1. Ch 7 Continuous Probability Distributions
Chapter 7
Continuous Probability Distributions
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2. Ch 7
Goals
Understand the difference between discrete and continuous distributions
Compute the mean and the standard deviation for a uniform distribution
Compute probabilities using the uniform distribution
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3. Goals List the characteristics of the: Define and calculate z values
Normal probability distribution
Standard normal probability distribution
Define and calculate z values
Use the standard normal probability distribution to find area:
Above the mean
Below the mean
Between two values
Above one value
Below one value
Use the normal distribution to approximate the binomial probability distribution
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4. Probability Distribution
Ch 7
Probability Distribution
A listing of all the outcomes of an experiment and the probability associated with each outcome
Probability distributions are useful for making probability statements concerning the values of a random variable
Our goal is to find probability between two values:
Example: What is the probability that the daily water usage will lie between 15 and 25 gallons? A: 68%
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5. Probability Distribution
Ch 7
Probability Distribution
Discrete Probability Distributions (Chapter 6)
Based On Discrete Random Variables
We looked at:
Binomial Probability Distribution
Continuous Probability Distribution (Chapter 7)
Based On Continuous Random Variables:
We will look at:
Uniform Probability Distribution
Normal Probability Distribution
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6. Continuous Probability Distributions
Ch 7
Continuous Probability Distributions
These continuous probability distributions will be all about 
Area!!!!
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7. Continuous Probability Distribution:
Ch 7
Continuous Probability Distribution:
Uniform Probability Distribution
Within the interval 15 to 25 minutes, the time it takes to fill out a typical 1040EZ tax return at a VITA site tends to follow a uniform distribution
Random Variable is time (only possibilities within the interval)
Each value has same probability
Normal Probability Distribution
The weight distribution of a manufactured box of cereal tends to follow a normal distribution
Random Variable is box weight and will cover all possibilities
Each value has different probability
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8. Uniform Probability Distribution
Ch 7
Uniform Probability Distribution
Distributions shape is rectangular
Minimum value = a
Maximum value = b
a and b imply a range
Height of the distribution is constant (uniform) for all values between a and b
Implies all values in range are equally likely
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9. Ch 7
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10. What is the standard deviation of the wait time?
Ch 7
What is the mean wait time?
Suppose the time that you wait on the telephone for a live representative of your phone company to discuss your problem with you is uniformly distributed between 5 and 25 minutes.
a + b 2
m =
5+25 2 =
= 15
What is the standard deviation of the wait time?
(b-a)2 12
s =
(25-5)2 12 =
= 5.77
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11. The area from 10 to 25 minutes is 15 minutes. Thus:
Ch 7
What is the probability of waiting more than ten minutes?
The area from 10 to 25 minutes is 15 minutes. Thus:
P(10 < wait time < 25) = height*base = 1
(25-5)
*15 = .75
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12. The area from 15 to 20 minutes is 5 minutes. Thus:
Ch 7
What is the probability of waiting between 15 and 20 minutes?
The area from 15 to 20 minutes is 5 minutes. Thus:
P(15 < wait time < 20) = height*base = 1
(25-5)
*5 = .25
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13. Normal Probability Distribution Is All About Area! Total Area = 1.0
Ch 7
Normal Probability Distribution Is All About Area! Total Area = 1.0
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14. Normal Probability Distribution Formula
Ch 7
Normal Probability Distribution Formula
Awesome!
No, that’s o.k., we can use Appendix or Excel functions!
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15. The normal probability distribution is symmetrical about its mean
Ch 7
Characteristics of a Normal Probability Distribution (And Accompanying Normal Curve)
The normal curve is bell-shaped and has a single peak at the exact center of the distribution
The arithmetic mean, median, and mode of the distribution are equal and located at the peak
Thus half the area under the curve is above the mean and half is below it
The normal probability distribution is symmetrical about its mean
If we cut the normal curve vertically at this center value, the two halves will be mirror images
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16. The normal probability distribution is asymptotic
Ch 7
Characteristics of a Normal Probability Distribution (And Accompanying Normal Curve)
The normal probability distribution is asymptotic
The curve gets closer and closer to the X-axis but never actually touches it
The “tails” of the curve extend indefinitely in both directions
The Location of a normal distribution is determined by mean µ
The dispersion of a normal distribution is determined by the standard deviation 
Now Let’s Look At Some Pictures That Will Show
Relationships Amongst Various Means & Standard Deviations
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17. Characteristics of a Normal Distribution
curve is
symmetrical
Theoretically,
curve
extends to
infinity
Mean, median, and
mode are equal
There Is A Family Of Normal Probability Distributions
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18. Ch 7
Which One Is Normal?
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19. Equal Means, Unequal Standard Deviations
Ch 7
Equal Means, Unequal Standard Deviations
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20. Equal Means, Unequal Standard Deviations
Ch 7
Equal Means, Unequal Standard Deviations
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21. Equal Means, Unequal Standard Deviations
Ch 7
Equal Means, Unequal Standard Deviations
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22. Unequal Means, Equal Standard Deviations
Ch 7
Unequal Means, Equal Standard Deviations
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23. Unequal Means, Unequal Standard Deviations
Ch 7
Unequal Means, Unequal Standard Deviations
This Family Of Normal Probability Distributions
Is Unlimited In Number!
