chapter 5 the binomial probability distribution and related topics

Chapter 5Chapter 5TheThe

Binomial Probability Binomial Probability DistributionDistribution

and and Related TopicsRelated Topics

Essential Question:

How are the mean and the standard deviation determined

from a discrete probability distribution?

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.

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

Statistical Experiment

any process by which an observation

(or measurement)

is obtained

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

Random Variable

a quantitative variable that assumes a value determined by

chance

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

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

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

Probability Distribution

an assignment of probabilities to the specific values of the random variable or to a range of values of

the random variable

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.

Probability distribution for the rolling of an ordinary

fair die

Features of a Probability DistributionFeatures of a Probability Distribution

Probabilities must be between zero and one (inclusive)

Σ P(x) =1

Mean and standard deviation of a discrete probability distribution

Mean = μ = expectation or expected value, the long-run average

Formula:

Standard Deviation

Finding the mean:

Finding the standard deviation

Standard Deviation

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

8. The histogram:

Probability Distributions

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

Linear Functions

Linear Combinations

9.

Linear Functions and Combinations

a.

9.

Linear Functions and Combinations

b.

9. a.

Linear Functions and Combinations

Linear Functions and Combinations 9. a.

9.

Linear Functions and Combinations

b.

Homework Assignment

Chapter 5 Section 1

Pages 190 - 195Exercises: #1 - 19, odd

Exercises: # 2 - 18, even