sst5e_ppt_ch06.pptx

STATISTICS

INFORMED DECISIONS USING DATA

Fifth Edition, Global Edition

Chapter 6

6.1 Discrete Random Variables

Learning Objectives

1.

Distinguish between discrete and continuous random

variables

2.

Identify discrete probability distributions

3.

Graph discrete probability distributions

4.

Compute and interpret the mean of a discrete random

variable

6.1 Discrete Random Variables

6.1.1 Distinguish between Discrete and Continuous Random

Variables

6.1 Discrete Random Variables

6.1.1 Distinguish between Discrete and Continuous Random

Variables

6.1 Discrete Random Variables

6.1.1 Distinguish between Discrete and Continuous Random

Variables

6.1 Discrete Random Variables

6.1.1 Distinguish between Discrete and Continuous Random

Variables

EXAMPLE Distinguishing Between Discrete and

Continuous Random Variables

Determine whether the following random variables are discrete or continuous. State possible values for the random variable.

(a) The number of light bulbs that burn out in a room of 10 light bulbs in the next year.

Discrete; x = 0, 1, 2, …, 10

6.1 Discrete Random Variables

6.1.2 Identify Discrete Probability Distributions

The

probability distribution

of a discrete random variable

6.1 Discrete Random Variables

6.1.2 Identify Discrete Probability Distributions

EXAMPLE A Discrete

Probability Distribution

The table to the right shows the probability distribution for the random variable X, where X

represents the number of movies streamed on Netflix each month.

x P(x)

6.1 Discrete Random Variables

6.1.2 Identify Discrete Probability Distributions

Rules for a Discrete Probability Distribution

Let P(x) denote the probability that the random variable X equals x; then

1. ΣP(x) = 1

6.1 Discrete Random Variables

6.1.2 Identify Discrete Probability Distributions

EXAMPLE Identifying Probability Distributions

Is the following a probability distribution?

x P(x)

0 0.16

1 0.18

2 0.22

3 0.10

6.1 Discrete Random Variables

6.1.2 Identify Discrete Probability Distributions

EXAMPLE Identifying Probability Distributions

Is the following a probability distribution?

x P(x)

0 0.16

1 0.18

2 0.22

3 0.10

4 0.30

6.1 Discrete Random Variables

6.1.2 Identify Discrete Probability Distributions

EXAMPLE Identifying Probability Distributions

Is the following a probability distribution?

x P(x)

0 0.16

1 0.18

2 0.22

3 0.10

6.1 Discrete Random Variables

6.1.3 Graph Discrete Probability Distributions

EXAMPLE Drawing a Probability Histogram

Draw a probability histogram of the probability distribution to the right, which represents the

number of movies streamed on Netflix each month.

x P(x)

6.1 Discrete Random Variables

6.1.4 Compute and Interpret the Mean of a Discrete Random

Variable

The Mean of a Discrete Random Variable

6.1 Discrete Random Variables

6.1.4 Compute and Interpret the Mean of a Discrete Random

Variable

EXAMPLE Computing the Mean

of a Discrete Random Variable

Compute the mean of the probability distribution to the right, which

represents the number of DVDs a person rents from a video store during a single visit.

x P(x)

6.1 Discrete Random Variables

6.1.4 Compute and Interpret the Mean of a Discrete Random

Variable

Interpretation of the Mean of a Discrete Random

Variable

Suppose an experiment is repeated n independent times and the value of the random variable X is recorded. As the number of

6.1 Discrete Random Variables

6.1.4 Compute and Interpret the Mean of a Discrete Random

Variable

EXAMPLE Interpretation of the Mean of a Discrete Random Variable

The following data represent the number of DVDs rented by 100 randomly

selected customers in a single visit. Compute the mean number of DVDs rented.

