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Chapter 5: The Normal Distribution
MODELING CONTINUOUS VARIABLES
Histogram 6.1
Proportion
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Age
If we draw a curve through the tops of the bars in Histogram 6.1 and require the smoothed
curve to have total area under it equal to 1, we would have what is called a density
function, also called a density curve. The key idea when working with density functions
is that area under the curve, above an interval, corresponds to the proportion of units with
values in the interval.
NOTATION...
Since we will be discussing models for populations, the mean and standard deviation for a
density curve or model will be represented by

(mu) and

(sigma), respectively.
DEFINITION:
A density function is a (nonnegative) function or curve that describes the overall shape of
a distribution. The total area under the entire curve is equal to 1, and proportions are
measured as areas under the density function.
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Let's Do It! 1
Lifetime Density Function
Let the variable X represent the length of life, in years, for an electrical component. The following
figure is the density curve for the distribution of X.
(a)
What proportion of electrical components lasts longer than 6 years?
(b)
What proportion of electrical components lasts longer than 1 year?
(c)
Describe the shape of the distribution.
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Normal Distributions
General Notation
X is N(  , ) means that the variable or characteristic X is
normally distributed with mean

and standard deviation  .
A normal distribution

Point of
inflection


Three members of the family of normal distributions
Distribution #3:
Normal with a mean of 80
and a standard deviation of 5
Distribution #1:
Normal with a mean of 50
and a standard deviation of 10
20
30
40
50
Distribution #2:
Normal with a mean of 80
and a standard deviation of 10
60
70
80
90
100
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Example 1 IQ Scores
Problem
Let the variable X represent IQ scores of 12-year-olds. Suppose that the distribution of X is
normal with a mean of 100 and a standard deviation of 16—that is, X is
N(100, 16). Jessica is a 12-year-old and has an IQ score of 132.We would like to determine
the proportion of 12-year olds that have IQ scores less than Jessica’s score of 132.
Since the area under the density curve corresponds to proportion, we want to find the area
to the left of 132 under an N(100, 16) curve. Sketch this curve and show the corresponding
area that represents this proportion.
IQ Scores have a
normal distribution
with mean 100 and
standard deviation 16

area to the left of 132 = ?
68
84
100
116
132
IQ Score
DEFINITION:
If
X is N(  , ) ,
the standardized normal variable
Z
X

is
N 0 ,1.
DEFINITION:
The z-score or standard score for an observed value tells us how many standard
deviations the observed value is from the mean – that is, it tells us how far the observed
value is from the mean in standard-deviation units. It is computed as follows:
Z
X

= number of standard deviations that
X differs from the mean

If Z > 0, then the value of X is above (greater than) its mean.
If Z < 0, then the value of X is below (less than) its mean.
If Z = 0, then the value of X is equal to its mean.
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Example 2 Standard IQ Score
Recall the distribution of IQ scores for 12-year-olds—normally distributed with a mean of
100 and a standard deviation of 16.
(a)
Jessica had a score of 132. Compute Jessica’s standardized score.
(b)
Suppose Jessica has an older brother, Mike, who is 20 years old and has an IQ
score of 144. It wouldn’t make sense to directly compare Mike’s score of 144 to
Jessica’s score of 132. The two scores come from different distributions due to the
age difference. Assume that the distribution of IQ scores for 20-year-olds is normal
with a mean of 120 and a standard deviation of 20. Compute Mike’s standardized
score.
(c)
Relative to their respective age group, who had the higher IQ score—Jessica or
Mike?
Solution
Jessica's standard score =
Mike's standard score =
132  100
 2 .
16
144  120
 12
. .
20
Thus, relative to their respective age groups, Jessica has a higher IQ score than Mike

Identifying Outliers (Method 2)
CASE STUDY
Suppose female bank employee believes that her salary is low as a result of sex
discrimination. To substantiate her belief, she collects information on salaries of her male
counterparts in the banking business. She finds that their salary is approximately normal
with an average of $54,000 and a standard deviation of $2000. Her salary is $47,000. Does
this information support her claim of sex discrimination?
Rules for detecting outliers:
Box-Plot: outliers are data points beyond the lower and upper fences
Z-score: outliers have a z-score that is more than 3 standard deviations from the
mean, |z| > 3.
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How to Calculate Areas under a Normal Distribution
Example 3
Finding Proportions for the Standard Normal Distribution
Problem
Finding proportions under a normal distribution involves standardization and then finding
the corresponding proportion (area) under the standard normal distribution. Let’s first work
on finding areas under a standard normal N(0, 1) distribution.
(a)
Find the area under the standard normal distribution to the left of z = 1.22. Sketch
a picture of the corresponding area and use your TI to find the area.
Solution
The area to the left of is described below. z = 1.22

