Download Ch. 5

Chapter 5
Normal Probability Distributions
1
Chapter Outline
• 5.1 Introduction to Normal Distributions and the
Standard Normal Distribution
• 5.2 Normal Distributions: Finding Probabilities
• 5.3 Normal Distributions: Finding Values
• 5.4 Sampling Distributions and the Central Limit
Theorem
• 5.5 Normal Approximations to Binomial
Distributions
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Section 5.1
Introduction to Normal Distributions
3
Section 5.1 Objectives
• Interpret graphs of normal probability distributions
• Find areas under the standard normal curve
4
Properties of a Normal Distribution
Continuous random variable
• Has an infinite number of possible values that can be
represented by an interval on the number line.
Hours spent studying in a day
0
3
6
9
12
15
18
21
24
The time spent
studying can be any
number between 0
and 24.
Continuous probability distribution
• The probability distribution of a continuous random
variable.
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Properties of Normal Distributions
Normal distribution
• A continuous probability distribution for a random
variable, x.
• The most important continuous probability
distribution in statistics.
• The graph of a normal distribution is called the
normal curve.
x
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Properties of Normal Distributions
1. The mean, median, and mode are equal.
2. The normal curve is bell-shaped and symmetric
about the mean.
3. The total area under the curve is equal to one.
4. The normal curve approaches, but never touches the
x-axis as it extends farther and farther away from the
mean.
Total area = 1
μ
x
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Properties of Normal Distributions
5. Between μ – σ and μ + σ (in the center of the curve),
the graph curves downward. The graph curves
upward to the left of μ – σ and to the right of μ + σ.
The points at which the curve changes from curving
upward to curving downward are called the
inflection points.
Inflection points
μ  3σ
μ  2σ
μσ
μ
μ+σ
μ + 2σ
μ + 3σ
x
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Means and Standard Deviations
• A normal distribution can have any mean and any
positive standard deviation.
• The mean gives the location of the line of symmetry.
• The standard deviation describes the spread of the
data.
μ = 3.5
σ = 1.5
μ = 3.5
σ = 0.7
μ = 1.5
σ = 0.7
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Example: Understanding Mean and
Standard Deviation
1. Which curve has the greater mean?
Solution:
Curve A has the greater mean (The line of symmetry
of curve A occurs at x = 15. The line of symmetry of
curve B occurs at x = 12.)
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Example: Understanding Mean and
Standard Deviation
2. Which curve has the greater standard deviation?
Solution:
Curve B has the greater standard deviation (Curve
B is more spread out than curve A.)
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Example: Interpreting Graphs
The heights of fully grown white oak trees are normally
distributed. The curve represents the distribution. What
is the mean height of a fully grown white oak tree?
Estimate the standard deviation.
Solution:
μ = 90 (A normal
curve is symmetric
about the mean)
σ = 3.5 (The inflection
points are one standard
deviation away from
the mean)
12
The Standard Normal Distribution
Standard normal distribution
• A normal distribution with a mean of 0 and a standard
deviation of 1.
Area = 1
3
2
1
z
0
1
2
3
• Any x-value can be transformed into a z-score by
using the formula
Value - Mean
x-
z

Standard deviation

13
The Standard Normal Distribution
• If each data value of a normally distributed random
variable x is transformed into a z-score, the result will
be the standard normal distribution.
Normal Distribution

