Transcript Essential Statistics 1/e

7-1
Chapter
7
Continuous Distributions
Continuous Variables
Describing a Continuous Distribution
Uniform Continuous Distribution
Normal Distribution
Standard Normal Distribution
Normal Approximation to the Binomial (Optional)
Normal Approximation to the Poisson (Optional)
Exponential Distribution
McGraw-Hill/Irwin
© 2008 The McGraw-Hill Companies, Inc. All rights reserved.
7-3
Continuous Variables
 Events as Intervals
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Discrete Variable – each value of X has its own
probability P(X).
Continuous Variable – events are intervals and
probabilities are areas underneath smooth
curves. A single point has no probability.
7-4
Describing a Continuous Distribution
 PDFs and CDFs
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Probability Density Function (PDF) –
For a continuous
random variable,
the PDF is an
equation that shows
the height of the
curve f(x) at each
possible value of X
over the range of X.
Normal PDF
7-5
Describing a Continuous Distribution
 PDFs and CDFs
Continuous PDF’s:
• Denoted f(x)
• Must be nonnegative
• Total area under
curve = 1
• Mean, variance and
shape depend on
the PDF parameters
• Reveals the shape
of the distribution
Normal PDF
7-6
Describing a Continuous Distribution
 PDFs and CDFs
Continuous CDF’s:
• Denoted F(x)
• Shows P(X < x), the
cumulative proportion
of scores
• Useful for finding
probabilities
Normal CDF
7-7
Describing a Continuous Distribution
 Probabilities as Areas
Continuous probability functions are smooth curves.
• Unlike discrete
distributions, the
area at any
single point = 0.
• The entire area under
any PDF must be 1.
• Mean is the balance
point of the distribution.
7-8
Describing a Continuous Distribution
 Expected Value and Variance
7-9
Uniform Continuous Distribution
 Characteristics of the Uniform Distribution
7-10
Uniform Continuous Distribution
 Example: Anesthesia Effectiveness
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An oral surgeon injects a painkiller prior to
extracting a tooth. Given the varying
characteristics of patients, the dentist views the
time for anesthesia effectiveness as a uniform
random variable that takes between 15 minutes
and 30 minutes.
X is U(15, 30)
a = 15, b = 30, find the mean and standard
deviation.
7-11
Uniform Continuous Distribution
 Example: Anesthesia Effectiveness
a + b 15 + 30
m=
=
= 22.5 minutes
2
2
s=
(b – a)2 = (30 – 15)2 = 4.33 minutes
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Find the probability that the anesthetic takes between
20 and 25 minutes.
P(c < X < d) = (d – c)/(b – a)
P(20 < X < 25) = (25 – 20)/(30 – 15)
= 5/15 = 0.3333 or 33.33%
7-12
Normal Distribution
 Characteristics of the Normal Distribution
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Normal or Gaussian distribution was named for
German mathematician Karl Gauss (1777 –
1855).
Defined by two parameters, m and s
Denoted N(m, s)
Domain is – < X < + 
Almost all area under the normal curve is
included in the range m – 3s < X < m + 3s
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Normal Distribution
 Characteristics of the Normal Distribution
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Normal Distribution
 What is Normal?
A normal random variable should:
• Be measured on a continuous scale.
• Possess clear central tendency.
• Have only one peak (unimodal).
• Exhibit tapering tails.
• Be symmetric about the mean (equal tails).
7-15
Standard Normal Distribution
 Characteristics of the Standard Normal
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Since for every value of m and s, there is a
different normal distribution, we transform a
normal random variable to a standard normal
distribution with m = 0 and s = 1 using the
formula:
z= x–m
s
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Denoted N(0,1)
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Standard Normal Distribution
 Characteristics of the Standard Normal
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Standard Normal Distribution
 Finding Areas by using Standardized Variables
• Suppose John took an economics exam and
scored 86 points. The class mean was 75 with a
standard deviation of 7. What percentile is John
in (i.e., find P(X < 86)?
zJohn = x – m = 86 – 75 = 11/7 = 1.57
7
s
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So John’s score is 1.57 standard deviations about
the mean.