Download Distribution of Sample Means

Distribution of Sample Means
The distribution of the means of an infinite
number of samples of a certain size selected
from a population
Before, we talked about distributions of scores, but now
we are talking about the distribution of all possible
sample means of a specific size from a population
SAMPLING ERROR:
•The amount of error, or the difference, between a
sample statistic and the corresponding population
parameter
•If we start with a population, and take a number of
samples from that population, each of the sample
means will be different from the population mean in
varying degrees
CENTRAL LIMIT THEOREM
For samples of size n taken from a population
with mean, µ, and a standard deviation, σ,
€
€
–
µX = µ
–
σX =
σ
n
– As n increases, the sampling distribution approaches
the normal distribution
– Holds for all population distributions
• in other words, as long as your sample size is large (30
or above usually) the sampling distribution will be
normal even if the population (parent distribution) from
which it was sampled is not
1
Central Limit Theorem
Population
σ = 20
µ =€
100
Sampling Distribution
Though the population
distribution is skewed the
sampling distribution will still be
normal as long as you have a
large enough n .
20
σX =
n
€
µX = 100
€
Distribution of Sample Means
STANDARD ERROR OF THE MEAN: The
standard deviation of the sample means
Formula:
σx =
σ
n
• Like the standard deviation, the standard
error of the mean gives an average distance
from all of the sample means to the
population mean
• The σ x tells us how much error, on
average, you would expect between a sample
mean and a population mean
2
Distribution of Sample Means
• As n (sample size) gets larger, the distribution of
sample means will approximate a normal distribution
-- when n>30, the shape of the sampling distribution
is almost perfectly normal regardless of the shape of
the original distribution
• The mean of the sampling distribution of the mean is
represented by
µx
or
X
• The mean of the sample means (
population mean (µ)
µx
) will equal the
µX = µ
Distribution of Sample Means
LAW OF LARGE NUMBERS:
• The larger the sample, the more accurately the
sample represents the population
• Therefore, the larger the sample size, the
smaller the standard error
3
From the previous class example.
POPULATION
20
30
50
40
1
4
1
p(30) =
4
1
p(40) =
4
1
p(50) =
4
p(20) =
Sampling Distribution of the Mean
Rectangular
Distribution
0.3
0.25
€
p( )
0.2
0.15
0.1
0.05
0
20
30
40
50
Sample Means
LAW OF LARGE NUMBERS:
• For samples of size 2.
Sampling Distribution of the Mean
0.3
0.25
0.1
0.05
0
20
25
30
35
40
45
50
•For samples of size 3.
Sampling Distribution of the Mean
Sample Means
p( )
p( )
0.2
0.15
14
12
10
8
6
4
2
0
20
20.3 26.7
30
33.3 36.7
40
43.3 46.7
50
Sample Means
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Distribution of Sample Means
EXAMPLE:
What is the standard error for a sample of
n=25, σ=15?
σ
15
15
σx =
=
=
=3
n
25 5
• The average distance of a sample mean from
the population mean is 3 points
• As n increases, the distribution approaches
the normal
• Knowing or assuming the population mean
and standard deviation, we can use the table
of the standard normal distribution to
determine the probability of obtaining
sample means within any interval or beyond
any point
• Thus, we can test simple hypotheses about
sample means (called the z-test)
– Calculate z-score for sample mean, zobt
– Compare to critical z-score, zcrit
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Distribution of Sample Means / z-tests
• The primary use of the distribution of sample
means is to find the probability associated
with getting a specific sample value
• Because the distribution of sample means is
normal, we can use the z-score table to find
probabilities associated with given samples
• This is called a z-test
Formula:
z=
x−µ
σx
where
σx =
σ
n
Distribution of Sample Means / z-tests
METHOD 1:
Step 1: Convert sample mean to zobt
Step 2: Compare to zcrit,
which is either z α for a 1-tailed test, or
zα/2 for a 2-tailed test
Step 3: Decide whether or not to reject H0
METHOD 2:
Step 1: Convert sample mean to z
Step 2: Determine the probability from the ztable
Step 3: Compare to α
Step 4: Decide whether or not to reject H0
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Distribution of Sample Means / z-tests
EXAMPLE: A population of heights has a µ=68
and σ=4. What is the probability of selecting
a sample of size n=25 that has a mean of 70
or greater?
