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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
- Applications

Normal Distribution

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

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

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

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

Z-scores

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

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

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

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

Converting Raw Scores to Z-scores

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

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

Applications

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?

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?

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!