Download The Mean of the Sampling Distribution

Review of Statistical Terms
•
•
•
•
Population
Sample
Parameter
Statistic
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1
Population
the set of all measurements (either
existing or conceptual) under
consideration
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2
Sample
a subset of measurements from a
population
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3
Statistic
a numerical descriptive measure of
a sample
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4
Parameter
a numerical descriptive measure of
a population
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5
We use a statistic to make
inferences about a population
parameter.
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6
Principal types of inferences
• Estimate the value of a population
parameter
• Formulate a decision about the value of a
population parameter
• Make a prediction about the value of a
statistical variable.
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7
Sampling Distribution
a probability distribution for the
sample statistic we are using
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8
Example of a Sampling
Distribution
Select samples with two elements
each (in sequence with
replacement) from the set
{1, 2, 3, 4, 5, 6}.
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9
Constructing a Sampling
Distribution of the Mean for
Samples of Size n = 2
List all samples and compute the mean of each
sample.
sample:
mean:
sample:
mean
{1,1}
{1,2}
{1,3}
{1,4}
{1,5}
1.0
1.5
2.0
2.5
3.0
{1,6}
{2,1}
{2,2}
…
3.5
1.5
4
...
There are 36 different samples.
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10
Sampling Distribution of the Mean
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x
p
1.0
1.5
2.0
2.5
3.0
3.5
4.0
4.5
5.0
5.5
6.0
1/36
2/36
3/36
4/36
5/36
6/36
5/36
4/36
3/36
2/36
1/36
11
Sampling Distribution
Histogram
6
36
1
36
|
1
|
|
|
|
|
|
|
|
|
|
|
1.5 2 2.5 3 3.5 4 4.5 5 5.5 6
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12
Let x be a random variable
with a normal distribution with
mean  and standard deviation
. Let x be the sample mean
corresponding to random
samples of size n taken from
the distribution.
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13
Facts about sampling
distribution of the mean:
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14
Facts about sampling
distribution of the mean:
• The
x
distribution is a normal distribution.
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15
Facts about sampling
distribution of the mean:
• The x distribution is a normal distribution.
• The mean of the x distribution is  (the
same mean as the original distribution).
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16
Facts about sampling
distribution of the mean:
• The x distribution is a normal distribution.
• The mean of the x distribution is  (the
same mean as the original distribution).
• The standard deviation of the x
distribution is  n (the standard deviation
of the original distribution, divided by the
square root of the sample size).
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17
We can use this theorem to
draw conclusions about means
of samples taken from normal
distributions.
If the original distribution is
normal, then the sampling
distribution will be normal.
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18
The Mean of the Sampling
Distribution
x
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19
The mean of the sampling
distribution is equal to the
mean of the original
distribution.
x  
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20
The Standard Deviation of the
Sampling Distribution
x
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21
The standard deviation of the
sampling distribution is equal to the
standard deviation of the original
distribution divided by the square
root of the sample size.

x 
n
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22
The time it takes to drive between
cities A and B is normally
distributed with a mean of 14
minutes and a standard deviation
of 2.2 minutes.
• Find the probability that a trip between
the cities takes more than 15 minutes.
• Find the probability that mean time of
nine trips between the cities is more than
15 minutes.
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23
Mean = 14 minutes, standard
deviation = 2.2 minutes
• Find the probability that a trip between
the cities takes more than 15 minutes.
15  14
z
 0.45
14 15
2.2
P( z  0.45)  1.00  0.6736  0.3264
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Find this
area
24
Mean = 14 minutes, standard
deviation = 2.2 minutes
• Find the probability that mean time of
nine trips between the cities is more than
15 minutes.
 x    14
 2.2
x 

 0.73
n
9
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25
Mean = 14 minutes, standard
deviation = 2.2 minutes
• Find the probability that mean time of
nine trips between the cities is more than
15 minutes.
Find this
area
15  14
z
 1.37
0.73
14 15
P( z  1.37 )  0.5  0.4147  0.0853
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26
What if the Original
Distribution Is Not Normal?
Use the Central Limit Theorem.
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27
Central Limit Theorem
If x has any distribution with mean  and
standard deviation , then the sample
mean based on a random sample of size n
will have a distribution that approaches
the normal distribution (with mean  and
standard deviation  divided by the
square root of n) as n increases without
bound.
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28
How large should the sample
size be to permit the
application of the Central
Limit Theorem?
In most cases a sample size of
n = 30 or more assures that the
distribution will be approximately
normal and the theorem will apply.
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29
Central Limit Theorem
• For most x distributions, if we use a
sample size of 30 or larger, the x
distribution will be approximately
normal.
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30
Central Limit Theorem
• The mean of the sampling distribution is
the same as the mean of the original
distribution.
• The standard deviation of the sampling
distribution is equal to the standard
deviation of the original distribution
divided by the square root of the sample
size.
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31
Central Limit Theorem
Formula
x  
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32
Central Limit Theorem
Formula

x 
n
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33
Central Limit Theorem
Formula
z 
x  

x
x
x  

 / n
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34
Application of the Central
Limit Theorem
Records indicate that the packages shipped by a
certain trucking company have a mean weight of
510 pounds and a standard deviation of 90
pounds. One hundred packages are being shipped
today. What is the probability that their mean
weight will be:
a.
b.
c.
more than 530 pounds?
less than 500 pounds?
between 495 and 515 pounds?
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35
Are we authorized to use the
Normal Distribution?
Yes, we are attempting to
draw conclusions about
means of large samples.
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36
Applying the Central Limit
Theorem
What is the probability that their mean weight will
be more than 530 pounds?
Consider the distribution of sample means:
 x  510 ,  x  90 / 100  9
P( x > 530): z = 530 – 510 = 20 = 2.22
9
9
.0132
P(z > 2.22) = _______
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37
Applying the Central Limit
Theorem
What is the probability that their mean weight
will be less than 500 pounds?
P( x < 500): z = 500 – 510 = –10 = – 1.11
9
9
.1335
P(z < – 1.11) = _______
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38