the normal distribution - statpower normal distribution.pdf · distribution of iq scores...
TRANSCRIPT
![Page 1: The Normal Distribution - Statpower Normal Distribution.pdf · Distribution of IQ Scores Normal(100,15) 0.000 0.004 0.008 0.012 0.016 0.020 0.024 0.028 55 70 85 100 115 130 145. Problem](https://reader030.vdocuments.net/reader030/viewer/2022041110/5f1009447e708231d447208a/html5/thumbnails/1.jpg)
The Normal Distribution
James H. Steiger
![Page 2: The Normal Distribution - Statpower Normal Distribution.pdf · Distribution of IQ Scores Normal(100,15) 0.000 0.004 0.008 0.012 0.016 0.020 0.024 0.028 55 70 85 100 115 130 145. Problem](https://reader030.vdocuments.net/reader030/viewer/2022041110/5f1009447e708231d447208a/html5/thumbnails/2.jpg)
Types of Probability Distributions
There are two fundamental types of probability distributions
DiscreteContinuous
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Discrete Probability Distributions
These are used to model a situation where The number of events (things that can happen) is countable (i.e., can be placed in correspondence with integers)Probabilities sum to 1Each event has a probability
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A Discrete Probability FunctionDiscrete Uniform (1,6) Distribution
0.000
0.167
0 1 2 3 4 5 6 7
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Continuous Probability Distributions
These are used to model situations where the number of things that can happen is not countableProbability of a particular outcome cannot be definedInterval probabilities can be defined: Probability of an interval is the area under the probability density curve between the endpoints of the interval
![Page 6: The Normal Distribution - Statpower Normal Distribution.pdf · Distribution of IQ Scores Normal(100,15) 0.000 0.004 0.008 0.012 0.016 0.020 0.024 0.028 55 70 85 100 115 130 145. Problem](https://reader030.vdocuments.net/reader030/viewer/2022041110/5f1009447e708231d447208a/html5/thumbnails/6.jpg)
Probability Density FunctionUniform(0,6) Distribution
0.000
0.167
-2 -1 0 1 2 3 4 5 6 7 8
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The Normal Distribution FamilyStandard Normal Distribution
0.0
0.1
0.2
0.3
0.4
σ
34.13%
µ µ + σ
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The Standard Normal Distribution
Standard Normal Distribution
0.0
0.1
0.2
0.3
0.4
-3 -2 -1 0 1 2 3
σ
34.13%
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The Normal Curve Table
Z F(Z) Z F(Z)0.25 59.87 1.75 95.990.50 69.15 1.96 97.50.75 77.34 2.00 97.721.00 84.13 2.25 98.781.25 89.44 2.326 99.001.50 93.32 2.50 99.381.645 95.00 2.576 99.50
![Page 10: The Normal Distribution - Statpower Normal Distribution.pdf · Distribution of IQ Scores Normal(100,15) 0.000 0.004 0.008 0.012 0.016 0.020 0.024 0.028 55 70 85 100 115 130 145. Problem](https://reader030.vdocuments.net/reader030/viewer/2022041110/5f1009447e708231d447208a/html5/thumbnails/10.jpg)
Some Typical “Routine” Problems
Distribution of IQ ScoresNormal(100,15)
0.000
0.004
0.008
0.012
0.016
0.020
0.024
0.028
55 70 85 100 115 130 145
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Some Typical “Routine” Problems
1.What percentage of people have IQ scores between 100 and 115?2.What percentage of people have IQ scores greater than 130?3.What percentage of people have IQ scores between 70 and 85?4. What is the percentile rank of an IQ score of 122.5?5. What IQ score is at the 99th percentile?