6.1 Discrete Random Variables

6.1 Discrete Random Variables

6.1.4 Compute and Interpret the Mean of a Discrete Random

Variable

6.1 Discrete Random Variables

6.1.5 Compute and Interpret the Mean of a Discrete Random

Variable

Because the mean of a random variable represents what we

would expect to happen in the long run, it is also called the

6.1 Discrete Random Variables

6.1.5 Compute and Interpret the Mean of a Discrete Random

Variable

EXAMPLE Computing the Expected Value of a Discrete Random Variable

A term life insurance policy will pay a beneficiary a certain sum of money upon the death of the policy holder. These policies have premiums that must be paid annually. Suppose a life insurance company sells a $250,000 one year term life insurance policy to a 49-year-old female for $530. According to the National Vital Statistics Report, Vol. 47, No. 28, the probability the female will survive the year is 0.99791. Compute the expected value of this policy to the insurance company.

blank x P(x)

Survives 530 0.99791

6.1 Discrete Random Variables

6.1.5 Compute the Standard Deviation of a Discrete Random

Variable

Standard Deviation of a Discrete Random Variable

6.1 Discrete Random Variables

6.1.5 Compute the Standard Deviation of a Discrete Random

Variable

EXAMPLE Computing the

Variance and Standard Deviation

of a Discrete Random Variable

Compute the variance and standard deviation of the following probability distribution which represents the

number of DVDs a person rents from a video store during a single visit.

Remember, the mean that we found was 1.49.

x P(x)

6.1 Discrete Random Variables

6.1.5 Compute the Standard Deviation of a Discrete Random

Variable

6.2 The Binomial Probability Distribution

Learning Objectives

1.

Determine whether a probability experiment is a binomial

experiment

2.

Compute probabilities of binomial experiments

3.

Compute the mean and standard deviation of a binomial

random variable

6.2 The Binomial Probability Distribution

6.2.1 Determine Whether a Probability Experiment is a Binomial Experiment (1 of 5)

Criteria for a Binomial Probability Experiment

An experiment is said to be a

binomial experiment

if

1. The experiment is performed a fixed number of times. Each repetition of the experiment is called a trial.

2. The trials are independent. This means the outcome of one trial will not affect the outcome of the other trials.

6.2 The Binomial Probability Distribution

6.2.1 Determine Whether a Probability Experiment is a Binomial Experiment (2 of 5)

Notation Used in the Binomial Probability Distribution

• Let p denote the probability of success so that 1 − p is the probability of failure.

• Let X be a binomial random variable that denotes the number of successes in n independent trials of the experiment.

6.2 The Binomial Probability Distribution

6.2.1 Determine Whether a Probability Experiment is a Binomial Experiment (3 of 5)

EXAMPLE Identifying Binomial Experiments

(a) A player rolls a pair of fair die 10 times. The number X of 7’s rolled is recorded.

6.2 The Binomial Probability Distribution

6.2.1 Determine Whether a Probability Experiment is a Binomial Experiment (4 of 5)

EXAMPLE Identifying Binomial Experiments

(b) The 11 largest airlines had an on-time percentage of 84.7% in November, 2001 according to the Air Travel Consumer

Report. In order to assess reasons for delays, an official with the FAA randomly selects flights until she finds 10 that were not on time. The number of flights X that need to be selected is recorded.

6.2 The Binomial Probability Distribution

6.2.1 Determine Whether a Probability Experiment is a Binomial Experiment (5 of 5)

EXAMPLE Identifying Binomial Experiments

(c) In a class of 30 students, 55% are female. The instructor randomly selects 4 students. The number X of females selected is recorded.

6.2 The Binomial Probability Distribution

6.2.2 Compute Probabilities of Binomial Experiments

EXAMPLE Constructing a Binomial Probability Distribution According to the Air Travel Consumer Report, the 11 largest air carriers had an on-time percentage of 79.0% in May, 2008.

Suppose that 4 flights are randomly selected from May, 2008 and the number of on-time flights X is recorded.

6.2 The Binomial Probability Distribution

6.2.2 Compute Probabilities of Binomial Experiments

EXAMPLE Constructing a

Binomial Probability Distribution Compute probability of x = 0

6.2 The Binomial Probability Distribution

6.2.2 Compute Probabilities of Binomial Experiments

EXAMPLE Constructing a

Binomial Probability Distribution Compute probability of x = 1

6.2 The Binomial Probability Distribution

6.2.2 Compute Probabilities of Binomial Experiments

EXAMPLE Constructing a

Binomial Probability Distribution Compute probability of x = 2

6.2 The Binomial Probability Distribution

6.2.2 Compute Probabilities of Binomial Experiments

EXAMPLE Constructing a

Binomial Probability Distribution Compute probability of x = 3

6.2 The Binomial Probability Distribution

6.2.2 Compute Probabilities of Binomial Experiments

Binomial Probability Distribution Compute probability of x = 4

(i.e., 4 successes).