z

x
x-
Standard Normal
Distribution

1
0
z
• Use the Standard Normal Table to find the
cumulative area under the standard normal curve.
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Properties of the Standard Normal
Distribution
1. The cumulative area is close to 0 for z-scores close
to z = 3.49.
2. The cumulative area increases as the z-scores
increase.
Area is
close to 0
z = 3.49
z
3
2
1
0
1
2
3
15
Properties of the Standard Normal
Distribution
3. The cumulative area for z = 0 is 0.5000.
4. The cumulative area is close to 1 for z-scores close
to z = 3.49.
Area
is close to 1
z
3
2
1
0
1
z=0
Area is 0.5000
2
3
z = 3.49
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Example: Using The Standard Normal Table
Find the cumulative area that corresponds to a z-score of
1.15.
Solution:
Find 1.1 in the left hand column.
Move across the row to the column under 0.05
The area to the left of z = 1.15 is 0.8749.
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Example: Using The Standard Normal Table
Find the cumulative area that corresponds to a z-score of
-0.24.
Solution:
Find -0.2 in the left hand column.
Move across the row to the column under 0.04
The area to the left of z = -0.24 is 0.4052.
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Finding Areas Under the Standard
Normal Curve
1. Sketch the standard normal curve and shade the
appropriate area under the curve.
2. Find the area by following the directions for each
case shown.
a. To find the area to the left of z, find the area that
corresponds to z in the Standard Normal Table.
2.
The area to the left
of z = 1.23 is 0.8907
1. Use the table to find the
area for the z-score
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Finding Areas Under the Standard
Normal Curve
b. To find the area to the right of z, use the Standard
Normal Table to find the area that corresponds to
z. Then subtract the area from 1.
2. The area to the
left of z = 1.23
is 0.8907.
3. Subtract to find the area
to the right of z = 1.23:
1  0.8907 = 0.1093.
1. Use the table to find the
area for the z-score.
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Finding Areas Under the Standard
Normal Curve
c. To find the area between two z-scores, find the
area corresponding to each z-score in the
Standard Normal Table. Then subtract the
smaller area from the larger area.
2. The area to the
left of z = 1.23
is 0.8907.
3. The area to the
left of z = 0.75
is 0.2266.
4. Subtract to find the area of
the region between the two
z-scores:
0.8907  0.2266 = 0.6641.
1. Use the table to find the
area for the z-scores.
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Example: Finding Area Under the
Standard Normal Curve
Find the area under the standard normal curve to the left
of z = -0.99.
Solution:
0.1611
0.99
z
0
From the Standard Normal Table, the area is
equal to 0.1611.
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Example: Finding Area Under the
Standard Normal Curve
Find the area under the standard normal curve to the
right of z = 1.06.
Solution:
1  0.8554 = 0.1446
0.8554
z
0
1.06
From the Standard Normal Table, the area is equal to
0.1446.
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Example: Finding Area Under the
Standard Normal Curve
Find the area under the standard normal curve between
z = 1.5 and z = 1.25.
Solution:
0.8944 0.0668 = 0.8276
0.8944
0.0668
1.50
0
1.25
z
From the Standard Normal Table, the area is equal to
0.8276.
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Section 5.1 Summary
• Interpreted graphs of normal probability distributions
• Found areas under the standard normal curve
25
Test1 data before and after the curve:
Section 5.2
Normal Distributions: Finding
Probabilities
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Section 5.2 Objectives
• Find probabilities for normally distributed variables
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Probability and Normal Distributions
• If a random variable x is normally distributed, you
can find the probability that x will fall in a given
interval by calculating the area under the normal
curve for that interval.
μ = 500
σ = 100
P(x < 600) = Area
x
μ =500 600
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Probability and Normal Distributions
Normal Distribution
Standard Normal Distribution
μ = 500 σ = 100
μ=0 σ=1
P(x < 600)
x   600  500
z

1

100
P(z < 1)
z
x
μ =500 600
μ=0 1
Same Area
P(x < 600) = P(z < 1)
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Example: Finding Probabilities for
Normal Distributions
A survey indicates that people use their computers an
average of 2.4 years before upgrading to a new
machine. The standard deviation is 0.5 year. A computer
owner is selected at random. Find the probability that he
or she will use it for fewer than 2 years before
upgrading. Assume that the variable x is normally
distributed.
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Solution: Finding Probabilities for
Normal Distributions
Normal Distribution
μ = 2.4 σ = 0.5
Standard Normal Distribution
μ=0 σ=1
x   2  2.4
z

 0.80

0.5
P(x < 2)
P(z < -0.80)
0.2119
z
x
2 2.4
-0.80 0
P(x < 2) = P(z < -0.80) = 0.2119
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Example: Finding Probabilities for
Normal Distributions
A survey indicates that for each trip to the supermarket,
a shopper spends an average of 45 minutes with a
standard deviation of 12 minutes in the store. The length
of time spent in the store is normally distributed and is
represented by the variable x. A shopper enters the store.
Find the probability that the shopper will be in the store
for between 24 and 54 minutes.
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Solution: Finding Probabilities for
Normal Distributions
Normal Distribution
μ = 45 σ = 12
x-
Standard Normal Distribution
μ=0 σ=1
24 - 45
 -1.75

