Sampling Distributions - Mrs. Hamilton AP...
Transcript of Sampling Distributions - Mrs. Hamilton AP...
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Sampling
Distributions
Chapter 9
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First, a word from our textbook
A statistic is a numerical value computed from
a sample. EX. Mean, median, mode, etc.
A parameter is a numerical value determined
by the entire population and is assumed that
the value is fixed,unchanging and unknown.
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Introduction to Statistics and
Sampling Variability
Consider a small population consisting of the
board of directors of a day care center.
Board member and number of children: Jay Carol Allison Teresa Anselmo Bob Roxy Vishal
5 2 1 0 2 2 1 3
Find the average number of children for the entire
group of eight:
m = 2 children
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Discovery question ONE:
How is the parameter of the population
related to a sampling distribution based on
the population?
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Introduction to Statistics and
Sampling Variability
Board member and number of children: Jay Carol Allison Teresa Anselmo Bob Roxy Vishal
5 2 1 0 2 2 1 3
List all possible samples of size 2. Calculate the
average number of children represented by the
group.
Samples:
3.5x
Jay Carol
5 2
Jay Allison
5 1 3x
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Answer question ONE:
the average of all possible values for a
sampling distribution will equal the population
parameter
xm m
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Variability of a statistic
What is the relationship between the population parameter and each sample statistic?
The observed value of a statistic will vary from sample to sample. This fact is called sampling variability.
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Sampling distributions
If we calculated using only the first 3
columns of values, would we get the same
results? Explain.
How did the spread change from the
population to the sampling distribution?
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Definition
In summary, a sampling distribution is the
distribution of all possible values for a given
sample size for a fixed population.
Sampling distribution applet
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Discovery question TWO:
For a normal population, how will the shape
and spread of a sampling distribution change
as we increase the sample size?
Population distribution: m = 16 s = 5
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Discovery question TWO:
16X
m 16X
m
16X
m 16X
m
3.535X
s 1.581X
s
1.25X
s 1X
s
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Answer question TWO:
For a normal population, the shape of the
sampling distribution remains mound shaped
and symmetrical (taller/thinner)for all sample
sizes. We can conclude the sampling
distribution remains approximately normal.
The standard deviation for the sampling
distribution is equal to the population
standard deviation divided by the square root
of the sample size.
Xn
ss
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Sample means
parameter statistic
mean m x
standard deviation s s
Formulas:
xm m Xn
ss
sampling
distribution
xm
Xs
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Example ONE
The average sales price of a single-family
house in the United States is $243,756.
Assume that the sales prices are normally
distributed with a standard deviation of
$44,000.
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Draw the normal distribution. Within what
range would the middle 68% of the houses
fall?
$243,756
$243,756m
$44,000s
$287,756 $199,756
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Draw the sampling distribution for a sample
size of 4 houses. Within what range would the
middle 68% of the samples of size 4 houses
fall?
$243,756
$243,756xm
$44,000xs
$265,756 $221,756
$44,000
4xs
$22,000xs
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Draw the sampling distribution for a sample
size of 16 houses. Within what range would
the middle 68% of the samples of size 16
houses fall?
$243,756
$243,756xm
$254,756 $232,756
$44,000
16xs
$11,000xs
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Draw the sampling distribution for a sample
size of 25 houses. Within what range would
the middle 68% of the samples of size 25
houses fall?
$243,756
$243,756xm
$252,556 $234,956
$44,000
25xs
$8,800xs
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Example TWO
Suppose the mean room and board
expense per year at a certain four-year
college is $7,850. You randomly select 9
dorms offering room and board near the
college. Assume that the room and board
expenses are normally distributed with a
standard deviation of $1125.
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Draw the population distribution.
$7,850
$7,850m
1125s
$8,975 $6,725 $10,100 $11,225 $5,600 $4,475
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$7,850m
$1125s
$8,180
( 8180)P x
What is the probability that a randomly
dorm has room and board of less than
$8,180?
$7,850 $8,975 $6,725 $10,100 $11,225 $5,600 $4,475
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What is the probability that a randomly dorm
has room and board of less than $8,180?
$7,850m $1125s
( 8180)P x
Given normal distribution
xz
m
s
8180 7850
1125
0.29
.6141
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Draw the sampling distribution for a sample
size of 9 dorms.
$7,850
$7,850xm
1125$375
9xs
$8,225 $7,475 $8,600 $8,975 $7,100 $6,725
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What is the probability that the mean room
and board of the nine dorms is less than
$8,180?
$7,850
$7,850xm
1125$375
9xs
$8,225 $7,475 $8,600 $8,975 $7,100 $6,725
$8,180
( 8180)P x
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What is the probability that the mean room
and board of the nine dorms is less than
$8,180?
$7,850xm 1125
$3759
xs
( 8180)P x
Given normal distribution
xz
n
m
s
8180 7850
375
0.88
.8106
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What is the probability that the mean cost of a
sample of four dorms is more than $7,250?
$7,850xm 1125
$562.504
xs
( 7250)P x
Given normal distribution
xz
n
m
s
7250 7850
562.5
1.067
1 .1423 .8577
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Central Limit Theorem
Take a random sample of size n from any
population with mean m and standard
deviation s. When n is large, the sampling
distribution of the sample mean is close to
the normal distribution.
How large a sample size is needed depends
on the shape of the population distribution.
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Uniform distribution
Sample size 1
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Uniform distribution
Sample size 2
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Uniform distribution
Sample size 3
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Uniform distribution
Sample size 4
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Uniform distribution
Sample size 8
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Uniform distribution
Sample size 16
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Uniform distribution
Sample size 32
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Triangle distribution
Sample size 1
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Triangle distribution
Sample size 2
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Triangle distribution
Sample size 3
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Triangle distribution
Sample size 4
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Triangle distribution
Sample size 8
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Triangle distribution
Sample size 16
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Triangle distribution
Sample size 32
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Inverse distribution
Sample size 1
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Inverse distribution
Sample size 2
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Inverse distribution
Sample size 3
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Inverse distribution
Sample size 4
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Inverse distribution
Sample size 8
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Inverse distribution
Sample size 16
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Inverse distribution
Sample size 32
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Parabolic distribution
Sample size 1
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Parabolic distribution
Sample size 2
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Parabolic distribution
Sample size 3
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Parabolic distribution
Sample size 4
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Parabolic distribution
Sample size 8
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Parabolic distribution
Sample size 16
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Parabolic distribution
Sample size 32
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Loose ends
An unbiased statistic falls sometimes above
and sometimes below the actual mean, it
shows no tendency to over or underestimate.
As long as the population is much larger than
the sample (rule of thumb, 10 times larger),
the spread of the sampling distribution is
approximately the same for any size
population.
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Loose ends
As the sampling standard deviation continually decreases, what conclusion can we make regarding each individual sample mean with respect to the population mean m?
As the sample size increases, the mean of the observed sample gets closer and closer to m. (law of large numbers)