Download Equation Chapter 1 Section 1Normal Distribution Example

Normal Distribution Example
Suppose that the weight of adult males is normally distributed with a mean of 160 pounds and a
standard deviation of 30 pounds.
Part A
Compute the probability that a randomly selected adult male weighs less than 100 pounds.
Solution
1. Translate from words to symbols (i.e., model
a. Identify the underlying random variable of interest, say X
b. Assumptions about parameters of the underlying distribution, e.g.,
(central location) population mean, i.e., expected value of the underlying random variable X, i.e.,
 X  E  X  , is a certain number,
the population standard deviation of the underlying distribution, often denoted by  , is a certain
number, and
the shape, i.e., functional form, of the underlying distribution is a certain functional form, e.g.,
the Normal distribution.
c. "State" question mathematically in terms of the model
P a  X  b  
(1)
where a, b, and  are numbers, some of which are given and some of
which are sought (i.e., to find).
d. There are two general types of questions one can ask about the distribution of a
random variable according to what is given and what is to find in (1):
If a and b, are given to find  :
a) The question gives a specific interval of X, e.g., the interval
a  X  b
where a and b are specific numbers, (including, possibly,  and
 ) and
b) the questions asks for the probability  of that interval, as in
equation (1).
Copyright 1998, 2000, 2008 (c) All rights reserved, Golde I. Holtzman,Department of Statistic, Virginia Tech (VPI)
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If  is given to find a and b:
c) the question gives a specific probability  for an interval of X, as
in equation (1),
d) the questions asks for the numbers a and/or b of that interval,
a  X  b
(2)
2. Standardize. If a and b, are given to find  , use the z transformation
z
X  X
X
(3)
with  X  a and/or  X  b . I.e., compute
z1 
a  X
(4)
b  X
(5)
X
and/or
z2 
X
which rescales the interval from (2), to
a  X  b  z1  Z  z2
(6)
  P a  X  b  P z1  Z  z2 
(7)
so that
3. If a and b, are given to find  , solve (8) to use the continuous distribution function
(CDF)  FZ  z   P Z  z , e.g.,
P a  X  b  P  z1  Z  z2 
 P Z  z2   P Z  z1
 FZ  z2   FZ  z1 
4. If a and b, are given to find  , use the Normal table to look up FZ  z1  and/or FZ  z2 
or use
Copyright 1998, 2000, 2008 (c) All rights reserved, Golde I. Holtzman,Department of Statistic, Virginia Tech (VPI)
http://courseware.vt.edu/users/holtzman/STAT5605/normal02.doc
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(9)
JMP > File > New or Open > [highlight column] > Cols > Formula > Probability
> Normal Distribution  zi  .
See JMP > Help.
5.
6. If  is given to find a and b, apply the inverse transform
x    z
7. Translate from symbols to English.
8. Check
1. Translate
Let X be the weight (lbs) of a randomly
selected adult male.
Assume X ~ N(160, 900), i.e., assume
 = 160,  = 30, X ~ N (location, dispersion,
shape)
Find: P{X < 100}.
2. Standardize
Z = (X - )/ = (100 – 160)/30 = -60/30 = -2
3. Solve for Table
P{X < 100} = P{Z < -2} = F(-2)
4. Use Table (CDF) to find probabilities.
P{X < 100} = P{Z < -2} = F(-2) = .02275
5. Inverse trasform if necessary,
Not necessary
6. Translate from symbols to English
Copyright 1998, 2000, 2008 (c) All rights reserved, Golde I. Holtzman,Department of Statistic, Virginia Tech (VPI)
http://courseware.vt.edu/users/holtzman/STAT5605/normal02.doc
3
(10)
Assuming that the weight of adult males is normally distributed with a mean of 160 pounds and a
standard deviation of 30 pounds, the probability that a randomly selected male weighs less than
100 pounds is 0.02275.