![Page 12: The Normal Distribution - Statpower Normal Distribution.pdf · Distribution of IQ Scores Normal(100,15) 0.000 0.004 0.008 0.012 0.016 0.020 0.024 0.028 55 70 85 100 115 130 145. Problem](https://reader030.vdocuments.net/reader030/viewer/2022041110/5f1009447e708231d447208a/html5/thumbnails/12.jpg)
General Strategy for Normal Curve Problems
ALWAYS draw the pictureEstimate the answer from the picture, remembering:
SymmetryTotal area is 1.0Area to the left or right of center is .50
Convert to Z-score formCompute interval probability by subtraction
![Page 13: The Normal Distribution - Statpower Normal Distribution.pdf · Distribution of IQ Scores Normal(100,15) 0.000 0.004 0.008 0.012 0.016 0.020 0.024 0.028 55 70 85 100 115 130 145. Problem](https://reader030.vdocuments.net/reader030/viewer/2022041110/5f1009447e708231d447208a/html5/thumbnails/13.jpg)
Problem #1Distribution of IQ Scores
Normal(100,15)
0.000
0.004
0.008
0.012
0.016
0.020
0.024
0.028
55 70 85 100 115 130 145
![Page 14: The Normal Distribution - Statpower Normal Distribution.pdf · Distribution of IQ Scores Normal(100,15) 0.000 0.004 0.008 0.012 0.016 0.020 0.024 0.028 55 70 85 100 115 130 145. Problem](https://reader030.vdocuments.net/reader030/viewer/2022041110/5f1009447e708231d447208a/html5/thumbnails/14.jpg)
Problem #2Distribution of IQ Scores
Normal(100,15)
0.000
0.004
0.008
0.012
0.016
0.020
0.024
0.028
55 70 85 100 115 130 145
![Page 15: The Normal Distribution - Statpower Normal Distribution.pdf · Distribution of IQ Scores Normal(100,15) 0.000 0.004 0.008 0.012 0.016 0.020 0.024 0.028 55 70 85 100 115 130 145. Problem](https://reader030.vdocuments.net/reader030/viewer/2022041110/5f1009447e708231d447208a/html5/thumbnails/15.jpg)
Problem #2 (ctd)
130130 100 2.00
15Z −
= = +
![Page 16: The Normal Distribution - Statpower Normal Distribution.pdf · Distribution of IQ Scores Normal(100,15) 0.000 0.004 0.008 0.012 0.016 0.020 0.024 0.028 55 70 85 100 115 130 145. Problem](https://reader030.vdocuments.net/reader030/viewer/2022041110/5f1009447e708231d447208a/html5/thumbnails/16.jpg)
The Normal Curve Table
Z F(Z) Z F(Z)0.25 59.87 1.75 95.990.50 69.15 1.96 97.50.75 77.34 2.00 97.721.00 84.13 2.25 98.781.25 89.44 2.326 99.001.50 93.32 2.50 99.381.645 95.00 2.576 99.50
![Page 17: The Normal Distribution - Statpower Normal Distribution.pdf · Distribution of IQ Scores Normal(100,15) 0.000 0.004 0.008 0.012 0.016 0.020 0.024 0.028 55 70 85 100 115 130 145. Problem](https://reader030.vdocuments.net/reader030/viewer/2022041110/5f1009447e708231d447208a/html5/thumbnails/17.jpg)
Problem #3Distribution of IQ Scores
Normal(100,15)
0.000
0.004
0.008
0.012
0.016
0.020
0.024
0.028
55 70 85 100 115 130 145
![Page 18: The Normal Distribution - Statpower Normal Distribution.pdf · Distribution of IQ Scores Normal(100,15) 0.000 0.004 0.008 0.012 0.016 0.020 0.024 0.028 55 70 85 100 115 130 145. Problem](https://reader030.vdocuments.net/reader030/viewer/2022041110/5f1009447e708231d447208a/html5/thumbnails/18.jpg)
Problem #3
7070 100 2.00
15Z −
= = −
8585 100 1.00
15Z −
= = −
![Page 19: The Normal Distribution - Statpower Normal Distribution.pdf · Distribution of IQ Scores Normal(100,15) 0.000 0.004 0.008 0.012 0.016 0.020 0.024 0.028 55 70 85 100 115 130 145. Problem](https://reader030.vdocuments.net/reader030/viewer/2022041110/5f1009447e708231d447208a/html5/thumbnails/19.jpg)
The Normal Curve Table
Z F(Z) Z F(Z)0.25 59.87 1.75 95.990.50 69.15 1.96 97.50.75 77.34 2.00 97.721.00 84.13 2.25 98.781.25 89.44 2.326 99.001.50 93.32 2.50 99.381.645 95.00 2.576 99.50
![Page 20: The Normal Distribution - Statpower Normal Distribution.pdf · Distribution of IQ Scores Normal(100,15) 0.000 0.004 0.008 0.012 0.016 0.020 0.024 0.028 55 70 85 100 115 130 145. Problem](https://reader030.vdocuments.net/reader030/viewer/2022041110/5f1009447e708231d447208a/html5/thumbnails/20.jpg)
