1. Chapter 6 Introduction to Continuous Probability Distributions
Business Statistics:
A Decision-Making Approach
8th Edition
Chapter 6 Introduction to Continuous Probability Distributions
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2. Chapter Goals After completing this chapter, you should be able to:
Convert values from any normal distribution to a standardized z-score
Find probabilities using a normal distribution table
Apply the normal distribution to business problems
Recognize when to apply the uniform and exponential distributions
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3. Continuous Probability Distributions
A continuous random variable is a variable that can assume any value on a defined continuum (can assume an uncountable number of values) – see Chapter 5
thickness of an item
time required to complete a task
These can potentially take on any value, depending only on the ability to measure accurately.
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4. A B Types of Continuous Distributions Three types
Normal
Uniform
Exponential
A B
Involves determining the probability for a RANGE
of values rather than 1 particular incident or
outcome
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5. The Normal Distribution
‘Bell Shaped’
Symmetrical
Mean=Median=Mode
Location is determined by the
mean, μ
Spread is determined by the standard deviation, σ
The random variable has an infinite theoretical range:
+  to  
f(x) σ x μ
Mean
Median
Mode
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6. Many Normal Distributions
By varying the parameters μ and σ, we obtain different normal distributions
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7. The Normal Distribution Shape
f(x)
Changing μ shifts the distribution left or right.
Changing σ increases or decreases the spread. σ μ x
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8. Finding Normal Probabilities
Probability is measured by the area under the curve
f(x) P ( a  x  b ) a b x
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9. Probability as Area Under the Curve
The total area under the curve is 1.0, and the curve is symmetric, so half is above the mean, half is below
f(x)
0.5
0.5 μ x
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10. The Standard Normal Distribution
Also known as the “z” distribution
Mean is defined to be 0
Standard Deviation is 1
f(z) 1 z
Values above the mean have positive z-values Values below the mean have negative z-values
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11. The Standard Normal
Any normal distribution (with any mean and standard deviation combination) can be transformed into the standard normal distribution (z)
Need to transform x units into z units
Where x is any point of interest
Can use the z value to determine probabilities
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12. Translation to the Standard Normal Distribution
Translate from x to the standard normal (the “z” distribution) by subtracting the mean of x and dividing by its standard deviation:
z is the number of standard deviations units that
x is away from the population mean
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13. Example
If x is distributed normally with mean of 100 and standard deviation of 50, the z value for x = 250 is
This says that x = 250 is three standard deviations (3 increments of 50 units) above the mean of 100.
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14. Comparing x and z units 100 250 x 3.0 z μ = 100 σ = 50
3.0 z
Note that the distribution is the same, only the scale has changed. We can express the problem in original units (x) or in standardized units (z)
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15. The Standard Normal Table
The Standard Normal table in the textbook (Appendix D)
Gives the probability from the mean (zero)
up to a desired value for z
0.4772
Example:
P(0 < z < 2.00) = z
2.00
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16. The Standard Normal Table
(continued)
The Standard Normal Table gives the probability between the mean and a certain z value
The z value ALWAYS refers to the area between some value (-z or +z) and the mean
Since the distribution is symmetrical, the Standard Normal Table only displays probabilities for ½ of the full distribution
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17. The Standard Normal Table
(continued)
The column gives the value of z to the second decimal point
The row shows the value of z to the first decimal point
The value within the table gives the probability from z = 0 up to the desired z value .
2.0
.4772
P(0 < z < 2.00) =
2.0
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18. General Procedure for Finding Probabilities
Determine m and s
Define the event of interest
e.g., P(x > x1)
Convert to standard normal
Use the table to find the probability
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19. z Table Example
Suppose x is normal with mean 8.0 and standard deviation Find P(8 < x < 8.6)
Calculate z-values: 8
8.6 x
0.12 Z
P(8 < x < 8.6)
= P(0 < z < 0.12)
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20. z Table Example
(continued)
Suppose x is normal with mean 8.0 and standard deviation Find P(8 < x < 8.6)
= 8
 = 5
= 0
 = 1 x z 8
8.6
0.12
P(8 < x < 8.6)
P(0 < z < 0.12)
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21. Solution: Finding P(0 < z < 0.12)
Standard Normal Probability
Table (Portion)
P(8 < x < 8.6)
= P(0 < z < 0.12)
.02 z
.00
.01
0.0478
0.0
.0000
.0040
.0080
0.1
.0398
.0438
.0478
0.2
.0793
.0832
.0871 Z
0.3
.1179
.1217
.1255
0.00
0.12
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22. Finding Normal Probabilities
Suppose x is normal with mean 8.0 and standard deviation 5.0.
Now Find P(x < 8.6)
The probability of obtaining a value less than 8.6
P = 0.5 Z
8.0
8.6
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23. Finding Normal Probabilities
(continued)
Suppose x is normal with mean 8.0 and standard deviation 5.0.
