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

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Doing Statistics for Business Data, Inference,
and Decision MakingMarilyn K. PelosiTheresa M.
Sandifer
Chapter 6 Probability

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Doing Statistics 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 Sample Space is the set of all
• possible outcomes of an experiment.

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• TRY IT NOW!
• 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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Doing Statistics for Business

• TRY IT NOW!
• 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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• TRY IT NOW!
• 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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• TRY IT NOW!
• The Spinner Problem
• Calculating the Probability of A OR B
• (cont)
• Now find the same probability using the simple
  addition rule.
• Why are the two answers the same?

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• TRY IT NOW!
• 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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• TRY IT NOW!
• Quality Problems
• Using the General Addition Rule (cont)
• 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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• TRY IT NOW!
• Quality Problems
• Using the General Addition Rule (cont)
• 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(AB) 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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Doing Statistics for Business

• TRY IT NOW!
• 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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Doing Statistics for Business

• TRY IT NOW!
• Defective Diskettes
• Finding Interval Probabilities (cont)
• 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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• TRY IT NOW!
• 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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• TRY IT NOW!
• Defective Diskettes
• Creating a Probability Histogram
• (cont)
• 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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Doing Statistics for Business

• A Binomial Random Variable is the
• number of successes in n trials or in a
• sample of size of n.

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• TRY IT NOW!
• 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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• TRY IT NOW!
• 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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• TRY IT NOW!
• 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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• TRY IT NOW!
• Loan Defaults
• Solving Binomial Probability Problems
• (cont)
• 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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• TRY IT NOW!
• Loan Defaults
• Solving Binomial Probability Problems
• (cont)
• 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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• TRY IT NOW!
• 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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Doing Statistics for Business

• 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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Doing Statistics for Business

• Discovery Exercise 6.1
• Exploring the Binomial Distribution
• (cont)
• 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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Doing Statistics for Business

• Discovery Exercise 6.1
• Exploring the Binomial Distribution
• (cont)
• 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. Lets 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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• Figure 6.6. Probability
• Distribution for a
• Continuous Random
• Variable.

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• Figure 6.7. Probability
• represented by an area
• under the curve

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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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• Figure 6.8. Normal
• Probability Curve

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• Figure 6.9 Probability represented by an
  area under the curve

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Doing Statistics for Business

• TRY IT NOW!
• 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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Doing Statistics for Business

• TRY IT NOW!
• 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 6.10 Probability
• given by the standard
• normal table

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Figure 6.11 The Standard Normal Table
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• Figure 6.12 Comparison
• of Upper and Lower
• Probabilities

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• Figure 6.13
• Finding the area
• between two Z values

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• TRY IT NOW!
• 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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• TRY IT NOW!
• The Standard Normal Table
• Using the Table to Find Probabilities
• (cont)
• Find the probability that Z is between -1.36 and
  1.87.

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• TRY IT NOW!
• 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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• TRY IT NOW!
• Speed Reading
• Solving Normal Probability Problems
• (cont)
• 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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Doing Statistics for Business

• TRY IT NOW!
• 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.

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• Calculating Binomial Probabilities
• in Excel (cont)
• 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 (cont)
• 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 6.25 The Binomial Random Variable
  Dialog Box

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• Figure 6.26 Random 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 (cont)
• 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.