Luckily, One Of The Family Members May Be Used In All
Circumstances Where The Normal Distribution Is Applicable
Standard Normal Distribution
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24. Standard Normal Probability Distribution
Ch 7
Standard Normal Probability Distribution
The standard normal distribution (z distribution ) is a normal distribution with a mean of 0 and a standard deviation of 1
The percentage of area between two z-scores in any normal distribution is the same!
Standard deviation & terms may be different, but area will be the same!
Normal distributions can be converted to the standard normal distribution using z-values…
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25. Define And Calculate z-values
Ch 7
Define And Calculate z-values
Any normal distribution can be converted, or “standardized” to the standard normal distribution using z-values
Z-values:
Distance from the mean, measured in units of standard deviation
The Formula Is:
Z-values are also called:
Standard normal value
Z score
Z statistic
Standard normal deviate
Normal deviate
Remember Your Algebra
So That You Can Solve For
Any One Of The Variables
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26. Standard Normal Probability Distribution
Ch 7
Standard Normal Probability Distribution
0 means that there is no deviation from the mean!
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27. Convert Value From A Normal Distribution To A Z-score Example 1
Ch 7
Convert Value From A Normal Distribution To A Z-score Example 1
The bi-monthly starting salaries of recent MBA graduates follows the normal distribution with a mean of $2,000 and a standard deviation of $200
What is the z-value for a salary of $2,200?
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28. Convert Value From A Normal Distribution To A Z-score, Example 2
Ch 7
Convert Value From A Normal Distribution To A Z-score, Example 2
What is the z-value of $1,700?
A z-value of 1 indicates that the value of $2,200 is one standard deviation above the mean of $2,000 ( *200)
A z-value of –1.50 indicates that $1,700 is 1.5 standard deviation below the mean of $2000 (2000 – 1.5*200)
Now we can look at a graph 
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29. Ch 7
What is the probability that a foreman’s salary will fall between 1,700 and $2,200?
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30. But It Is Really All About Area Under The Curve Remember:
Ch 7
But It Is Really All About Area Under The Curve Remember:
Areas Under the Normal Curve
Empirical Rule (Normal Rule):
About 68% of the observations will lie within 1 σ of the mean
About 95% of the observations will lie within 2 σ of the mean
Virtually all the observations will be within 3 σ of the mean
Hints: Many statistical chores can be solved with this normal curve
Nevertheless, “The whole world does not fit into a normal curve”
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31. Empirical Rule (Normal Rule):
Ch 7
Empirical Rule (Normal Rule):
Between what two values do about 95% of the values occur?
What if you want to find the % of values that lie between z-scores 0 and 1.56?
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32. Table In Appendix Or On Inside Cover
Ch 7
Table In Appendix Or On Inside Cover
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33. Ch 7
Use The Standard Normal Probability Distribution To Find Area: (Table On Inside Back Cover)
z 
.475
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34. How Can We Use This Standard Normal Curve,
Ch 7
Find The Areas
z 
.475
How Can We Use This Standard Normal Curve,
And The Area Under It?
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35. Ch 7
Example 1
The daily water usage per person in New Providence, New Jersey is normally distributed
Mean = 20 gallons
Standard deviation = 5 gallons
About 68% of those living in New Providence will use how many gallons of water?
+/- 1 standard deviation will give us:
About 68% of the daily water usage will lie between 15 and 25 gallons
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36. Ch 7
Example 2
What is the probability that a person from New Providence selected at random will use between 20 and 24 gallons per day?
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37. Use The Table In The Back Of The Book And Look Up .80
Ch 7
Use The Table In The Back Of The Book And Look Up .80
The area under a normal curve between a z-value of 0 and a z-value of 0.80 is
We conclude that percent of the residents use between 20 and 24 gallons of water per day
See the following diagram:
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38. Ch 7 - 5 . 4 3 2 1 x f ( r a l i t b u o n : m = ,
Area =.2881
“28.81% of the residents use between 20 and 24 gallons of water per day” z
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39. Ch 7
Example 3
What percent of the population use between 18 and 26 gallons per day?
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40. Example 3 The area associated with a z-value of –0.40 is .1554
Ch 7
Example 3
The area associated with a z-value of –0.40 is .1554
Because the curve is symmetrical. look up .40 on the right
The area associated with a z-value of 1.20 is .3849
= .5403
We conclude that percent of the residents use between 18 and 26 gallons of water per day
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41. Ch 7
Example 4
Professor Mann has determined that the scores in his statistics course are approximately normally distributed with a mean of 72 and a standard deviation of 5
He announces to the class that the top 15 percent of the scores will earn an A
What is the lowest score a student can earn and still receive an A?
= .35  This is the area under the curve!
You must look into table and find the value closest to .35
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42. Table In Appendix Or On Inside Cover
Ch 7
Table In Appendix Or On Inside Cover
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43. Solve for X, The Score You Need To Get An A
Ch 7
Solve for X, The Score You Need To Get An A
The result is the score that separates students that earned an A from those that earned a B
Those with a score of 77.2 or more earn an A
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44. Example 5 What is the probability of selecting a shift foreman
Ch 7
Example 5
What is the probability of
selecting a shift foreman
whose salary is
between $790 & $1200?
