Download §6.1--Introducing Normally Distributed Variables

§6.1–Introducing Normally Distributed Variables
Tom Lewis
Fall Term 2009
Tom Lewis ()
§6.1–Introducing Normally Distributed Variables
Fall Term 2009
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Outline
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The standard normal curve
2
Normal curves
3
Standardizing
Tom Lewis ()
§6.1–Introducing Normally Distributed Variables
Fall Term 2009
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The standard normal curve
The standard normal curve
The standard normal curve is the bell-shaped curve given by
1
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φ(x) = √ e −x /2 .
2π
The total area trapped between the curve and the x-axis is 1.
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The standard normal curve
Standard normal random variables
A variable x is said to have a standard normal distribution if the
probability that it lies within a specified range is the area under the
standard normal curve over that same range.
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§6.1–Introducing Normally Distributed Variables
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The standard normal curve
Standard normal random variables
A variable x is said to have a standard normal distribution if the
probability that it lies within a specified range is the area under the
standard normal curve over that same range.
Example
The probability that a normal random variable has a value between 1 and
2 is the area under the normal curve between 1 and 2, as pictured below.
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Normal curves
Normal curves
A normal curve is any bell-shaped curve given by
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φ(x) = √ e −(x−µ) /(2σ) .
σ 2π
µ is called the mean parameter and σ is called the standard deviation
parameter. The total area trapped between the curve and the x-axis is 1.
Tom Lewis ()
§6.1–Introducing Normally Distributed Variables
Fall Term 2009
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Normal curves
Normal curves
A normal curve is any bell-shaped curve given by
1
2
φ(x) = √ e −(x−µ) /(2σ) .
σ 2π
µ is called the mean parameter and σ is called the standard deviation
parameter. The total area trapped between the curve and the x-axis is 1.
Problem
Study the effects of σ and µ on the shape of a normal curve.
Tom Lewis ()
§6.1–Introducing Normally Distributed Variables
Fall Term 2009
5/7
Normal curves
Normal curves
A normal curve is any bell-shaped curve given by
1
2
φ(x) = √ e −(x−µ) /(2σ) .
σ 2π
µ is called the mean parameter and σ is called the standard deviation
parameter. The total area trapped between the curve and the x-axis is 1.
Problem
Study the effects of σ and µ on the shape of a normal curve.
Normal random variables
A variable x is said to have a normal distribution (with mean µ and
standard deviation σ) if the probability that it lies within a specified range
is the area under the corresponding normal curve over that same range.
Tom Lewis ()
§6.1–Introducing Normally Distributed Variables
Fall Term 2009
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Standardizing
Standardizing
If x has a normal distribution with mean µ and standard deviation σ, then
the standardized variable
x −µ
z=
σ
is a standard normal random variable.
Tom Lewis ()
§6.1–Introducing Normally Distributed Variables
Fall Term 2009
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Standardizing
Standardizing
If x has a normal distribution with mean µ and standard deviation σ, then
the standardized variable
x −µ
z=
σ
is a standard normal random variable.
The equal area principle
The area trapped beneath a normal curve with mean µ and standard
deviation σ over the interval [a, b] is the same as the area trapped beneath
the standard normal curve over the interval [(a − µ)/σ, (b − µ)/σ].
Tom Lewis ()
§6.1–Introducing Normally Distributed Variables
Fall Term 2009
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Standardizing
Problem
Scores on the SAT (math) test are normally distributed with a mean of
500 and a standard deviation of 100.
Tom Lewis ()
§6.1–Introducing Normally Distributed Variables
Fall Term 2009
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Standardizing
Problem
Scores on the SAT (math) test are normally distributed with a mean of
500 and a standard deviation of 100.
Represent the probability of that a randomly selected student will
score between 550 and 650 as an area under a normal curve with
µ = 500 and σ = 100.
Tom Lewis ()
§6.1–Introducing Normally Distributed Variables
Fall Term 2009
7/7
Standardizing
Problem
Scores on the SAT (math) test are normally distributed with a mean of
500 and a standard deviation of 100.
Represent the probability of that a randomly selected student will
score between 550 and 650 as an area under a normal curve with
µ = 500 and σ = 100.
Represent the probability of that a randomly selected student will
score between 550 and 650 as an area under a standard normal curve.
Tom Lewis ()
§6.1–Introducing Normally Distributed Variables
Fall Term 2009
7/7