Download Topic 15: Normal Distributions

Chapter 6
Normal Probability Distributions
Concepts
Normal distributions
How do you handle large data sets?
Normal curve
Standard normal distribution
Empirical rule
Standardized score (z-score, standardized unit)
Standard normal probability table
Notations:
P(Z  b) or P(Z < b)

P(Z  a) or P(Z > a)
P(a  Z  b) or P(a < Z < b)
Applications
Survival skills
You must know how to use the standard normal table.
Please read the following chapter:
Chapters 6, Opentax
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Chapter 6
Normal Probability Distributions
Normal distributions
Characteristics of a normal distribution (Theory)
 A normal distribution forms a continuous, symmetric, single-peaked, bell-shaped normal
curve.
 The mean, median, and mode of a normal distribution are equal and are located at the center of
the distribution.
 A normal distribution is symmetric about the mean  (or x ).
 Area to the left of the mean = area to the right of the mean = 0.5
Note:
In practice, a normal distribution comes from a data set.
 Data values are approximately normally distributed.
 All normal distributions have the same overall shape.
Normal curve



describes the overall pattern of a data set
is always on or above the horizontal axis
has an area exactly 1 (or 100%) underneath it.
Basic property of probability: the sum of all probabilities is exactly 1.0.
The total area under the normal curve must be 1.
The exact normal curve for a particular normal distribution is described by giving its
 mean  (or x )
 standard deviation  (or s)
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Chapter 6
Normal Probability Distributions
Standard normal distribution
A normal distribution with mean  and standard deviation  can be transformed to the standard
normal distribution.
The standard normal distribution is a normal distribution with a:
 mean  = 0
 standard deviation  = 1
The standard normal curve is symmetric about the mean 0.
We have infinite number of different normal distributions.
We have only one standard normal distribution.
Applications of the standard normal distribution
The standard normal distribution curve can be used to solve many practical problems:
Heights, birth weights, gestation period, IQ scores
Manufacturing industry
Ready-to-wear clothes,
Product warranties, ….
Medical applications (blood pressures, diabetes,…)
Educational testing scores
Insurance industry
Condition for use
 The data (or variable) must be normally distributed or approximately normally distributed, or
 Sample size  30
 For this topic, what type of data are we talking about?
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Chapter 6
Normal Probability Distributions
Standardizing data values
All normal distributions are the same if we measure in units of standard deviation s (or ) about
the mean x (or ) as center.
Key points
We have a data set with a mean ( x or ) and a standard deviation (s or ).
The data values are normally distributed.
We want to measure the distance between a data value and the mean in units of standard deviation.
Standardized value (standardized score, or z-score)
A standardized value measures the distance between a data value and the mean, measured in units
of standard deviation.
How to standardize a data value x
For a sample
If x is a data value from a sample that has mean x and standard deviation s, the standardized value
of x is:
z=
xx
x value - sample mean
=
s
sample standard deviation
 Conversion formula
To change an x value into a standardized value z, use the conversion formula. 
A standardized vale (or z-score) tells how many standard deviations a data value x falls away from
the mean, and in which direction.


A positive z-score indicates that a data value is larger than the mean.
A negative z-score indicates that a data value is smaller than the mean.
For a population
If x is a data value from a population that has mean  and standard deviation , the standardized
value of x is
x - value  population mean x  
z
=
Standard deviation

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Chapter 6
Normal Probability Distributions
How to use the standard normal table (to find probabilities)
Areas under the standard normal curve are probabilities. To find the area under the standard
normal curve, we must first use the conversion formula to find the standardized value z of the data
value x.
The area under the standard normal curve for a specific range of values represents either
 The probability that a randomly selected data value from the population with the
characteristics described by the range, or
 The proportion of the population with the characteristics described by the range.
 P(Z < b)
 P(Z < b) = P(Z  b)
This notation denotes that the probability is the area under the standard normal curve to the left of
a specific value (or cutoff point) b.
 P(Z > a) = 1 – P(Z  a)
 P(Z  a) = P(Z > a)
This notation denotes that the probability is the area under the standard normal curve to the right
of a specific value a.
 P(a < Z < b) = P(Z < b) – P(Z < a)
This notation denotes that the probability is the area under the standard normal curve that lies
between the specific values a and b.
Example
In each case, draw a sketch of the given information and the area (probability) under the standard
normal curve. Use the normal table to find each area (or probability).
[a] P(Z < –2.25)
[b] P(Z > 1.77)
[c] P(–2.25 < Z < 1.77)
Solution
[a] P(Z < –2.25) =
[b] P(Z > 1.77) =
[c] P(–2.25 < Z < 1.77) =
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Chapter 6
Normal Probability Distributions
Example (Source: Workshop Statistics)
Suppose that the IQ scores of students in a certain college follow a normal distribution with mean
115 and standard deviation 12.
[a] Draw a sketch of this distribution.
[b] Shade in the area corresponding to the proportion of students with an IQ below 100.
[c] Use the normal model to determine the proportion of students with an IQ below 100.
Step 1: Use the conversion formula to change x value (= 100) to a standardized value z.
Step 2: Use the normal table to find the proportion.
[d] Find the proportion of these undergraduates having IQs greater than 130.
Step 1: Use the conversion formula to change x value (= 130) to a standardized value z.
Step 2: Use the normal table to find the proportion.
[e] Find the proportion of these undergraduates having IQs between 110 and 130.
Hint: Use P(a < Z < b)
[f] With his IQ of 75, Forrest Gump would have a higher IQ than what percentage of these
undergraduates?
[g] Determine how high must one’s IQ must be to be in the top 1% of all IQs at this college.
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