12
x -  54 - 45
z2 

 0.75

12
z1 
P(24 < x < 54)

P(-1.75 < z < 0.75)
0.7734
0.0401
x
24
45 54
z
-1.75
0 0.75
P(24 < x < 54) = P(-1.75 < z < 0.75)
= 0.7734 – 0.0401 = 0.7333
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Example: Finding Probabilities for
Normal Distributions
Find the probability that the shopper will be in the store
more than 39 minutes. (Recall μ = 45 minutes and
σ = 12 minutes)
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Solution: Finding Probabilities for
Normal Distributions
Normal Distribution
μ = 45 σ = 12
z
P(x > 39)
Standard Normal Distribution
μ=0 σ=1
x-


39 - 45
 -0.50
12
P(z > -0.50)
0.3085
z
x
39 45
-0.50 0
P(x > 39) = P(z > -0.50) = 1– 0.3085 = 0.6915
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Example: Finding Probabilities for
Normal Distributions
If 200 shoppers enter the store, how many shoppers
would you expect to be in the store more than 39
minutes?
Solution:
Recall P(x > 39) = 0.6915
200(0.6915) =138.3 (or about 138) shoppers
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Example: Using Technology to find
Normal Probabilities
Assume that cholesterol levels of men in the United
States are normally distributed, with a mean of 215
milligrams per deciliter and a standard deviation of 25
milligrams per deciliter. You randomly select a man
from the United States. What is the probability that his
cholesterol level is less than 175? Use a technology tool
to find the probability.
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In fact, modern humans are the only adult mammals, excluding some domesticated
animals, with a mean LDL level over 80 mg/dl and a total cholesterol over 160 mg/dl
15 and 16 (Fig. 1). Thus, although an LDL level of 50 to 70 mg/dl seems excessively low
by modern American standards, it is precisely the normal range for individuals living the
lifestyle and eating the diet for which we are genetically adapted.
Solution: Using Technology to find
Normal Probabilities
Must specify the mean, standard deviation, and the xvalue(s) that determine the interval.
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Section 5.2 Summary
• Found probabilities for normally distributed variables
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Section 5.3
Normal Distributions: Finding
Values
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Section 5.3 Objectives
• Find a z-score given the area under the normal curve
• Transform a z-score to an x-value
• Find a specific data value of a normal distribution
given the probability
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Finding values Given a Probability
• In section 5.2 we were given a normally distributed
random variable x and we were asked to find a
probability.
• In this section, we will be given a probability and we
will be asked to find the value of the random variable
x.
5.2
x
z
probability
5.3
44
Example: Finding a z-Score Given an
Area
Find the z-score that corresponds to a cumulative area of
0.3632.
Solution:
0.3632
z
z 0
45
Solution: Finding a z-Score Given an
Area
• Locate 0.3632 in the body of the Standard Normal
Table.
The z-score
is -0.35.
• The values at the beginning of the corresponding row
and at the top of the column give the z-score.
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Example: Finding a z-Score Given an
Area
Find the z-score that has 10.75% of the distribution’s
area to its right.
Solution:
1 – 0.1075
= 0.8925
0.1075
z
0
z
Because the area to the right is 0.1075, the
cumulative area is 0.8925.
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Solution: Finding a z-Score Given an
Area
• Locate 0.8925 in the body of the Standard Normal
Table.
The z-score
is 1.24.
• The values at the beginning of the corresponding row
and at the top of the column give the z-score.
48
Example: Finding a z-Score Given a
Percentile
Find the z-score that corresponds to P5.
Solution:
The z-score that corresponds to P5 is the same z-score that
corresponds to an area of 0.05.
0.05
z
0
z
The areas closest to 0.05 in the table are 0.0495 (z = -1.65)
and 0.0505 (z = -1.64). Because 0.05 is halfway between the
two areas in the table, use the z-score that is halfway
between -1.64 and -1.65. The z-score is -1.645.
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Transforming a z-Score to an x-Score
To transform a standard z-score to a data value x in a
given population, use the formula
x = μ + zσ
50
Example: Finding an x-Value
The speeds of vehicles along a stretch of highway are
normally distributed, with a mean of 67 miles per hour
and a standard deviation of 4 miles per hour. Find the
speeds x corresponding to z-sores of 1.96, -2.33, and 0.
Solution: Use the formula x = μ + zσ
• z = 1.96: x = 67 + 1.96(4) = 74.84 miles per hour
• z = -2.33: x = 67 + (-2.33)(4) = 57.68 miles per hour
• z = 0:
x = 67 + 0(4) = 67 miles per hour
Notice 74.84 mph is above the mean, 57.68 mph is
below the mean, and 67 mph is equal to the mean.
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Example: Finding a Specific Data Value
Scores for a civil service exam are normally distributed,
with a mean of 75 and a standard deviation of 6.5. To be
eligible for civil service employment, you must score in
the top 5%. What is the lowest score you can earn and
still be eligible for employment?
Solution:
1 – 0.05
= 0.95
0
75
5%
?
?
z
x
An exam score in the top 5%
is any score above the 95th
percentile. Find the z-score
that corresponds to a
cumulative area of 0.95.
52
Solution: Finding a Specific Data Value
From the Standard Normal Table, the areas closest to
0.95 are 0.9495 (z = 1.64) and 0.9505 (z = 1.65).
Because 0.95 is halfway between the two areas in the
table, use the z-score that is halfway between 1.64 and
1.65. That is, z = 1.645.
5%
0
75
1.645
?
z
x
53
Solution: Finding a Specific Data Value
Using the equation x = μ + zσ
x = 75 + 1.645(6.5) ≈ 85.69
5%
0
1.645
75 85.69
z
x
The lowest score you can earn and still be eligible
for employment is 86.
54
Section 5.3 Summary
• Found a z-score given the area under the normal
curve
• Transformed a z-score to an x-value
• Found a specific data value of a normal distribution
given the probability
55
Section 5.4
Sampling Distributions and the
Central Limit Theorem
56
Section 5.4 Objectives
• Find sampling distributions and verify their properties
• Interpret the Central Limit Theorem
• Apply the Central Limit Theorem to find the
probability of a sample mean
57
Sampling Distributions
Sampling distribution
• The probability distribution of a sample statistic.
• Formed when samples of size n are repeatedly taken
from a population.
• e.g. Sampling distribution of sample means
58
Properties of Sampling Distributions of
Sample Means
1. The mean of the sample means,  x , is equal to the
population mean μ.
x  
2. The standard deviation of the sample means,  x , is
equal to the population standard deviation, σ
divided by the square root of the sample size, n.
x 

