Download §6.2--Area Under the Standard Normal Curve

§6.2–Area Under the Standard Normal Curve
Tom Lewis
Fall Term 2009
Tom Lewis ()
§6.2–Area Under the Standard Normal Curve
Fall Term 2009
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Fall Term 2009
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Outline
1
The cumulative distribution function
2
The zα notation
Tom Lewis ()
§6.2–Area Under the Standard Normal Curve
The cumulative distribution function
Cumulative distribution
For z ∈ (−∞, ∞), let Φ(z) denote the area under the standard
normal curve over the region from −∞ to z.
Thus Φ(z) represents the probability that a standard normal random
variable takes on a value less than z.
Values for Φ have been tabulated; see Table II.
R will calculate values of Φ through the pnorm() function.
Tom Lewis ()
§6.2–Area Under the Standard Normal Curve
Fall Term 2009
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The cumulative distribution function
Problem
Find the area under the standard normal curve from −∞ to 1.31
Find the probability that a standard normal variable has a value less
than −.24
What is the probability that a standard normal variable assumes a
value between −1.32 and 2.01?
What is the probability that a standard normal random variable
exceeds 1.12.
If the probability that a standard normal random variable is greater
than z is .18, then what is z?
Tom Lewis ()
§6.2–Area Under the Standard Normal Curve
Fall Term 2009
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The zα notation
The zα notation
The symbol zα is used to denote the z-score that has area α to its right
under the standard normal curve. In other words,
1 − Φ(zα ) = α
or Φ(zα ) = 1 − α.
Using R
The R function qnorm() can be used to calculate zα values. Given an
number 0 < p < 1, qnorm(p) calculates the z value such that the area
under the standard normal curve from −∞ up to z is p.
Tom Lewis ()
§6.2–Area Under the Standard Normal Curve
Fall Term 2009
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The zα notation
Problem
Use Table II and R in the following problems:
Find z.05
Find z.65
Find the value a > 0 such that the area trapped under the standard
normal curve over the symmetric interval [−a, a] is .88.
Tom Lewis ()
§6.2–Area Under the Standard Normal Curve
Fall Term 2009
6/6