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Doing Statistics for Business Data, Inference, and Decision Making Marilyn K. Pelosi Theresa M. Sandifer

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### DoingStatistics for BusinessData, Inference, and Decision MakingMarilyn K. PelosiTheresa M. Sandifer

Chapter 6

Probability

DoingStatistics for Business

Chapter 6 Objectives

- Basic Probability Rules
- Random Variables and Probability Distributions
- The Binomial Probability Distribution
- The Normal Probability Distribution

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Probability is measure of how likely it

is that something will occur.

An Experiment is any action whose outcomes are recordable data.

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The Spinner Problem

Writing out the Sample Space

An experiment consists of spinning the different spinners pictured below:

Write down the sample space for this experiment.

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An event, A, is an outcome or a set of outcomes

that are of interest to the experimenter.

The probability of an event A, P(A), is a

measure of the likelihood that an event A will

occur.

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The Spinner Problem

Classical Definition of Probability

In the previous exercise you found the sample space

for the spinner example to be:

S = {1A, 1B, 1C, 2A, 2B, 2C, 3A, 3B, 3C}

Let A be the event that the first spinner lands on an odd number. Find

P (A).

Let B be the event that the second spinner is a vowel. Find P (B).

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The complement of an event A, denoted A´, is

the set of all outcomes in the sample space, S,

that do not correspond to the event A.

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The event A OR B describes when either A

happens or B happens or they both happen.

The event A and B is the event that A and B both occur.

Two events A and B are said to be mutually exclusive if they have no outcomes in common.

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The Spinner Problem

Calculating the Probability of A OR B

The sample space for the experiment of spinning the two spinners is:

S = {1A, 1B, 1C, 2A, 2B, 2C, 3A, 3B, 3C}

Let A be the event that the first spinner comes up a 1 and let B be the

event that the first spinner comes up a 3.

Find the probability that A OR B occurs using the sample space.

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The Spinner Problem

Calculating the Probability of A OR B

(con’t)

Now find the same probability using the simple addition rule.

Why are the two answers the same?

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Quality Problems

Using the General Addition Rule

The company that manufactures cardboard boxes collected

data on the defect type and production shift. The data are

summarized in the contingency table below:

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Quality Problems

Using the General Addition Rule (con’t)

If a box has more than one defect, then it is classified by the more serious

of the defects only. Suppose that a box from the sample is selected at

random and examined more closely. What is the probability that the box

has a color defect?

What is the probability that the box was produced during the second shift?

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Quality Problems

Using the General Addition Rule (con’t)

Is it possible for the selected box to have a color defect and to have

been produced on the second shift? If so, what is the probability?

What is the probability that the selected box will have a color defect or

will have been produced on the second shift?

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The conditional probability of an event A given an event B is

P(A|B) = P(A and B)

P(B)

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Two events are independent if the probability that one event occurs on any given trial of an experiment is not affected or changed by the occurrence of the other event.

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A Random Variable, X, is a quantitative

variable whose value varies according to

the rules of probability.

The Probability Distribution of a random variable, X, written as p (x), gives the probability that the random variable will take on each of its possible values.

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Defective Diskettes

Finding Interval Probabilities

A company that sells computer diskettes in bulk packages for

a warehouse club outlet knows that the number of defective diskettes

in a package is a random variable with the probability distribution given

below:

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Defective Diskettes

Finding Interval Probabilities (con’t)

Find the probability that a package of the diskettes will contain

at least 3 defective disks.

Find the probability that the package will contain between 2 and 5

defective diskettes.

Find the probability that the number of defective diskettes will be at most

2.

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Defective Diskettes

Creating a Probability Histogram

The company that sells computer diskettes in bulk packages for

warehouse clubs would like to have a picture of how the number of

defective diskettes in a package behaves. The probability distribution

is given below:

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Defective Diskettes

Creating a Probability Histogram

(con’t)

Create a probability histogram for the number of defective diskettes.

Use the probability histogram to describe the distribution of the number

of defective diskettes in a package.

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A Binomial Random Variable is the

number of successes in n trials or in a

sample of size of n.