n
• Called the standard error of the mean.
59
Example: Sampling Distribution of
Sample Means
The population values {1, 3, 5, 7} are written on slips of
paper and put in a box. Two slips of paper are randomly
selected, with replacement.
a. Find the mean, variance, and standard deviation of
the population.
Solution:
Mean:
x

4
N
2

(
x


)
Variance:  2 
5
N
Standard Deviation:   5  2.236
61
Example: Sampling Distribution of
Sample Means
b. Graph the probability histogram for the population
values.
Solution:
Probability Histogram of
Population of x
P(x)
0.25
Probability
All values have the
same probability of
being selected (uniform
distribution)
x
1
3
5
7
Population values
62
Example: Sampling Distribution of
Sample Means
c. List all the possible samples of size n = 2 and
calculate the mean of each sample.
Solution:
Sample
1, 1
1, 3
1, 5
1, 7
3, 1
3, 3
3, 5
3, 7
x
1
2
3
4
2
3
4
5
Sample
5, 1
5, 3
5, 5
5, 7
7, 1
7, 3
7, 5
7, 7
x
3
4
5
6
4
5
6
7
These means
form the
sampling
distribution of
sample means.
63
Example: Sampling Distribution of
Sample Means
d. Construct the probability distribution of the sample
means.
Solution:
f
Probability
x x f Probability
1
1
0.0625
2
3
4
5
2
3
4
3
0.1250
0.1875
0.2500
0.1875
6
7
2
1
0.1250
0.0625
64
Example: Sampling Distribution of
Sample Means
e. Find the mean, variance, and standard deviation of
the sampling distribution of the sample means.
Solution:
The mean, variance, and standard deviation of the
16 sample means are:
x  4
2
5

 x2  
 2.5  x  2.5  1.581
2
n
These results satisfy the properties of sampling
distributions of sample means.
x    4
x 

