Download Section 5 - Introduction Handout

Probability and Statistics
Normal Distributions
Chapter 6
Section 5
The Central Limit Theorem
Essential Question: What is the importance of the Central Limit Theorem in statistical
inference?
Student Objectives: The student will explain the underlying meaning of the Central Limit
Theorem.
The student will use sample estimates to construct an appropriate
sampling distribution for the sample mean.
The student will calculate the probability of a single event and
compare the answer to the probability based on the mean of a
small random sample.
Terms:
Central Limit Theorem
Mu of x bar ( µ x )
Sample mean ( x )
Sample standard deviation ( ! x )
Standard error
Unbiased
Variability of the statistic
Theorems
Central Limit Theorem
As the sample size continue to increase closer and closer
to the population size the following statements are true.
1. The sample will have a normal distribution.
(
2. The sample mean will approach the population mean. lim x = µ
n!N
)
3. The standard deviation will take on the intermediate value of the
population standard deviation divided by the square root of the sample
size. However, because of the increasing sample size this value will
"
#
&
approach zero. % lim " x = lim
= 0(
n!N
n!N
$
'
n
4. The probability of an interval that contains the population mean
(
)
will approach 1. lim P ( x1 < µ < x2 ) = 1
n!N
5. The probability of an interval that does NOT contain the population
(
) (
mean will approach 0. lim P ( x1 < µ ) = 0 or lim P ( x > µ ) = 0
n!N
n!N
)
Shown below is a graphical representation of what happens to the
normal bell shaped curve as the sample size increases closer and
closer to the population size.
Graphing Calculator Skills:
None
Formulas:
You are to use following formula (1) for a single x value.
You are to use formula (2) for a given sample size.
You may only use these formulas!
DO NOT USE THE BOOK FORMULA!
Formula (1)
Formula (2)
x " µ'
$
P& z !
)
%
# (
$
n ( x " µ)'
P& z !
)
#
%
(
x " µ'
$
P& z <
)
%
# (
x " µ'
$
P& z *
)
%
# (
or
x " µ'
$
P& z >
)
%
# (
$
n ( x " µ)'
P& z <
)
#
%
(
$
n ( x " µ)'
P& z *
)
#
%
(
or
$
n ( x " µ)'
P& z >
)
#
%
(
Sample Questions:
1.
Suppose that it is known that the time spent by customers in the local coffee shop is
normally distributed with a mean of 24 minutes and a standard deviation of 6 minutes.
a.
Find the probability that an individual customer will spend more than 26
minutes in the coffee shop.
b.
Find the probability that a random sample of 9 customers will have a mean
stay of more than 26 minutes in the coffee shop.
c.
Find the probability that a random sample of 64 customers will have a mean
stay of more than 26 minutes in the coffee shop.
d.
Find the probability that a random sample of 100 customers will have a mean
stay of more than 26 minutes in the coffee shop.
2.
e.
Find the probability that a random sample of 144 customers will have a mean
stay of more than 26 minutes in the coffee shop.
f.
Explain what is happening to your answers in parts a through e by using the
Central Limit Theorem.
Suppose that it is known that the time spent by customers in the local coffee shop is
normally distributed with a mean of 24 minutes and a standard deviation of 6 minutes.
a.
Find the probability that an individual customer will spend between 22 to 25
minutes in the coffee shop.
b.
Find the probability that a random sample of 9 customers will have a mean
stay in the coffee shop between 22 to 25 minutes.
c.
Find the probability that a random sample of 64 customers will have a mean
stay in the coffee shop between 22 to 25 minutes.
d.
Find the probability that a random sample of 100 customers will have a mean
stay in the coffee shop between 22 to 25 minutes.
e.
Find the probability that a random sample of 144 customers will have a
mean stay in the coffee shop between 22 to 25 minutes.
f.
Explain what is happening to your answers in parts a through e by using the
Central Limit Theorem.
Homework Assignment:
Pages 303 - 307
Pages 303 - 307
Exercises: #1 - 19, odd
Exercises: #2 - 20, even
SAMPLE QUESTIONS ANSWERS
1.
Suppose that it is known that the time spent by customers in the local coffee shop is
normally distributed with a mean of 24 minutes and a standard deviation of 6 minutes.
a.
Find the probability that an individual customer will spend more than 26
minutes in the coffee shop.
P ( x > 26 )
26 ! 24 %
"
P$ z >
'
#
6 &
2%
"
P$ z > '
#
6&
P ( z > 0.33)
0.5000 ! 0.1293
0.3707
0.33
b.
Find the probability that a random sample of 9 customers will have a mean
stay of more than 26 minutes in the coffee shop.
