose two values are taken randomly
from two populations with means μ1 and μ2
respectively and variances σ12 and σ22.
Properties of the Sampling Distribution
of the Difference
Suppose two values are taken randomly
from two populations with means μ1 and μ2
respectively and variances σ12 and σ22.
Then the sampling distribution of the
difference of the two values has mean:
μdifference = μ1 - μ2
Properties of the Sampling Distribution
of the Difference
Suppose two values are taken randomly
from two populations with means μ1 and μ2
respectively and variances σ12 and σ22.
If the two values were selected
independently, the variance of the difference
is:
σ2difference = σ12 + σ22
The shapes of the sampling distributions of
the sum and difference depend on the
shapes of the two original populations.
The shapes of the sampling distributions of
the sum and difference depend on the
shapes of the two original populations.
If both populations are normally distributed,
so are the sampling distributions of the
sum and the difference.