n

5 2.236

 1.581
2
2
65
Example: Sampling Distribution of
Sample Means
f. Graph the probability histogram for the sampling
distribution of the sample means.
Solution:
P(x)
Probability
0.25
Probability Histogram of
Sampling Distribution of x
0.20
0.15
0.10
0.05
x
2
3
4
5
6
The shape of the
graph is symmetric
and bell shaped. It
approximates a
normal distribution.
7
Sample mean
66
The Central Limit Theorem
1. If samples of size n  30, are drawn from any
population with mean =  and standard deviation = ,
x

then the sampling distribution of the sample means
approximates a normal distribution. The greater the
sample size, the better the approximation.
xx
x x
x x x
x x x x x

x
67
The Central Limit Theorem
2. If the population itself is normally distributed,
x

the sampling distribution of the sample means is
normally distribution for any sample size n.
xx
x x
x x x
x x x x x

x
68
The Central Limit Theorem
• In either case, the sampling distribution of sample
means has a mean equal to the population mean.
x  
• The sampling distribution of sample means has a
variance equal to 1/n times the variance of the
population and a standard deviation equal to the
population standard deviation divided by the square
root of n.
2
 x2 
x 

n

n
Variance
Standard deviation (standard
error of the mean)
69
The Central Limit Theorem
1.
Any Population Distribution
Distribution of Sample Means,
n ≥ 30
2.
Normal Population Distribution
Distribution of Sample Means,
(any n)
70
Example: Interpreting the Central Limit
Theorem
Phone bills for residents of a city have a mean of $64
and a standard deviation of $9. Random samples of 36
phone bills are drawn from this population and the mean
of each sample is determined. Find the mean and
standard error of the mean of the sampling distribution.
Then sketch a graph of the sampling distribution of
sample means.
71
Solution: Interpreting the Central Limit
Theorem
• The mean of the sampling distribution is equal to the
population mean
 x    64
• The standard error of the mean is equal to the
population standard deviation divided by the square
root of n.
x 

n

9
 1.5
36
72
Solution: Interpreting the Central Limit
Theorem
• Since the sample size is greater than 30, the sampling
distribution can be approximated by a normal
distribution with
 x  1.5
 x  64
73
Example: Interpreting the Central Limit
Theorem
The heights of fully grown white oak trees are normally
distributed, with a mean of 90 feet and standard
deviation of 3.5 feet. Random samples of size 4 are
drawn from this population, and the mean of each
sample is determined. Find the mean and standard error
of the mean of the sampling distribution. Then sketch a
graph of the sampling distribution of sample means.
74
Solution: Interpreting the Central Limit
Theorem
• The mean of the sampling distribution is equal to the
population mean
x    90
• The standard error of the mean is equal to the
population standard deviation divided by the square
root of n.
x 

n

3.5
 1.75
4
75
Solution: Interpreting the Central Limit
Theorem
• Since the population is normally distributed, the
sampling distribution of the sample means is also
normally distributed.
 x  1.75
 x  90
76
Probability and the Central Limit
Theorem
• To transform x to a z-score
x  x x  
Value-Mean
z



Standard Error
x
n
77
Example: Probabilities for Sampling
Distributions
The graph shows the length of
time people spend driving each
day. You randomly select 50
drivers age 15 to 19. What is the
probability that the mean time
they spend driving each day is
between 24.7 and 25.5 minutes?
Assume that σ = 1.5 minutes.
79
Solution: Probabilities for Sampling
Distributions
From the Central Limit Theorem (sample size is greater
than 30), the sampling distribution of sample means is
approximately normal with
 x    25
x 

n

1.5
 0.21213
50
80
Solution: Probabilities for Sampling
Distributions
Normal Distribution
Standard Normal Distribution
μ = 25 σ = 0.21213 x -  24.7 - 25
μ=0 σ=1
z1 

 -1.41

1.5
n
50
P(-1.41 < z < 2.36)
P(24.7 < x < 25.5)
z2 
x-

n

25.5 - 25
 2.36
1.5
50
0.9909
0.0793
x
24.7
25
25.5
z
-1.41
0
2.36
P(24 < x < 54) = P(-1.41 < z < 2.36)
= 0.9909 – 0.0793 = 0.9116
81
Example: Probabilities for x and x
A bank auditor claims that credit card balances are
normally distributed, with a mean of $2870 and a
standard deviation of $900.
1. What is the probability that a randomly selected
credit card holder has a credit card balance less than
$2500?
Solution:
You are asked to find the probability associated with
a certain value of the random variable x.
82
Solution: Probabilities for x and x
Normal Distribution
μ = 2870 σ = 900
P(x < 2500)
z
Standard Normal Distribution
μ=0 σ=1
x-


