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Chapter 5Chapter 5TheThe
Binomial Probability Binomial Probability DistributionDistribution
and and Related TopicsRelated Topics
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Essential Question:
How are the mean and the standard deviation determined
from a discrete probability distribution?
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Student Objectives:
The student will distinguish between random and discrete random variables.
The student will graph discrete probability distributions.
The student will compute the mean and standard deviation for a discrete probability distribution.
The student will compute the mean and standard deviation for a linear function of a random variable x.
The student will compute the mean and standard deviation for a linear combination of two independent random variables.
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Terms:
Continuous random variable
Discrete random variable
Linear function of a random variable
Linear function of two independent random variables
Mean
Probability Distribution
Random variable
Standard deviation
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Statistical Experiment
any process by which an observation
(or measurement)
is obtained
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Examples of Statistical Experiments
• Counting the number of books in the College Library
• Counting the number of mistakes on a page of text
• Measuring the amount of rainfall in your state during the month of June
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Random Variable
a quantitative variable that assumes a value determined by
chance
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Discrete Random VariableA discrete random variable is a
quantitative random variable that can take on only a finite number of values or a
countable number of values.
Example: the number of books in the College Library
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Continuous Random Variable
A continuous random variable is a quantitative random variable that can
take on any of the countless number of values in a line interval or a
measurable amount.
Example: the amount of rainfall in your state during the month of June
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Continuous vs DiscreteDirections: In problems #1 - 7, identify each of the following as either a discrete
or continuous random variable.
1. The number of people who are in a car.
2. The number of miles you drive in one week.
3. The weight of a box of cereal.
4. The number of boxes of cereal you buy in one year.
5. The length of time you spend eating your lunch.
6. The number of patients on a psychiatric ward in one day.
7. The volume of blood that is transfused during an operation.
Continuous
Discrete
Continuous
Discrete
Continuous
Continuous
Discrete
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Probability Distribution
an assignment of probabilities to the specific values of the random variable or to a range of values of
the random variable
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Probability Distribution of a Discrete Random
Variable
• A probability is assigned to each value of the random variable.
• The sum of these probabilities must be 1.
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Probability distribution for the rolling of an ordinary
fair die
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Features of a Probability DistributionFeatures of a Probability Distribution
Probabilities must be between zero and one (inclusive)
Σ P(x) =1
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Mean and standard deviation of a discrete probability distribution
Mean = μ = expectation or expected value, the long-run average
Formula:
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Standard Deviation
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Finding the mean:
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Finding the standard deviation
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Standard Deviation
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8. In a personality inventory test for passive-aggressive traits, the possible scores are:
1 = extremely passive2 = moderately passive3 = neither4 = moderately aggressive5 = extremely aggressiveThe test was administered to a group of 110 people and the results were as follows:
Construct a probability distribution table, calculate the expected value (the mean) and
the standard deviation. Use a histogram to graph the probability distribution.
x (score) 1 2 3 4 5
f (frequency) 19 23 32 26 10
x f P(x)
1 19
2 23
3 32
4 26
5 10
Sum:
Probability Distributions
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8. The histogram:
Probability Distributions
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8. The chart:
Probability Distributions
x f P(x) xP(x) x - µ (x - µ)2 (x - µ)2P(x)
1 19 0.1727 0.1727 -1.8636 3.4731 0.5999
2 23 0.2091 0.4182 -0.8636 0.7459 0.1560
3 32 0.2909 0.8727 0.1364 0.0186 0.0054
4 26 0.2364 0.9455 1.1364 1.2913 0.3052
5 10 0.0909 0.4545 2.1364 4.5640 0.4149
Sum: 110 1.0000 2.8636 1.4814
1 = extremely passive2 = moderately passive3 = neither4 = moderately aggressive5 = extremely aggressive
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Linear Functions
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Linear Combinations
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9.
Linear Functions and Combinations
a.
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9.
Linear Functions and Combinations
b.
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9. a.
Linear Functions and Combinations
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Linear Functions and Combinations 9. a.
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9.
Linear Functions and Combinations
b.
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Homework Assignment
Chapter 5 Section 1
Pages 190 - 195Exercises: #1 - 19, odd
Exercises: # 2 - 18, even