Problem #4Distribution of IQ Scores
Normal(100,15)
0.000
0.004
0.008
0.012
0.016
0.020
0.024
0.028
55 70 85 100 115 130 145
![Page 21: The Normal Distribution - Statpower Normal Distribution.pdf · Distribution of IQ Scores Normal(100,15) 0.000 0.004 0.008 0.012 0.016 0.020 0.024 0.028 55 70 85 100 115 130 145. Problem](https://reader030.vdocuments.net/reader030/viewer/2022041110/5f1009447e708231d447208a/html5/thumbnails/21.jpg)
Problem #4
122.5122.5 100 1.50
15Z −
= =
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The Normal Curve Table
Z F(Z) Z F(Z)0.25 59.87 1.75 95.990.50 69.15 1.96 97.50.75 77.34 2.00 97.721.00 84.13 2.25 98.781.25 89.44 2.326 99.001.50 93.32 2.50 99.381.645 95.00 2.576 99.50
![Page 23: The Normal Distribution - Statpower Normal Distribution.pdf · Distribution of IQ Scores Normal(100,15) 0.000 0.004 0.008 0.012 0.016 0.020 0.024 0.028 55 70 85 100 115 130 145. Problem](https://reader030.vdocuments.net/reader030/viewer/2022041110/5f1009447e708231d447208a/html5/thumbnails/23.jpg)
Problem #5Distribution of IQ Scores
Normal(100,15)
0.000
0.004
0.008
0.012
0.016
0.020
0.024
0.028
55 70 85 100 115 130 145
![Page 24: The Normal Distribution - Statpower Normal Distribution.pdf · Distribution of IQ Scores Normal(100,15) 0.000 0.004 0.008 0.012 0.016 0.020 0.024 0.028 55 70 85 100 115 130 145. Problem](https://reader030.vdocuments.net/reader030/viewer/2022041110/5f1009447e708231d447208a/html5/thumbnails/24.jpg)
Problem #5
Find Z-score corresponding to 99th
percentile.Convert to a raw score.
![Page 25: The Normal Distribution - Statpower Normal Distribution.pdf · Distribution of IQ Scores Normal(100,15) 0.000 0.004 0.008 0.012 0.016 0.020 0.024 0.028 55 70 85 100 115 130 145. Problem](https://reader030.vdocuments.net/reader030/viewer/2022041110/5f1009447e708231d447208a/html5/thumbnails/25.jpg)
Relative Tail Probability Problems
These problems involve analyzing relative probabilities of extreme events in two populations. Such problems:
Are seldom found in textbooksAre often relevant to real world problemsSometimes produce results that are counterintuitive
![Page 26: The Normal Distribution - Statpower Normal Distribution.pdf · Distribution of IQ Scores Normal(100,15) 0.000 0.004 0.008 0.012 0.016 0.020 0.024 0.028 55 70 85 100 115 130 145. Problem](https://reader030.vdocuments.net/reader030/viewer/2022041110/5f1009447e708231d447208a/html5/thumbnails/26.jpg)
Relative Tail Probability #1
Group A has a mean of 100 and a standard deviation of 15. Group B has a mean of 100 and a standard deviation of 17. What is the relative likelihood of finding a person with a score above 145 in Group B, relative to Group A?
![Page 27: The Normal Distribution - Statpower Normal Distribution.pdf · Distribution of IQ Scores Normal(100,15) 0.000 0.004 0.008 0.012 0.016 0.020 0.024 0.028 55 70 85 100 115 130 145. Problem](https://reader030.vdocuments.net/reader030/viewer/2022041110/5f1009447e708231d447208a/html5/thumbnails/27.jpg)
The Normal Curve Table
Z Area Above Z2.6 .00472.65 .004052.7 .00353.0 .00135Answer: About 3 to 1.
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Relative Tail Probability #2
Group A has a mean of 100 and a standard deviation of 15. Group B has a mean of 100 and a standard deviation of 17. What is the relative likelihood of finding a person with a score above 160 in Group B, relative to Group A?
![Page 29: The Normal Distribution - Statpower Normal Distribution.pdf · Distribution of IQ Scores Normal(100,15) 0.000 0.004 0.008 0.012 0.016 0.020 0.024 0.028 55 70 85 100 115 130 145. Problem](https://reader030.vdocuments.net/reader030/viewer/2022041110/5f1009447e708231d447208a/html5/thumbnails/29.jpg)
The Normal Curve Table
Z Area Above Z3.529 .0002084.00 .0000317
Answer: About 6.57 to 1.