2500 - 2870
 -0.41
900
P(z < -0.41)
0.3409
x
2500 2870
z
-0.41
0
P( x < 2500) = P(z < -0.41) = 0.3409
83
Example: Probabilities for x and x
2. You randomly select 25 credit card holders. What is
the probability that their mean credit card balance
is less than $2500?
Solution:
You are asked to find the probability associated with
a sample mean x.
 x    2870
x 

n

900
 180
25
84
Solution: Probabilities for x and x
Normal Distribution
μ = 2870 σ = 180
z
Standard Normal Distribution
μ=0 σ=1
x-

n

2500 - 2870
 -2.06
900
25
P(z < -2.06)
P(x < 2500)
0.0197
x
2500
2870
z
-2.06
0
P( x < 2500) = P(z < -2.06) = 0.0197
85
Solution: Probabilities for x and x
• There is a 34% chance that an individual will have a
balance less than $2500.
• There is only a 2% chance that the mean of a sample
of 25 will have a balance less than $2500 (unusual
event).
• It is possible that the sample is unusual or it is
possible that the auditor’s claim that the mean is
$2870 is incorrect.
86
Section 5.4 Summary
• Found sampling distributions and verify their
properties
• Interpreted the Central Limit Theorem
• Applied the Central Limit Theorem to find the
probability of a sample mean
87
Section 5.5
Normal Approximations to Binomial
Distributions
88
Section 5.5 Objectives
• Determine when the normal distribution can
approximate the binomial distribution
• Find the correction for continuity
• Use the normal distribution to approximate binomial
probabilities
89
Normal Approximation to a Binomial
• The normal distribution is used to approximate the
binomial distribution when it would be impractical to
use the binomial distribution to find a probability.
Normal Approximation to a Binomial Distribution
• If np  5 and nq  5, then the binomial random
variable x is approximately normally distributed with
 mean μ = np
 standard deviation σ  npq
90
Normal Approximation to a Binomial
• Binomial distribution: p = 0.25
• As n increases the histogram approaches a normal
curve.
91
Example: Approximating the Binomial
Decide whether you can use the normal distribution to
approximate x, the number of people who reply yes. If
you can, find the mean and standard deviation.
1. Fifty-one percent of adults in the U.S. whose New
Year’s resolution was to exercise more achieved
their resolution. You randomly select 65 adults in
the U.S. whose resolution was to exercise more
and ask each if he or she achieved that resolution.
92
Solution: Approximating the Binomial
• You can use the normal approximation
n = 65, p = 0.51, q = 0.49
np = (65)(0.51) = 33.15 ≥ 5
nq = (65)(0.49) = 31.85 ≥ 5
• Mean: μ = np = 33.15
• Standard Deviation: σ  npq  65  0.51 0.49  4.03
93
Example: Approximating the Binomial
Decide whether you can use the normal distribution to
approximate x, the number of people who reply yes. If
you can find, find the mean and standard deviation.
2. Fifteen percent of adults in the U.S. do not make
New Year’s resolutions. You randomly select 15
adults in the U.S. and ask each if he or she made a
New Year’s resolution.
94
Solution: Approximating the Binomial
• You cannot use the normal approximation
n = 15, p = 0.15, q = 0.85
np = (15)(0.15) = 2.25 < 5
nq = (15)(0.85) = 12.75 ≥ 5
• Because np < 5, you cannot use the normal
distribution to approximate the distribution of x.
95
Correction for Continuity
• The binomial distribution is discrete and can be
represented by a probability histogram.
• To calculate exact binomial probabilities, the
binomial formula is used for each value of x and the
results are added.
• Geometrically this corresponds to adding the areas of
bars in the probability histogram.
96
Correction for Continuity
• When you use a continuous normal distribution to
approximate a binomial probability, you need to
move 0.5 unit to the left and right of the midpoint to
include all possible x-values in the interval
(correction for continuity).
Exact binomial probability
P(x = c)
c
Normal approximation
P(c – 0.5 < x < c + 0.5)
c– 0.5 c c+ 0.5
97
Example: Using a Correction for
Continuity
Use a correction for continuity to convert the binomial
intervals to a normal distribution interval.
1. The probability of getting between 270 and 310
successes, inclusive.
Solution:
• The discrete midpoint values are 270, 271, …, 310.
• The corresponding interval for the continuous normal
distribution is
269.5 < x < 310.5
98
Example: Using a Correction for
Continuity
Use a correction for continuity to convert the binomial
intervals to a normal distribution interval.
2. The probability of getting at least 158 successes.
Solution:
• The discrete midpoint values are 158, 159, 160, ….
• The corresponding interval for the continuous normal
distribution is
x > 157.5
99
Example: Using a Correction for
Continuity
Use a correction for continuity to convert the binomial
intervals to a normal distribution interval.
3. The probability of getting less than 63 successes.
Solution:
• The discrete midpoint values are …,60, 61, 62.
• The corresponding interval for the continuous normal
distribution is
x < 62.5
100
Using the Normal Distribution to
Approximate Binomial Probabilities
In Words
1. Verify that the binomial
distribution applies.
2. Determine if you can use
the normal distribution to
approximate x, the binomial
variable.
3. Find the mean  and
standard deviation for the
distribution.
In Symbols
Specify n, p, and q.
Is np  5?
Is nq  5?
  np
  npq
101
Using the Normal Distribution to
Approximate Binomial Probabilities
In Words
4. Apply the appropriate
continuity correction.
Shade the corresponding
area under the normal
curve.
5. Find the corresponding zscore(s).
6. Find the probability.
In Symbols
Add or subtract 0.5
from endpoints.
z
x-