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Loan Defaults

Recognizing a Binomial Random Variable

While the Chamber of Commerce is concerned about the problems

of small businesses, it must also be sensitive to the problems that the

lending institutions have when issuing credit. One of the problems that

banks have with small businesses is default on loan payments. It is

estimated that approximately 20% of all small business with less than

50 employees are at least six months behind in loan payments.

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Loan Defaults

Recognizing a Binomial Random Variable

The Chamber of Commerce that surveyed the small businesses of

a city wants to look at this problem in more detail. It finds that of the

1536 small businesses in the city, 965 have less than 50 employees. It

randomly selects 25 of these small businesses, checks their credit histories

and counts the number of companies in the sample of 25 that are at least

six months behind in loan payments. Does this qualify as a binomial

probability distribution?

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Loan Defaults

Solving Binomial Probability Problems

The Chamber of Commerce that is checking credit problems of small

businesses estimated that 20% of all small businesses were at least six

months behind in load payments. The Chamber of commerce took a

random sample of 25 small businesses and counted that number of the

businesses that were at least six months behind in loan payments.

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Loan Defaults

Solving Binomial Probability Problems

(con’t)

Define a success for this problem.

Describe the random variable, X, in words.

Find the parameters of the binomial distribution for this problem.

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Loan Defaults

Solving Binomial Probability Problems

(con’t)

Find the probability that in the sample of 25 businesses less than 6 were

at least 6 months behind in loan payments.

Find the probability that between 4 and 9 inclusive were at least 6 months

behind in loan payments.

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Loan Defaults

Calculating the Mean & Standard

Deviation of a Binomial Random Variable

The Chamber of Commerce that was looking at the loan defaults

for small businesses wants to know the mean and standard deviation for

the binomial random variable with n = 25 and = 0.20.

Find the mean and standard deviation of the number of small businesses in

25 that will default on their loans.

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Discovery Exercise 6.1

Exploring the Binomial Distribution

Dear Mom and Dad: Send Cash

According to USA Today, 70% of college students receive

spending money from their parents when at school.

For this exercise, you will need to simulate selecting 30 samples of 5

students from this population of college students and observe whether

they receive spending money from their parents. Consider the successful

outcome to be “receives money” with = 0.70 and the failure outcome to

be “does not receive money.”

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Discovery Exercise 6.1

Exploring the Binomial Distribution

(con’t)

If your instructor does not provide you with a method, you can take ten

pieces (small) of paper and write an S on 7 of them and an F on 3 of them.

Put the papers in a bag or other container and select one at random to

simulate an observation. Note: Be sure to replace the the paper each time

or will not always be 0.70.

Record an S when you select a student who receives money from his/her

parents and a F when you select a student who does not receive spending

money from his/her parents.

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Discovery Exercise 6.1

Exploring the Binomial Distribution

(con’t)

For each sample, record the number of successes you sampled.

In the last outcome compute a running estimate of . Remember that is

the probability of a successful outcome. In this case, is known to be

0.70. Let’s see how close the estimate gets to 0.70 as the sample size

increases. So, after the first sample is selected your estimate of is

simply the number of successes divided by 5. After the second sample is

selected, your estimate of is the number of successes in both samples

divided by 10 and so forth.

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Figure 6.3 Effects of changing p when n is fixed

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Figure 6.4 Effects of Changing n for a fixed value of p

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Figures 6.5 Binomial Distribution for large values of n

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A Probability Density Function, f(x), is

a smooth curve that represents the

probability distribution of a continuous

random variable.

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For a Normal Random Variable, the

parameter is the mean of the normal

random variable, X, and is the

standard deviation.

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Food Expenditures

Looking at the Normal Curve

The amount of money that a person working in a large city spends each

week for lunch is a normally distributed random variable. For professional

and management personnel the random variable has a mean of $35 and a

standard deviation of $5. For hourly employees the mean is $30 with a

standard deviation of $2. Sketch the normal curves for each of the two

random variables on the same graph.

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A Z Random Variable is normally

distributed with a mean of 0 and a

standard deviation of 1, Z ~ N(0,1).

A Standard Normal Table is a table of

probabilities for a Z random variable.