Now Find P(x < 8.6)
0.0478
0.5000
P(x < 8.6)
= P(z < 0.12)
= P(z < 0) + P(0 < z < 0.12)
= = Z
0.00
0.12
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24. Upper Tail Probabilities
Suppose x is normal with mean 8.0 and standard deviation 5.0.
Now Find P(x > 8.6) Z
8.0
8.6
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25. Upper Tail Probabilities
(continued)
Now Find P(x > 8.6)…
P(x > 8.6) = P(z > 0.12) = P(z > 0) - P(0 < z < 0.12)
= =
0.0478
0.5000
0.4522 Z Z
0.12
0.12
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26. Lower Tail Probabilities
Suppose x is normal with mean 8.0 and standard deviation 5.0.
Now Find P(7.4 < x < 8) Z
8.0
7.4
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27. Lower Tail Probabilities
(continued)
Now Find P(7.4 < x < 8)…the probability between 7.4 and the mean of 8
The Normal distribution is symmetric, so we use the same table even if z-values are negative:
P(7.4 < x < 8)
= P(-0.12 < z < 0) =
0.0478 Z
8.0
7.4
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28. Normal Probabilities in PHStat
We can use Excel and PHStat to quickly
generate probabilities for any normal
distribution
We will find P(8 < x < 8.6) when x is
normally distributed with mean 8 and
standard deviation 5
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29. Select desired options and enter values
PHStat Dialogue Box
Select desired options and enter values
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30. PHStat Output
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31. Recall Tchebyshev from Chpt. 3
Empirical Rules
What can we say about the distribution of values around the mean if the distribution is normal?
f(x)
μ ± 1σ covers about 68% of x’s σ σ
Recall Tchebyshev from Chpt. 3 x
μ-1σ μ
μ+1σ
68.26%
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32. μ ± 2σ covers about 95% of x’s μ ± 3σ covers about 99.7% of x’s
The Empirical Rule
(continued)
μ ± 2σ covers about 95% of x’s
μ ± 3σ covers about 99.7% of x’s 3σ 3σ 2σ 2σ μ x μ x
95.44%
99.72%
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33. Importance of the Rule If a value is about 2 or more standard
deviations away from the mean in a normal
distribution, then it is far from the mean
The chance that a value that far or farther
away from the mean is highly unlikely, given
that particular mean and standard deviation
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34. The Uniform Distribution
The uniform distribution is a probability distribution that has equal probabilities for all possible outcomes of the random variable
Referred to as the distribution of “little information”
Probability is the same for ANY interval of the same width
Useful when you have limited information about how the data “behaves” (e.g., is it skewed left?)
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35. The Uniform Distribution
(continued)
The Continuous Uniform Distribution:
f(x) =
where
f(x) = value of the density function at any x value
a = lower limit of the interval of interest
b = upper limit of the interval of interest
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36. The Mean and Standard Deviation for the Uniform Distribution
The mean (expected value) is:
The standard deviation is
where
a = lower limit of the interval from a to b
b = upper limit of the interval from a to b
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37. Steps for Using the Uniform Distribution
Define the density function
Define the event of interest
Calculate the required probability
f(x) x
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38. Uniform Distribution Example: Uniform Probability Distribution
Over the range 2 ≤ x ≤ 6: 1
f(x) = = for 2 ≤ x ≤ 6
6 - 2
f(x)
.25 x 2 6
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39. Uniform Distribution Example: Uniform Probability Distribution
Over the range 2 ≤ x ≤ 6:
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40. The Exponential Distribution
Used to measure the time that elapses between two occurrences of an event (the time between arrivals)
Examples:
Time between trucks arriving at a dock
Time between transactions at an ATM Machine
Time between phone calls to the main operator
Recall l = mean for Poisson (see Chpt. 5)
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41. The Exponential Distribution
The probability that an arrival time is equal to or less than some specified time a is
where 1/ is the mean time between events and e =
NOTE: If the number of occurrences per time period is Poisson with mean , then the time between occurrences is exponential with mean time 1/  and the standard deviation also is 1/l
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42. Exponential Distribution
(continued)
Shape of the exponential distribution
f(x)
 = 3.0
(mean = .333)
 = 1.0
(mean = 1.0)
= 0.5
(mean = 2.0) x
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43. Example
Example: Customers arrive at the claims counter at the rate of 15 per hour (Poisson distributed). What is the probability that the arrival time between consecutive customers is less than five minutes?
Time between arrivals is exponentially distributed with mean time between arrivals of 4 minutes (15 per 60 minutes, on average)
1/ = 4.0, so  = .25
P(x < 5) = 1 - e-a = 1 – e-(.25)(5) =
There is a 71.35% chance that the arrival time between consecutive customers is less than 5 minutes
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44. Using PHStat
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45. Chapter Summary Reviewed key continuous distributions
normal
uniform
exponential
Found probabilities using formulas and tables
Recognized when to apply different distributions
Applied distributions to decision problems
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Printed in the United States of America.
Copyright ©2011 Pearson Education, Inc. publishing as Prentice Hall