Use the Standard
Normal Table.
102
Example: Approximating a Binomial
Probability
Fifty-one percent of adults in the U. S. whose New
Year’s resolution was to exercise more achieved their
resolution. You randomly select 65 adults in the U. S.
whose resolution was to exercise more and ask each if
he or she achieved that resolution. What is the
probability that fewer than forty of them respond yes?
(Source: Opinion Research Corporation)
Solution:
• Can use the normal approximation (see slide 89)
μ = 65∙0.51 = 33.15 σ  65  0.51  0.49  4.03
103
Solution: Approximating a Binomial
Probability
• Apply the continuity correction:
Fewer than 40 (…37, 38, 39) corresponds to the
continuous normal distribution interval x < 39.5
Normal Distribution
μ = 33.15 σ = 4.03
P(x < 39.5)
z
x-

Standard Normal
μ=0 σ=1

39.5 - 33.15
 1.58
4.03
P(z < 1.58)
0.9429
x
μ =33.15
39.5
z
μ =0
1.58
P(z < 1.58) = 0.9429
104
Example: Approximating a Binomial
Probability
A survey reports that 86% of Internet users use
Windows® Internet Explorer ® as their browser. You
randomly select 200 Internet users and ask each whether
he or she uses Internet Explorer as his or her browser.
What is the probability that exactly 176 will say yes?
(Source: 0neStat.com)
Solution:
• Can use the normal approximation
np = (200)(0.86) = 172 ≥ 5 nq = (200)(0.14) = 28 ≥ 5
μ = 200∙0.86 = 172
σ  200  0.86  0.14  4.91
105
Solution: Approximating a Binomial
Probability
• Apply the continuity correction:
Exactly 176 corresponds to the continuous normal
distribution interval 175.5 < x < 176.5
Normal Distribution
Standard Normal
μ = 172 σ = 4.91 x -  175.5 - 172
μ=0 σ=1
z 

 0.71
1
P(175.5 < x < 176.5)

