Transcript Ch 7.2 Applications of the Normal Distribution

Modular 12
Ch 7.2 Part II to 7.3
Ch 7.2 Part II Applications of the Normal Distribution
Objective B : Finding the Z-score for a given probability
Objective C : Probability under a Normal Distribution
Objective D : Finding the Value of a Normal Random
Variable
Ch 7.3 Assessing Normality
Ch 7.4 The Normal Approximation to the Binomial
Probability Distribution (Skip)
Objective A : Continuity Correction
Objective B : A Normal Approximation to the Binomial
Ch 7.2 Applications of the Normal Distribution
Objective B : Finding the Z-score for a given probability
Area  0 . 5
Area  0 . 5
Area  0 . 5
Example 1 : Find the Z-score such that the area under the standard
normal curve to its left is 0.0418.
 1 . 73 )  0.0418
P ( Z  _____
From Table V
0 . 03
0 . 0418
 1 .7
Z ?
0
0 . 0418
Example 2 : Find the Z-score such that the area under the standard
normal curve to its right is 0.18.
From Table V
0 . 92
P ( Z  ______
)  0.18
0 . 02
1  0 . 18
 0 . 82
0 . 18
0 .9
0 . 8186
0 . 8212
(Closer to
0.82)
Z ?
Example 3 : Find the Z-score such that separates the middle 70%.
 1 . 04  Z  _____
1 . 04 )  0.70
P ( _____
70 %
Z ?
From Table V
0 . 04
Z ?
15 %
15 %
or 0 . 15
or 0 . 15
Z   1 . 04
 1 .0
Z  1 . 04 ( Due to symmetry )
0 . 1515
0 .1 4 9 2
(Closer to
0.15)
Ch 7.2 Part II Applications of the Normal Distribution
Objective B : Finding the Z-score for a given probability
Objective C : Probability under a Normal Distribution
Objective D : Finding the Value of a Normal Random
Variable
Ch 7.3 Assessing Normality
Ch 7.4 The Normal Approximation to the Binomial
Probability Distribution
Objective A : Continuity Correction
Objective B : A Normal Approximation to the Binomial
Ch 7.2 Applications of the Normal Distribution
Objective C : Probability under a Normal Distribution
Step 1 : Draw a normal curve and shade the desired area.
Step 2 : Convert the values of
X
to
Z
– scores using
Z 
X 

.
Step 3 : Use Table V to find the area to the left of each Z – score found
in Step 2.
Step 4 : Adjust the area from Step 3 to answer the question if necessary.
Example 1 : Assume that the random variable X is normally distributed
with mean   50 and a standard deviation   7 .
(Note: This is not a standard normal curve because   0
and   1 .)

(a) P ( X  58 )
Z 


X 

58  50
50
7

8
 1 . 14
58
X
0 .8 7 2 9
7
P ( Z  1 . 14 )
From Table V
 0 .8 7 2 9
0 1 . 14
Z
(b) P ( 45  X  63 )
X  63
X  45
Z 

X 

X 
Z 
45  50
0 . 2389
63  50

7
5
7
Z   0 . 71
0 . 9686

7



13
7
Z  1 . 86
P (  0 . 71  Z  1 . 86 )
From Table V
 0 . 71  0 . 2389
1 . 86  0 . 9686
 0 . 9686  0 . 2389
 0 . 7297 ( the whole blue area )
 0 . 71
0
1 . 86
Z
Example 2 : GE manufactures a decorative Crystal Clear 60-watt light
bulb that it advertises will last 1,500 hours. Suppose that
the lifetimes of the light bulbs are approximately normal
distributed, with a mean of 1,550 hours and a standard
deviation of 57 hours, what proportion of the light bulbs
will last more than 1650 hours?

P ( X  1650 )
X  1650 ,   1550 ,   57
Z 

0 . 0401
X 

0
1650  1550
57

0 . 9599
100
57
Z  1 . 75
P ( Z  1 . 75 )
From Table V
1 . 75  0 . 9599
 1  0 . 9599
 0 . 0401
1 . 75
Z
Ch 7.2 Part II Applications of the Normal Distribution
Objective B : Finding the Z-score for a given probability
Objective C : Probability under a Normal Distribution
Objective D : Finding the Value of a Normal Random
Variable
Objective E : Applications
Ch 7.3 Assessing Normality
Ch 7.4 The Normal Approximation to the Binomial
Probability Distribution
Objective A : Continuity Correction
Objective B : A Normal Approximation to the Binomial
Ch 7.2 Applications of the Normal Distribution
Objective D : Finding the Value of a Normal Random
Variable
Step 1 : Draw a normal curve and shade the desired area.
Step 2 : Find the corresponding area to the left of the cutoff score if
necessary.
Step 3 : Use Table V to find the Z – score that corresponds to the area to
the left of the cutoff score.
Step 4 : Obtain x from
Z
by the formula
Z 
X 

or
x    z
.
Ch 7.2 Applications of the Normal Distribution
Objective D : Find the Value of a Normal Distribution
Example 1 : The reading speed of 6th grade students is approximately
normal (bell-shaped) with a mean speed of 125 words per
minute and a standard deviation of 24 words per minute.
(a) What is the reading speed of a 6th grader whose reading speed is at
the 90% percentile?
  125 ,   24 , Z  1 . 28
90 %
or 0 . 9
Solve for X
Z  1 . 28
From Table V
0 . 08
Z 
1 . 28 
X 

X  125
24
1 . 28 ( 24 )  X  125
1 .2
0 . 8997
(Closer to
0.9)
0 . 9015
X  1 . 28 ( 24 )  125
X  155 . 72
words per minute
(b) Determine the reading rates of the middle 95% percentile.
  125 ,   24 , solve for X
95 %
or 0 . 95
Z   1.96
Z ?
Z ?
2 .5 %
2 .5 %
or 0 . 025
or 0 . 025
Z   1.96
Z  1.96 due to sym m etry
Z 
 1.96 
X 

X  125
24
 1.96 (24)  X  125
From Table V
0 .0 6
X   1.96 (24)  125
X  7 7 .9 6
 1 .9
0 . 025
words per
minute
Z  1 .9 6
Z 
1.96 
X 

X  125
24
1.96 (24)  X  125
X  1.96 (24)  125
X  1 7 2 .0 4
words per
minute
Ch 7.2 Part II Applications of the Normal Distribution
Objective B : Finding the Z-score for a given probability
Objective C : Probability under a Normal Distribution
Objective D : Finding the Value of a Normal Random
Variable
Ch 7.3 Assessing Normality
Ch 7.4 The Normal Approximation to the Binomial
Probability Distribution
Objective A : Continuity Correction
Objective B : A Normal Approximation to the Binomial
Ch 7.3 Assessing Normality
Ch 7.3 Normality Plot
Recall: A set of raw data is given, how would we know the data has
a normal distribution? Use histogram or stem leaf plot.
Histogram is designed for a large set of data.
For a very small set of data it is not feasible to use histogram to
determine whether the data has a bell-shaped curve or not.
We will use the normal probability plot to determine whether the
data were obtained from a normal distribution or not. If the data
were obtained from a normal distribution, the data distribution shape
is guaranteed to be approximately bell-shaped for n is less than 30.
Perfect normal curve. The curve is
aligned with the dots.
Almost a normal curve. The dots are
within the boundaries.
Not a normal curve. Data is outside the
boundaries.
Example 1: Determine whether the normal probability plot indicates
that the sample data could have come from a population
that is normally distributed.
(a)
Not a normal curve.
The sample data do not come from a normally distributed
population.
(b)
A normal curve.
The sample data comes from a normally distribute population.