P ( x > 26 )
"
( 26 ! 24 ) 9 %
P$ z >
'
6
#
&
2 ( 3) %
"
P$ z >
#
6 '&
1.00
6%
"
P$ z > '
#
6&
P ( z > 1.00 )
0.5000 ! 0.3413
0.1587
c.
Find the probability that a random sample of 64 customers will have a mean
stay of more than 26 minutes in the coffee shop.
P ( x > 26 )
"
( 26 ! 24 ) 64 %
P$ z >
'
6
#
&
2 (8)%
"
P$ z >
#
6 '&
2.67
16 %
"
P$ z > '
#
6&
P ( z > 2.67 )
0.5000 ! 0.4962
0.0038
d.
Find the probability that a random sample of 100 customers will have a mean
stay of more than 26 minutes in the coffee shop.
P ( x > 26 )
"
( 26 ! 24 ) 100 %
P$ z >
'
6
#
&
2 (10 ) %
"
P$ z >
#
6 '&
3.33
20 %
"
P$ z > '
#
6&
P ( z > 3.33)
0.5000 ! 0.4996
0.0004
e.
Find the probability that a random sample of 144 customers will have a mean
stay of more than 26 minutes in the coffee shop.
P ( x > 26 )
"
( 26 ! 24 ) 144 %
P$ z >
'
6
#
&
2 (12 ) %
"
P$ z >
#
6 '&
4.00
24 %
"
P$ z > '
#
6&
P ( z > 4.00 )
0.5000 ! 0.4999
0.0001
f.
Explain what is happening to your answers in parts a through e by using the
Central Limit Theorem.
The Central Limit Theorem states that if the probability interval does not
contain the population mean and the sample size continues increase closer
and closer to the population size then the probability will continue to decrease
and get closer and closer to 0.
2.
Suppose that it is known that the time spent by customers in the local coffee shop is
normally distributed with a mean of 24 minutes and a standard deviation of 6 minutes.
a.
Find the probability that an individual customer will spend between 22 to 25
minutes in the coffee shop.
P ( 22 ! x ! 25 )
25 " 24 &
# 22 " 24
P%
!z!
(
$ 6
6 '
1&
# "2
P%
!z! (
$ 6
6'
P ( "0.33 ! z ! 0.17 )
-0.33
b.
0.1293 + 0.0675
0.1968
0.17
Find the probability that a random sample of 9 customers will have a mean
stay in the coffee shop between 22 to 25 minutes.
(
P ( 22 ! x ! 25 )
)
# 22 " 24 9
( 25 " 24 ) 9 &(
P%
!z!
6
6
%$
('
# ( "2 ) ( 3)
(1)( 3) &
P%
!z!
$ 6
6 ('
-1.00
0.50
3&
# "6
P%
!z! (
$ 6
6'
P ( "1.00 ! z ! 0.50 )
0.3413 + 0.1915
0.5328
c.
Find the probability that a random sample of 64 customers will have a mean
stay in the coffee shop between 22 to 25 minutes.
(
P ( 22 ! x ! 25 )
)
# 22 " 24 64
25 " 24 ) 64 &
(
P%
!z!
(
6
6
%$
('
# ( "2 ) ( 8 )
(1)( 8 ) &
P%
!z!
$
6
6 ('
8&
# "16
P%
!z! (
$ 6
6'
-2.67
d.
P ( "2.67 ! z ! 1.33)
1.33
0.4962 + 0.4082
0.9044
Find the probability that a random sample of 100 customers will have a mean
stay in the coffee shop between 22 to 25 minutes.
(
P ( 22 ! x ! 25 )
)
# 22 " 24 100
25 " 24 ) 100 &
(
P%
!z!
(
6
6
%$
('
# ( "2 ) (10 )
(1)(10 ) &
P%
!z!
$
6
6 ('
-3.33
1.67
10 &
# "20
P%
!z! (
$ 6
6'
P ( "3.33 ! z ! 1.67 )
0.4996 + 0.4525
0.9521
e.
Find the probability that a random sample of 144 customers will have a
mean stay in the coffee shop between 22 to 25 minutes.
(
P ( 22 ! x ! 25 )
)
# 22 " 24 144
25 " 24 ) 144 &
(
P%
!z!
(
6
6
%$
('
# ( "2 ) (12 )
(1)(12 ) &
P%
!z!
$
6
6 ('
-4.00
2.00
12 &
# "24
P%
!z! (
$ 6
6'
P ( "4.00 ! z ! 2.00 )
0.4999 + 0.4772
0.9771
f.
Explain what is happening to your answers in parts a through e by using the
Central Limit Theorem.
The Central Limit Theorem states that if the probability interval contains the
population mean and the sample size continues to increase closer and closer
to the population size then the probability will continue to increase and get closer
and closer to 1.