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## Normal Distribution

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### Normal Distribution MATH 102 Contemporary Math S. Rook Overview Section 15.4 in the textbook: Normal distribution Z-scores Converting raw scores to z-scores ... – PowerPoint PPT presentation

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Title: Normal Distribution

1
Normal Distribution
• MATH 102
• Contemporary Math
• S. Rook

2
Overview
• Section 15.4 in the textbook
• Normal distribution
• Z-scores
• Converting raw scores to z-scores
• Applications

3
Normal Distribution
4
Normal Distribution
• When a sufficient number of data are collected,
the resulting histogram becomes nearly symmetric
• i.e. split into two equal halves
• By connecting the tops of the bars of the
histogram, of the data we obtain a bell-

shaped curve more commonly
known as the Normal
Curve
• If a set of data can be modeled
by a normal curve,
we can
calculate the proportion of the
data in ANY interval

5
Properties of the Normal Distribution
• Given that a set of data follows a normal
distribution, the following properties apply to
the resulting normal curve
• It is bell-shaped
• Its highest point is the mean
• It is symmetric with respect to the mean
• The total area under it is 1
• Approximately
• 68 of the data lies within 1
standard deviation
of the mean
• 95 of the data lies within 2
standard deviations
of the mean
• 99.7 of the data lies within 3
standard deviations
of the mean

6
Normal Distribution (Example)
• Ex 1 Assume we have a normal distribution with
a mean of 10 and a standard deviation of 2. Use
the 68-95-99.7 rule to find the percentage of
values in the desired interval
• a) Between 10 and 12
• b) Above 14
• c) Below 10

7
Normal Distribution (Example)
• Ex 2 Assume we have a normal distribution with
a mean of 12 and a standard deviation of 3. Use
the 68-95-99.7 rule to find the percentage of
values in the desired interval
• a) Above 6
• b) Below 9
• c) Between 15 and 18

8
Z-scores
9
Z-scores
• Examples 1 and 2 allowed us to determine the
percentage of data values that lay within 1, 2,
or 3 standard deviations of the mean
• Z-scores allow us to determine the percentage of
data that lie within ANY number of standard
deviations (sds) of the mean of a standard normal
distribution
• e.g. z 2.3 refers to 2.3 sds to the RIGHT of
the mean and z -1.7 refers to 1.7 sds to the
LEFT of the mean
• The standard normal distribution has a mean of 0
and a standard deviation of 1

10
Calculating Percentage/Proportion of Area
• To calculate the percentage or proportion of area
under a standard normal curve, we use the
standard normal table
• A z-score such as z -1.92 is looked up by
finding the first two digits (-1.9) in the rows
and then the value of the hundredths (0.02) in
the columns
• e.g. What is the value associated by looking up
z -1.92?
• The area obtained represents the proportion of
data values that lie to the LEFT of (below) the
given z-score

11
Calculating Percentage/Proportion of Area
(Continued)
• To find the percentage or proportion of area
• Below a z-score simply look up the z-score in
the table
• Above a z-score look up the z-score and
subtract it from 1
• Recall that the area underneath the entire normal
curve is 1
• Between two z-scores look up both z-scores and
subtract the smaller from the larger
• It may help to draw the normal curve

12
Z-scores (Example)
• Ex 3 Assume a data set follows the normal
distribution and use the standard normal table to
find the specified percentage of data values
under the standard normal curve
• a) Below z -1.35 d) Above z -2.73
• b) Above z 2.00 e) Below z 0.08
• c) Between z -0.98 f) Between z 1.01
• and z 1.72 and z 1.99

13
Converting Raw Scores to Z-scores
14
Converting Raw Scores to Z-scores
• Finding the proportion of data values using
z-scores works ONLY with data adhering to a
standard normal distribution
• However, given that the data comes from a normal
distribution and the distributions mean sd, we
can convert a value into a z-score using the
formula
where
• µ is the mean of the data
• s is the sd of the data
• x is any value

15
Converting Raw Scores to Z-scores (Example)
• Ex 4 Given a mean, a standard deviation, and a
raw score from a data set that comes from a
normal distribution, find the corresponding
z-score
• a) µ 80, s 5, x 87
• b) µ 21, s 4, x 14
• c) µ 38, s 10.3, x 48

16
Applications
17
Applications (Example)
• Ex 5 A machine fills bags of candy, but due to
slight irregularities in the operation of the
machine, not every bag gets exactly the same
number of pieces. Assume that the number of
pieces per bag has a mean of 200 and a standard
deviation of 2. Assuming a normal distribution,
what proportion of the bags will have
• a) Less than 197 pieces of candy?
• b) More than 204 pieces of candy?
• c) Between 199 and 201 pieces of candy?

18
Applications (Example)
• Ex 6 A supervisor observes and records the
amount of time his employees take for lunch.
Suppose the length of the average lunch is 42.5
minutes with a standard deviation of 5.8 minutes.
Assuming a normal distribution, what proportion
of the supervisors employees have a lunch break
• a) Less than 50 minutes?
• b) More than 40 minutes?
• c) Between 43 to 45 minutes?

19
Summary
• After studying these slides, you should know how
to do the following
• Use the 68-95-99.7 rule to find the percentage of
values in a desired interval of a normal
distribution
• Compute the proportion of the standard normal
curve using z-scores
• Compute a z-score given the mean, standard
deviation, and a raw score from a normal
distribution
• Solve application problems involving normal
distributions