area to the
left of z=1.22
is 0.8888
0
z=1.22
Z
Using TI:
(b)
Find the area under the standard normal distribution to the right of z = 1.22.
Solution
We already know the area to the left of z = 1.22, and we know that the total area
under any density to he right of 0 is equal to 0.5. So the area to the right of z =
1.22 must be 0.5 - 0.3888 = 0.1112. We could also find the area to the left of z =
-1.22. By the symmetry of the standard normal distribution, the area to the right
of z = 1.22 is equal to the area to the left of z = -1.22.
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Let's Do It! 2 6.2More Standard Normal Areas
(a)
Find the area under the standard normal distribution between z = 0 and z = 1.22.
Sketch the area and use your calculator to find the area.

Z
0
(b)
Find the area under the standard normal distribution to the left of z = -2.55. Sketch
the area and use your calculator to find the area.

Z
0
(c)
Find the area under the standard normal distribution
between z = -1.22 and z = 1.22. Sketch the area and use your calculator to find the
area.

0
Z
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Let's Do It! 3 6.3IQ Scores
We will continue with the model for IQ score of 12-year-olds. In answering the following
questions, remember to use the symmetry of the normal distribution and the fact that the
total area under the curve is 1. It may also be very useful to draw a picture of the area you
are trying to find so you can establish a frame of reference (for example, should it be larger
or smaller than 50%?) and see the way to approach getting the answer. If you will be using
Table II, you will need to first compute the corresponding z-scores.
X = IQ score (12-year-olds) has a
(a)
N 100,16  distribution.
What proportion of the 12-year-olds has IQ scores below 84? Sketch it.

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(b)
68
84
100
116
132
148
IQ Score
What proportion of the 12-year-olds has IQ scores 84 or more? Sketch it.

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(c)
68
84
100
116
132
148
IQ Score
What proportion of the 12-year-olds has IQ scores between 84 and 116? Sketch
it.

52
68
84
100
116
132
148
IQ Score
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Example The Top 1% of the IQ Distribution
Problem


Recall the N 100,16 model for IQ score of 12-year-olds. What IQ score must a 12-yearold have to place in the top 1% of the distribution of IQ scores?
(a)
Draw a picture to show what IQ score you are trying to find.
(b)
What percentile do you want to find for the IQ distribution?
(c)
Find the percentile using Table E in reverse or your calculator.
Again it may be helpful to draw a picture:

The area to the left is 0.99
100
?
IQ Score
Many calculators have the ability to find various percentiles of a normal distribution. The TI has a
built-in function called invNorm under the DIST menu. You must first specify the desired are to the
left, then the mean and the standard deviation for the normal distribution. The steps for finding the
99th percentile of our N(100,16) distribution are as follows:
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Let's Do It! 4 6. 7 Hours per Week
According to a study, men in the US devote an average of 16hrs per week to house work.
Assume that the number of hours men devote to house work is normally distributed with a
standard deviation of 3.5.
a. Suppose that the lower 10% of men on the distribution devote fewer than x hours
per week. Find the value of x.
b. Suppose the upper 5% of men on the distribution devote more than x hrs per week.
Find the value of x.
Let's Do It! 5 Middle portion of the normal distribution
If one-person household spends an average of $40 per week on groceries, find the
maximum and minimum dollar amounts spent per week for the middle 50% of one-person
household. Assume that the standard deviation is $5 and that the amount spent is normally
distributed.
Let's Do It! 6 7Freestyle Swim Times
The finishing times for 11–12-year-old male swimmers performing the 50-yard freestyle are
normally distributed with a mean of 35 seconds and a standard deviation of 2 seconds.
(a)
The sponsors of a swim meet decide to give certificates to all 11–12-year-old
male swimmers who finish their 50-yard race in under 32 seconds. If there are 50
such swimmers entered in the 50-yard freestyle event, approximately how many
certificates will be needed?
(b)
In what amount of time must a swimmer finish to be in the “top” fastest 2% of the
distribution of finishing times?
HW posted, Prepare for Quiz
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