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45. Example 5 Find Z-scores Look up area under the curve in the tables
Ch 7
Example 5
Find Z-scores
Look up area under the curve in the tables
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46. Table In Appendix Or On Inside Cover
Ch 7
Table In Appendix Or On Inside Cover
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47. Ch 7
Example 5
The probability of selecting a shift foreman whose salary is between $790 & $1200 is:
= .9593
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48. Example 6 What is the probability of selecting a shift foreman
Ch 7
Example 6
What is the probability of
selecting a shift foreman
whose salary is
less than $790?
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49. Example 6 Look up the area The area = .4821 .5 - .4821 = .0179
Ch 7
Example 6
Look up the area
The area = .4821
= .0179
The probability of selecting a shift foreman whose salary is less than $790 is .0179
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50. Ch 7
Summary
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51. Ch 7
Finding Area Under The Standard Normal Distribution – It’s All About Area!
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52. Ch 7
Finding Area Under The Standard Normal Distribution – Use Formulas & Tables
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53. Finding Area Under The Standard Normal Distribution – Four Situations
Ch 7
Finding Area Under The Standard Normal Distribution – Four Situations
If you wish to find the area between 0 and z (or – z), then you can look up the value directly in the table
If you wish to find the area beyond z (or –z), then locate the probability of z in the table and subtract it from .50
If you wish to find the area between two points on different sides of the mean, determine the z-values and add the corresponding areas
If you wish to find the area between two points on the same side of the mean, determine the z-values and subtract the smaller area from the larger
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54. Ch 7
Use The Normal Distribution To Approximate The Binomial Probability Distribution
The normal distribution (a continuous distribution) yields a good approximation of the binomial distribution (a discrete distribution) for large values of n
Let’s Remember the conditions that must be met before we can run a binomial experiment
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55. Ch 7
For An Experiment To Be Binomial It Must Satisfy The Following Conditions:
A random variable (x) counts the # of successes in a fixed number of trials (n)
Trials make up the experiment
Each trail must be independent of the previous trial
Outcome of one trial does not affect the outcome of any other trial
An outcome on each trial of an experiment is classified into one of two mutually exclusive categories:
Success or
Failure
The probability of success stays the same for each trial (so does the probability of failure)
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56. Normal Approximating The Binomial
Ch 7
Normal Approximating The Binomial
The normal probability distribution is generally a good approximation to the binomial probability distribution when:
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57. Mean & Variance of the Binomial Distribution
Ch 7
Mean & Variance of the Binomial Distribution
Empirical experiments have shown these to be acceptable estimates
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58. Continuity Correction Factor
Ch 7
Continuity Correction Factor
The continuity correction factor of .5 is used to extend the continuous value of x one-half unit in either direction
The correction compensates for estimating a discrete distribution by a continuous distribution
When you use discrete numbers, you have “gaps” – you need to take an average that will yield a number between
We will simply estimate by adding or subtracting the value .5
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59. Continuity Correction Factor
Ch 7
Continuity Correction Factor
For the probability at least x occur, use the area above (x - .5)
For the probability that more than x occur, use the area above (x + .5)
For the probability that x or fewer occur, use the area below (x + .5)
For the probability that fewer than x occur, use the area below (x - .5)
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60. Ch 7
Example
A recent study by a marketing research firm showed that 15% of American households owned a video camera.
For a sample of 200 homes, What is the probability that less than 40 homes in the sample have video cameras?
Step 1: Binomial?
Fixed Trails = yes, n = 200
Independent = yes
S/F  Success = have video camera, Failure = don’t have
 Constant = .15
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61. Ch 7
Example
Step 2 : Can we approximate binomial distribution with a standard normal distribution?
Step 3 : Calculate μ & σ for the binomial distribution
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62. Step 4: Calculate z-value (.5 correction)
Ch 7
Step 4: Calculate z-value (.5 correction)
What is the probability that less than 40 homes in the sample have video cameras?
We use the correction factor, so X is 39.5
The value of z is 1.88
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63. Ch 7
Step 5: Look Up Area
From Appendix the area between 0 and 1.88 on the z scale is .4699
So the area to the left of 1.88 is = .9699
The likelihood that less than 40 of the 200 homes have a video camera is about 97%
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64. - 5 . 4 3 2 1 f ( x r a l i t b u o n : m = , s2
Ch 7
EXAMPLE 5
Area = =.9699
z=1.88
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65. Ch 7
Summarize Chapter 7
Understand the difference between discrete and continuous distributions
Compute the mean and the standard deviation for a uniform distribution
Compute probabilities using the uniform distribution
List the characteristics of the:
Normal probability distribution
Standard normal probability distribution
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66. Summarize Chapter 7 Define and calculate z values
Use the standard normal probability distribution to find area:
Above the mean
Below the mean
Between two values
Above one value
Below one value
Use the normal distribution to approximate the binomial probability distribution
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