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?

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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
• Additional Practice
• See problems in Section 15.4
• Next Lesson
• Study for the Final Exam!