x P(x)

0 0.00194

6.2 The Binomial Probability Distribution

6.2.2 Compute Probabilities of Binomial Experiments

Binomial Probability Distribution Function

6.2 The Binomial Probability Distribution

6.2.2 Compute Probabilities of Binomial Experiments

Phrase Math Symbol

“at least” or “no less than”

or “greater than or equal to” ≥ “more than” or “greater than” > “fewer than” or “less than” < “no more than” or “at most”

6.2 The Binomial Probability Distribution

6.2.2 Compute Probabilities of Binomial Experiments

EXAMPLE Using the Binomial Probability Distribution

Function

According to the Experian Automotive, 35% of all car-owning households have three or more cars.

6.2 The Binomial Probability Distribution

6.2.2 Compute Probabilities of Binomial Experiments

EXAMPLE Using the Binomial Probability Distribution

Function

According to the Experian Automotive, 35% of all car-owning households have three or more cars.

6.2 The Binomial Probability Distribution

6.2.2 Compute Probabilities of Binomial Experiments

EXAMPLE Using the Binomial Probability Distribution

Function

According to the Experian Automotive, 35% of all car-owning households have three or more cars.

6.2 The Binomial Probability Distribution

6.2.3 Compute the Mean and Standard Deviation of a

Binomial Random Variable

Mean (or Expected Value) and Standard Deviation of a

Binomial Random Variable

A binomial experiment with n independent trials and probability of success p has a mean and standard deviation given by the

6.2 The Binomial Probability Distribution

6.2.3 Compute the Mean and Standard Deviation of a

Binomial Random Variable

EXAMPLE Finding the Mean and Standard Deviation of a

Binomial Random Variable

According to the Experian Automotive, 35% of all car-owning

households have three or more cars. In a simple random sample of 400 car-owning households, determine the mean and standard

6.2 The Binomial Probability Distribution

6.2.4 Graph a Binomial Probability Distribution

EXAMPLE Graphing Binomial Probability Distributions

(a)

Graph the binomial probability distribution with

n

= 8 and

p

= 0.15.

(b)

Graph the binomial probability distribution with

n

= 8 and

p

= 0. 5.

(c)

Graph the binomial probability distribution with

n

= 8 and

6.2 The Binomial Probability Distribution

6.2 The Binomial Probability Distribution

6.2 The Binomial Probability Distribution

6.2 The Binomial Probability Distribution

6.2.4 Graph a Binomial Probability Distribution

EXAMPLE Graphing Binomial Probability Distributions

6.2 The Binomial Probability Distribution

6.2 The Binomial Probability Distribution

6.2.4 Graph a Binomial Probability Distribution

6.2 The Binomial Probability Distribution

6.2 The Binomial Probability Distribution

6.2.4 Graph a Binomial Probability Distribution

6.2 The Binomial Probability Distribution

6.2 The Binomial Probability Distribution

6.2.4 Graph a Binomial Probability Distribution

For a fixed,

p

,

as the number of trials

n

in a binomial

experiment increases, the probability distribution of the random

variable

X

becomes bell-shaped.

6.2 The Binomial Probability Distribution

6.2.4 Graph a Binomial Probability Distribution

Use the Empirical Rule to identify unusual observations in a

binomial experiment.

The Empirical Rule states that in a bell-shaped distribution

about 95% of all observations lie within two standard

6.2 The Binomial Probability Distribution

6.2.4 Graph a Binomial Probability Distribution

Use the Empirical Rule to identify unusual observations in a

binomial experiment.

95% of the observations lie between

μ

−

2

σ

and

μ

+ 2

σ

.

Any observation that lies outside this interval may be

6.2 The Binomial Probability Distribution

6.2.4 Graph a Binomial Probability Distribution

EXAMPLE Using the Mean, Standard Deviation and Empirical Rule to Check for Unusual Results in a Binomial Experiment According to the Experian Automotive, 35% of all car-owning households have three or more cars. A researcher believes this percentage is higher than the percentage reported by Experian Automotive. He conducts a simple random sample of 400

6.2 The Binomial Probability Distribution

6.2.4 Graph a Binomial Probability Distribution