Download 6-7A Lecture

Section 6 – 7A: Sampling Distribution of the Sample Means X
To Create a Sampling Distribution of the Sample Means X
take every possible sample of size n from the distribution of x values
and then find the mean of each sample. Each mean is listed as an x a
The Distribution of the Sample Means X is the collection of all the sample means x a
x 1 + x 2 + ....+ x n
= x1
n
x1
x2
x3
x4
x5
.
.
.
xa
x
the population
of all x values
x 2 + x 5 + ....+ x n
= x2
n
x1
x 3 + x 4 + ....+ x n
= x3
n
x3
x 5 + x 7 + ....+ x n
= x4
n
:
:
:
Take all the possible samples
of size n
from the distribution of x values
and
find the mean of each sample
the mean of the x values = µ x
the SD of the x values = σ x
x2
x4
x5
.
.
.
xa
X
the population of all the
possible means of samples
of size n taken
from the distribution of x
values
The mean of X is written as ux
and
ux = ux
The standard deviation X is written as σ x
and
σ
σx = x
n
When is X Normal
1. The distribution of all the possible sample means X is normal if the original population of x
values is known to be normal.
2. The distribution of all the possible sample means X is normal if the sample size that created the
X distribution is greater than 30. ( n > 30 )
6 – 7A Central Limit Theorem Lecture
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© 2012 Eitel
Using the Standard Normal Distribution (Z curve)
to find Probabilities
for a Sampling Distribution of the Means X
Assumptions that must be true before you can use a Standard Normal Distribution (Z) to
answer questions about a Sampling Distribution of X
Original Distribution of x values
1. The original distribution of x values is a collection of data values ( x values) that represents
measurements of a population parameter. The distribution of all the x values in the population
will be called the “original distribution”.
Create a new distribution of Sample Means X
2.
A new distribution exits that contains all the possible sample means of size n. A sample of
size n is taken from the original population of x values and the mean of that first sample is
found ( x 1 ). A second sample of size n is taken from the original population of x values and
the mean of that second sample is found ( x 2 ). A third sample of size n is taken from the
original population of x values and the mean of that third sample is found ( x 3 ). If we take
every possible different sample of size n from the population of x values and compute the
mean of each different sample we would have a new population of all the possible
sample means. This new population is called the distribution of the sample means X .
X Must be Normal
3. The Distribution of Sample Means X must be normal. X will be normal if the original
distribution of x values is normal OR the sample size n is greater than 30. n > 30.
6 – 7A Central Limit Theorem Lecture
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© 2012 Eitel
The relationships between the original distribution of x values
and the Distribution of Sample Means X
The Original distribution of x values
must be Normal or n > 30
The Distribution of Sample Means
_
µ x = µx
X
σx
σx =
n
1. The original distribution of x values will have a mean of µ x and a standard deviation of σ x . Both
of these values will be known.
µ x is known
σ x is known
x
2. The mean of the distribution of all possible sample means X has the symbol µ x and is
equal to the mean of the original x distribution.
µ x = µx
3. The standard deviation of the distribution of all possible sample means X has the symbol
σ x and is found by dividing the standard deviation of the original distribution of x values
divided by the square root of the sample size:
σ
σx =
n
The Relationships between X values and Z values
4. Every x 1 value in the Distribution of Sample Means X has a unique corresponding z value in
the Standard Normal Distribution (Z). Convert any x 1 value to its corresponding Z1 value in a
Standard Normal Distribution by the formula below.
Standard Normal Distribution Z
Distribution of Sample Means X
Convert x 1 to its unique Z value by
z1 =
x 1 = ?? µ x = µx
σ
σx = x
n
_
X
6 – 7A Central Limit Theorem Lecture
(x 1 − µ x )
⎛σx ⎞
⎜
⎟
⎝ n⎠
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z1 = ??
0
1
Z
© 2012 Eitel
Relationships between the areas under
a Distribution of the Sample Means curve X
and the areas under a Standard Normal Z Curve
1. If an x 1 value has a corresponding z 1 value then the area to the left of x 1 is equal to the area
to the left of the corresponding z1 value.
The area to
the left of
z1
The area to
the left of
x1
_
X
x1
Z
z1
2. If an x 1 value has a corresponding z1 value then the area to the right of x 1 is equal to the area
to the right of the corresponding z1 value.
The area to
the right of
The area to
the right of
z1
x1
_
X
x1
z1
Z
3. If two x 1 and x 2 values have corresponding z1 and z2 values then the area between x 1 and x 2
is equal to the area between the corresponding z1 and z2 values.
The yellow area between z1 and z2
The area between x 1 and x 2
x1
x2
6 – 7A Central Limit Theorem Lecture
_
X
z1
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z2
Z
© 2012 Eitel
The Relationship between the area under a Distribution of Sample Means X
and Probabilities about selecting n values from a Normal Distribution of x values
1. The area in the left tail of the Distribution of Sample Means X represents the probability of
selecting n values from a Normal Distribution of x values and having the mean of those n
values x be less than the x 1 value.
P( x < x1 )
The yellow area
to the left of x 1
represents
P( x < x1 )
_
X
x1
2. The area in the right tail of the Distribution of Sample Means X represents the probability of
selecting n values from a Normal Distribution of x values and having the mean of those n
values x be more than the x 1 value.
P( x > x1 )
The yellow area
to the right of x 1
represents
P( x > x1 )
_
X
x1
3. The area between two values x 1 and x 2 from a Distribution of Sample Means X
represents the probability of selecting n values from a Normal Distribution of x values and
having the mean of those n values x be between the two X values x 1 and x 2 .
P( x1 < x < x 2 )
The yellow area between x 1 and x 2 represents
P( x1 < x < x 2 )
x1
6 – 7A Central Limit Theorem Lecture
x2
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_
X
© 2012 Eitel