1. Normal distribution and probability
Introduction
Normal Curve
Mean
Introducing Standard Deviation
Standard Deviation and the Normal curve
The Normal Distribution
The Standard Normal Distribution
Expected Value
Inverse Normal Problems
Probability Exam Questions
Probability exam Answers
Probability
Theoretical Probability
Equally Likely Outcomes
Probability Trees
Sampling without replacement
Venn diagrams
Experimental probability
Long-Run relative frequency
Simulation
describe a simulation
Simulation Question
Simulation Exam Question
L2 Normal Distribution & Probability
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2. L2 Normal Distribution & Probability
Introduction
L2 Normal Distribution & Probability
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3. L2 Normal Distribution & Probability
Introduction
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4. L2 Normal Distribution & Probability
The Normal Curve
Many sets of data collected in nature, industry, business and other situations fit what is called a normal distribution.
This means:
If the frequency of each piece of data were graphed, the curve would fit a symmetrical bell shape.
Most measurements are near the middle
The further away from the middle value (mean), the less frequent the occurrence.
The general shape looks like this
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5. L2 Normal Distribution & Probability
The Normal Curve
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6. L2 Normal Distribution & Probability
Mean
The mean is just the average of the numbers.
It is easy to calculate: Just add up all the numbers, then divide by how many numbers there are.
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7. Introducing Standard Deviation
The Standard Deviation (σ) is a measure of how spread out numbers are.
You and your friends have just measured the heights of your dogs (in millimeters):
The heights (at the shoulders) are: 600mm, 470mm, 170mm, 430mm and 300mm.
Let’s find out the Mean and the Standard Deviation.
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8. Introducing Standard Deviation
Answer:
so the average height is 394 mm. Let's plot this on the chart:
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9. Introducing Standard Deviation
Calculate the Standard Deviation using your calculators.
Now we can show which heights are within one Standard Deviation (147mm) of the Mean:
So, using the Standard Deviation we have a "standard" way of knowing what is normal, and what is extra large or extra small.
Rottweillers are tall dogs. And Dachsunds are a bit short .
L2 Normal Distribution & Probability
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10. Standard Deviation and the Normal Curve
One standard deviation away from the mean in either direction on the horizontal axis (the red area on the graph) accounts for somewhere around 68 percent of the people in this group. “Likely/probable”
Two standard deviations away from the mean (the red and green areas) account for roughly 95 percent of the people.
“Very likely/very probable”
Three standard deviations (the red, green and blue areas) account for about 99 percent of the people. “Almost certain”
The total area under the curve adds to 1 (or 100%)
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11. The Normal Distribution
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12. The Normal Distribution
95%
65km/h and 125km/h
16%
105km/h
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13. The Standard Normal Distribution
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14. The Standard Normal Distribution
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15. The Standard Normal Distribution
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16. The Standard Normal Distribution
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17. The Standard Normal Distribution
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18. The Standard Normal Distribution
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19. The Standard Normal Distribution
Notation
For the same question, calculate the percentage of cars travelling more than 97km/h?
What is the probability that X is greater than 97km/h
It is the probability that Z is greater than 0.2
This is equal to 1 – probability that Z is greater than 0.2
This is equal to 1 –
Which is 42.07%
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20. L2 Normal Distribution & Probability
Expected Value
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21. Inverse Normal Problems
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22. Inverse Normal Problems
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23. Inverse Normal Problems
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24. Equally Likely Outcomes
Theta Exercise 22.01
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25. L2 Normal Distribution & Probability
Probability Trees
0.4
0.6
Kauri
Rimu
0.1
0.9
0.05
0.95
Returned
Not Returned
0.04
0.36
0.03
0.57
L2 Normal Distribution & Probability
M. Guttormson

26. L2 Normal Distribution & Probability
Probability Trees
0.4
0.6
Kauri
Rimu
0.1
0.9
0.05
0.95
Returned
Not Returned
0.04
0.36
0.03
0.57
L2 Normal Distribution & Probability
M. Guttormson

27. L2 Normal Distribution & Probability
Probability Trees
0.4
0.6
Kauri
Rimu
0.1
0.9
0.05
0.95
Returned
Not Returned
0.04
0.36
0.03
0.57
L2 Normal Distribution & Probability
M. Guttormson

28. L2 Normal Distribution & Probability
Probability Trees
Action
Where
Key Word
Multiply
Sideways along branches
“and”
Add
Vertically at ends of branches
“or”
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29. Sampling Without Replacement
3/7
Blue
Red
1/2
2/3
1/3
2/7
1/7
4/7
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30. L2 Normal Distribution & Probability
Venn Diagrams
Union (one or the other or both)
Intersection (both only)
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31. L2 Normal Distribution & Probability
Venn Diagrams
Complimentary Events
P(A) is the probability inside the circle.
P(A’) is the probability outside the circle.
P(A) + P(A’)=1
Example: Probability that it rains tomorrow is 0.4.
The probability that it does not rain tomorrow is 0.6
Mutually Exclusive Events
Theta Chapter 22 Exercise 22.04
Intersecting Events
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32. Long-run relative frequency
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33. L2 Normal Distribution & Probability
Simulation
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34. How to describe a simulation
Tool: Definition of the probability tool
Statement of how the tool models the situation (Assign)
“Generate random numbers using the random number key on a calculator and truncate. 12ran#+1=”
“Designate numbers to colours of tee-shirts in the appropriate proportions.
1, 2 = blue
3, 4, 5 = red
6 = green
7, 8 = purple
9, 10 = yellow
11, 12 = black
Trial: Definition of a trial
Definition of a successful outcome of the trial
“One trial would involve generating random numbers until there are seven outcomes representing each day of the week. Days need to be labelled Monday, Tuesday, Wednesday, Thursday, Friday, Saturday, and Sunday.” A successful trial would be if 3 or more of the same colour shirt appears in one trial (one week).
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35. How to describe a simulation
Results: Statement of how the results will be tabulated giving an example of a successful outcome and an unsuccessful outcome.
Statement of how many trials should be carried out.
“ A tick in the results column will indicate when 3 of the 7 random numbers generated represent the same colour tee-shirt. Repeat a minimum of 20 times.”
Calculation: Statement of how the calculation needed for the conclusion will be done:
Long run relative frequency = Number of successful results
Number of trials
Mean = Sum of trial results
The probabilities will be calculated using: Number of successful results 20
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36. L2 Normal Distribution & Probability
Simulation Question
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37. Simulation Exam Question “T-shirts”
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38. Simulation Exam Question “T-shirts”
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39. Simulation Exam Question “Fair Cop”
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40. Simulation Exam Question “Fair Cop”
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41. Probability exam Questions
QUESTION ONE
Andrew collected a large sample of chest measurements from students of his year level.
He found the chest measurements to be approximately normally distributed with a mean chest measurement of 86cm and a standard deviation of 3.5cm.
(a) Calculate the probability that a randomly selected student from Andrew’s year level
(i) would have a chest measurement between 86cm and 90cm
(ii) would have a chest measurement less than 87.5cm
(b) Andrew’s class has 34 students in it. How many of these students would you expect to have chest measurements of more than 89cm?
L2 Normal Distribution & Probability
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42. Probability exam Questions
L2 Normal Distribution & Probability
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43. Probability exam answers
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