Power & z-tests
Summary of Power:
• Power is the sensitivity of an experiment to
detect a real effect if there is one.
• Power is the probability the experiment will
result in rejecting Ho if the IV has a real
effect.
• Power + β = 1
• Power increases with N.
• Power increases with α.
• Power increases with effect size.
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Calculating Power
1.
2.
3.
4.
5.
Choose 1-tail or 2-tail test
Set alpha value
Select statistical test
Determine critical values to reject H0
Calculate the probability of obtaining
those values under a specified H1
– That is the power of the test under that
version of H1
Power & z-test : N
From previous example:
A population of heights has a µ=68 and σ=4. What is the probability
of selecting a sample of size n=25 that has a mean of 70 or greater?
Calculate the power of this experiment.
What if we had a sample or size n=100?
1. Choose 1-tail or 2-tail test
1-tail
2. Set alpha value
0.05
3. Select statistical test
z-test
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Power & z-test : N
4. Determine critical values to reject
H0
α=0.05  zcrit=1.645
σ
4
4
σx =
=
= = .80
n
25 5
n=25
n=100
4
σX =
= 0.4
100
X crit − µnull  X = µ + σ (z )
crit
null
crit
X
σX
€
X crit = 68 + 0.40(1.645)
= 68 + 0.80(1.645)
zcrit =
n=25
X crit
X crit = 69.32
X crit = 68.66
€
n=100
€
€
€
Power & z-test : N
5. Calculate the probability of obtaining
those values under a specified H1
–
That is the power of the test under that
version of H1
n=25
z=
X − µreall
z = crit
σX
p= 1-0.1949 = 0.8023
Power = 0.8023
n=25
3.5
€
3
2.5
68.66 − 70
z=
= −3.355
0.40
2
f(z)
f(z)
€
1.5
1
0.5
0
-4 -4
-2-2
00
22
44
66
zz
z= -0.855
69.32 − 70
= −0.855
0.80
8
8
n=100
p= 1-0.0005 = 0.9995
n=100
Power = 0.9995
€
9
Power & z-test : α
From previous example:
A population of heights has a µ=68 and σ=4. What is the probability
of selecting a sample of size n=25 that has a mean of 70 or greater?
Calculate the power of this experiment for α=
0.05 and α=0.01
1. Choose 1-tail or 2-tail test
1-tail
2. Set alpha value
0.05 & 0.01
3. Select statistical test
z-test
Power & z-test : α
4. Determine critical values to reject
H0
α=0.05  zcrit=1.645
α=0.01  zcrit=2.335
zcrit =
α=0.05
€
€
X crit − µnull  X = µ + σ (z )
crit
null
crit
X
σX
X crit = 68 + 0.80(2.335)
X crit = 68 + 0.80(1.645)
€
X crit = 69.87
X crit = 69.32
α=0.01
€
10
Power & z-test : α
5. Calculate the probability of obtaining
those values under a specified H1
–
That is the power of the test under that
version of H1
α=0.05
z=
X − µreall
z = crit
σX
p= 1-0.1949 = 0.8023
Power = 0.8023
α=0.05
3.5
€
3
2.5
α=0.01
69.87 − 70
z=
= −0.165
0.80
2
f(z)
f(z)
€
1.5
1
0.5
0
-4 -4
-2-2
00
22
44
66
zz
z= -0.855
69.32 − 70
= −0.855
0.80
8
8
p= 1-0.4325 = 0.5675
Power = 0.5675
α=0.01
€
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