4.91
x -  176.5 - 172
z2 

 0.92

4.91
P(0.71 < z < 0.92)
0.8212
0.7611
z
x
μ =172 176.5
175.5
μ =0 0.92
0.71
P(0.71 < z < 0.92) = 0.8212 – 0.7611 = 0.0601
106
Section 5.5 Summary
• Determined when the normal distribution can
approximate the binomial distribution
• Found the correction for continuity
• Used the normal distribution to approximate binomial
probabilities
107
Chapter 5: Normal Probability Distributions
Elementary Statistics:
Picturing the World
Fifth Edition
by Larson and Farber
Slide 4- 108
Find the probability using the standard
normal distribution.
P(z < 1.49)
A. 0.9319
B. 0.0681
C. 0.6879
D. 0.3121
Slide 5- 109
Find the probability using the standard
normal distribution.
P(z < 1.49)
A. 0.9319
B. 0.0681
C. 0.6879
D. 0.3121
Slide 5- 110
Find the probability using the standard
normal distribution.
P(z ≥ –2.31)
A. 0.0104
B. 0.0087
C. 0.9896
D. 0.9913
Slide 5- 111
Find the probability using the standard
normal distribution.
P(z ≥ –2.31)
A. 0.0104
B. 0.0087
C. 0.9896
D. 0.9913
Slide 5- 112
Find the probability using the standard
normal distribution.
P(–2.14 < z < 0.95)
A. 0.1170
B. 0.0681
C. 0.1873
D. 0.8127
Slide 5- 113
Find the probability using the standard
normal distribution.
P(–2.14 < z < 0.95)
A. 0.1170
B. 0.0681
C. 0.1873
D. 0.8127
Slide 5- 114
IQ scores are normally distributed with a
mean of 100 and a standard deviation of
15. Find the probability a randomly
selected person has an IQ score greater
than 120.
A. 0.9082
B. 0.0918
C. 0.6293
D. 0.3707
Slide 5- 115
IQ scores are normally distributed with a
mean of 100 and a standard deviation of
15. Find the probability a randomly
selected person has an IQ score greater
than 120.
A. 0.9082
B. 0.0918
C. 0.6293
D. 0.3707
IQ scores are normally distributed with a
mean of 100 and a standard deviation of
15. Find the probability a randomly
selected person has an IQ score between
100 and 120.
A. 0.9082
B. 0.0918
C. 0.4082
D. 0.5918
Slide 5- 117
IQ scores are normally distributed with a
mean of 100 and a standard deviation of
15. Find the probability a randomly
selected person has an IQ score between
100 and 120.
A. 0.9082
B. 0.0918
C. 0.4082
D. 0.5918
Slide 5- 118
Find the z-score that has 2.68% of the
distribution’s area to its right.
A. z = 0.9963
B. z = –1.93
C. z = –0.0037
D. z = 1.93
Slide 5- 119
Find the z-score that has 2.68% of the
distribution’s area to its right.
A. z = 0.9963
B. z = –1.93
C. z = –0.0037
D. z = 1.93
Slide 5- 120
IQ scores are normally distributed with a
mean of 100 and a standard deviation of
15. What IQ score represents the 98th
percentile?
A. 131
B. 69
C. 113
D. 145
Slide 5- 121
IQ scores are normally distributed with a
mean of 100 and a standard deviation of
15. What IQ score represents the 98th
percentile?
A. 131
B. 69
C. 113
D. 145
Slide 5- 122
A population has a mean of 80 and a
standard deviation of 12. Samples of size
36 are selected from the population.
Describe the sampling distribution of x .
A. Normal,  x  80,  x  2
B. Normal,  x  80,  x  12
C. Approximately normal,  x  80,  x  2
D. Approximately normal,  x  80,  x  12
Slide 5- 123
A population has a mean of 80 and a
standard deviation of 12. Samples of size
36 are selected from the population.
Describe the sampling distribution of x .
A. Normal,  x  80,  x  2
B. Normal,  x  80,  x  12
C. Approximately normal,  x  80,  x  2
D. Approximately normal,  x  80,  x  12
Slide 5- 124
American children watch an average of 25
hours of television per week with a
standard deviation of 8 hours. A random
sample of 40 children is selected. What is
the probability the mean number of hours
of television they watch per week is less
than 22?
A. 0.3520
B. 0.0089
C. 0.9911
D. 0.6480
Slide 5- 125
American children watch an average of 25
hours of television per week with a
standard deviation of 8 hours. A random
sample of 40 children is selected. What is
the probability the mean number of hours
of television they watch per week is less
than 22?
A. 0.3520
B. 0.0089
C. 0.9911
D. 0.6480
Slide 5- 126
Use a correction for continuity to convert
the following interval to a normal
distribution interval.
The probability of getting at least 80
successes
A. x > 80.5
B. x > 79.5
C. x < 80.5
D. x < 79.5
Slide 5- 127
Use a correction for continuity to convert
the following interval to a normal
distribution interval.
The probability of getting at least 80
successes
A. x > 80.5
B. x > 79.5
C. x < 80.5
D. x < 79.5
Slide 5- 128