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Speed Reading

Translating from X to Z

The number of pages of a statistics textbook that a student can read

in a given hour is a normally distributed random variable with a mean of

7 pages and a standard deviation of 1.5 pages. One of the professors who

uses the book wants to know the probability that a randomly selected

student can read more than 8.5 pages of the textbook in an hour. Draw a

picture that depicts the problem to be solved and find the Z values

necessary to solve the problem.

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Figure 6.11 The Standard Normal Table

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The Standard Normal Table

Using the Table to Find Probabilities

For each of the following question, draw a picture of what you

are trying to find BEFORE you use the table to find it.

Find the probability that a Z random variable takes on a value that is less

than 2.74.

Find the probability that a Z random variable is greater than 0.85.

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The Standard Normal Table

Using the Table to Find Probabilities

(con’t)

Find the probability that Z is between -1.36 and 1.87.

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Speed Reading

Solving Normal Probability Problems

The instructor who is interested in how many pages of the

statistics textbook that students can read in an hour knows that the

random variable is N(7, 1.5).

Find the probability that a student could read more than 11.5 pages

in an hour.

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Speed Reading

Solving Normal Probability Problems

(con’t)

The instructor was worried about the percentage of students who could

not finish reading a 5-page section in the given hour. What percentage

of the students is this?

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Figure 6.14 The “inverse” normal probability problem

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Figure 6.15 Bottom 10% of the normal distribution

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Figure 6.16 Normal Probability Table Containing the Probability 0.1000

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Speed Reading

Solving the Inverse Problem

The instructor who is interested in how fast students can read

the statistics textbook would like to identify the bottom 25% of

the class, in terms of the number of pages that they can read in an hour.

Find the number of pages per hour that defines the bottom 25% of the

students.

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Calculating Binomial Probabilities

in Excel

1.Position the cursor in an empty cell in the worksheet.

2. From the main toolbar, click on the Function Wizard icon. The Paste Function dialog box opens. Highlight Statistical for the function category and BINOMDIST for the function. Click OK and the dialog box for the BINOMDIST function opens.

3. Place the cursor in the textbox labeled Number_s. Type in the number “25.”

Doing Statistics for Business

Calculating Binomial Probabilities

in Excel (con’t)

4.Place the cursor in the textbox labeled Trials

and enter “50.”

5. Place cursor in the textbox labeled Probability_s. Type in “0.45.”

6. The last textbox lets you indicate what kind of probability you want. In this case we want P(X = 25), so set this value to False.

7. Click on OK and the probability - 0.087330027 - will appear in the cell which you started.

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Calculating Normal Probabilities

in using KaddStat

1.Fill in the mean, , and the standard deviation, ,

of the random variable in the appropriate textboxes.

2. From the section labeled Range of Interest, select the type of probability you want. For left and right tail probabilities, you will enter one value of X in the textbox labeled Value. If you select Left and Right, either inside or outside,the box will change and you will enter two values of X.

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Calculating Normal Probabilities

in KaddStat (con’t)

3. To solve the inverse normal problem, select Inverse Value type in the left tail area in the box labeled Cumulative Probability.

4. Fill in where you want the output to go and click OK to obtain the results.

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Generating Binomial Random Data in Excel

1.Select Binomial from the list of distributions and the dialog box will change to allow input of the appropriate parameters.

2. In the textbox for Number of Variables: type “1” and in the textbox for Number of Random Numbers: type “20.”

3. In the textbox labeled p Value enter the value for 0.70 & in the textbox for Number of Trials: enter “20.”

4. Specify where you want the output to appear and click OK.

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Figure 6.25The Binomial Random Variable Dialog Box

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Figure 6.26Random Data from Binomial Distribution

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Chapter 6 Summary

In this chapter you have learned:

- Probability is more than just flipping coins, spinning spinners, or gambling. It is important

in the study and development of statistics.

- Probability is the bridge between Descriptive Statistics and Inferential Statistics.
- In Descriptive Statistics we use different

techniques to describe sample data.

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Chapter 6 Summary(con’t)

- In Inferential Statistics we will make test hypotheses about the populations from which these samples came.
- Probability is the tool that allows us to reconcile what happened (descriptive) with what we think is true by determining how likely the outcomes of the experiment we perform are.

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