7. Check
Example, Part B
Compute the percentage of adult males that weigh more than 150 pounds.
1. Translate
Let X be the weight (lbs) of a randomly
selected adult male.
Assume X ~ N(160, 900), i.e., assume
 = 160,  = 30, X ~ N
Find: P{X > 150}.
2. Standardize
Z = (X - )/ = (150 – 160)/30 = -10/30 = 0.33
P{X > 150} = P{Z > -0.33}
3. Solve for Table
P{X > 150} = P{Z > -0.33}
= 1 - P{Z < -0.33} = 1 - F(-0.33)
4. Use Table
= 1 - .3707 = .6293
5. Inverse transform if necessary
6. Translate from symbols to English
Copyright 1998, 2000, 2008 (c) All rights reserved, Golde I. Holtzman,Department of Statistic, Virginia Tech (VPI)
http://courseware.vt.edu/users/holtzman/STAT5605/normal02.doc
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Assuming that the weight of adult males is normally distributed with a mean of 160 pounds and a
standard deviation of 30 pounds, 62.9% of adult males weigh more than 150 pounds.
Example, Part C
Compute the fraction of adult males that would be expected to weigh between 160 and 190
pounds.
1. Translate
Find: P{160 < X < 190}.
2. Standardize
Z1 = (X - )/ = (160 – 160)/30 = 0/30 = 0
Z2 = (X - )/ = (190 – 160)/30 = 30/30 = 1
P{160 < X < 190} = P{0 < Z < 1}
3. Solve for Table
P{160 < X < 190} = P{0 < Z < 1}
= P{Z < 1} - P{Z < 0} = F(1) - F(0)
4. Use Table (CDF) to find probabilities
= .8413 - .5000 = .3413
5. Inverse transform if necessary
Not necessary
6. Translate from symbols to English
Assuming that the weight of adult males is
normally distributed with a mean of 160
pounds and a standard deviation of 30
pounds, 34.13% of adult males weigh between 160 and 190 pounds.
7. Check
Copyright 1998, 2000, 2008 (c) All rights reserved, Golde I. Holtzman,Department of Statistic, Virginia Tech (VPI)
http://courseware.vt.edu/users/holtzman/STAT5605/normal02.doc
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Example, Part D
Find two weights, a and b, such that the weights of 95% of all adult males is between a and b,
and such that 2.5% weigh less than a, and
2.5% weigh more than b.
1. Translate
Find: a and b such that
P{a < X < b} = 0.95,
P{X < a} = P{X > b} = 0.025.
2. Standardize
Find: z0.025 and z0.975 such that
P{z0.025 < Z < z0.975 } = 0.95,
P{Z < z0.025} = P{Z > z0.975 } = 0.025.
3. Solve for Table
P{Z < z0.025} = .025 = F(z0.025)
P{Z < z0.975} = 0.975 = F(z 0.975)
4. Use Table to find quantiles.
z0.025  -1.96, z0.975   1.96
5. Inverse transform if necessary
a =  + z  = 160 – (1.96)(30) = 160 – 58.8 = 101.2
b =  + z  = 160 + (1.96)(30) = 160 + 58.8 = 218.8
6. Translate from symbols to English
Assuming that the weight of adult males is normally distributed with a mean of 160 pounds and a
standard deviation of 30 pounds, 95% of adult males weigh between 101.2 and 218.8 pounds,
2.5% weigh less than 101.2, and 2.5% weigh more than 218.8 pounds.
Copyright 1998, 2000, 2008 (c) All rights reserved, Golde I. Holtzman,Department of Statistic, Virginia Tech (VPI)
http://courseware.vt.edu/users/holtzman/STAT5605/normal02.doc
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7. Check
Copyright 1998, 2000, 2008 (c) All rights reserved, Golde I. Holtzman,Department of Statistic, Virginia Tech (VPI)
http://courseware.vt.edu/users/holtzman/STAT5605/normal02.doc
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