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					Doing Statistics for Business
Data, Inference, and Decision Making
           Marilyn K. Pelosi
          Theresa M. Sandifer
             Chapter 6
             Probability



                                       1
Doing Statistics for Business
Chapter 6 Objectives

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

                                          2
Doing Statistics for Business

  Probability is measure of how likely it
  is that something will occur.


   An Experiment is any action whose
   outcomes are recordable data.



                                            3
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  The Sample Space is the set of all
  possible outcomes of an experiment.




                                        4
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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.

                                                                            5
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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.


                                                   6
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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).
                                                                         7
Doing Statistics for Business

 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.




                                                   8
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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.
                                                 9
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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.

                                                                          10
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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?


                                                                11
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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:
                                  Shift
                 Defect      1      2     3   Total
                 Color       8      4     3    15
                 Printing    6      5     2    13
                 Skewness    0      2     0    2
                 Total       14    11     5    30




                                                             12
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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?

                                                                           13
Doing Statistics for Business
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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?

                                                                            14
Doing Statistics for Business
  The conditional probability of an event A
    given an event B is

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




                                              15
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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.




                                                  16
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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.

                                              17
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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:
                  x       0      1      2      3      4      5     6
                 p(x)   0.30   0.21   0.12   0.10   0.10   0.09   0.08




                                                                            18
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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.
                                                                          19
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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:

                  x       0      1      2      3      4      5     6
                 p(x)   0.30   0.21   0.12   0.10   0.10   0.09   0.08




                                                                         20
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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.
                                                                           21
Doing Statistics for Business

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




                                            22
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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.

                                                                          23
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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?

                                                                          24
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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.

                                                                          25
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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.

                                                                     26
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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.

                                                                            27
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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.
                                                                         28
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.”
                                                                      29
Doing Statistics for Business
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.

                                                                        30
Doing Statistics for Business
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.

                                                                     31
 Doing Statistics for Business
        Binomial Distribtution n = 10 and  =
                                                                                               Binomial Distribution n = 10 and  =
                        0.20
                                                                                                               0.50
    0.400
    0.300                                                                                0.400
                                                                                         0.300
p(x) 0.200
                                                                                   p(x) 0.200
    0.100
                                                                                         0.100
    0.000
                                                                                         0.000
             0   1   2   3   4   5   6       7   8   9   10
                                                                                                     0        1   2   3   4   5   6   7     8   9   10
                         Number of Successes
                                                                                                                      Number of Successes
                                                 Binomial Distribution n = 10 and  =
                                                                 0.80

                                             0.400
                                             0.300
                                      p(x)




                                             0.200
                                             0.100
                                             0.000
                                                     0   1    2    3   4   5   6     7     8     9       10
                                                                  Number of Successes

             Figure 6.3 Effects of changing  when
                        n is fixed                                                                                                                   32
Doing Statistics for Business
                                                                  Binomial Distribution n = 5 and  = 0.20
  Figure 6.4 Effects
  of Changing n for a                                          0.500
                                                               0.400

  fixed value of                                         p(x)
                                                               0.300
                                                               0.200
                                                               0.100
                                                               0.000
                                                                             0           1        2       3        4       5
                                                                                             Number of Successes



        Binomial Distribution n = 25 and  =                         Binomial Distribution n = 10 and  =
                        0.20                                                         0.20
                                                                 0.450
                                                                 0.400
    0.400                                                        0.350
                                                                 0.300
     0.300
                                                                 0.250
p(x) 0.200                                                p(x)   0.200
                                                                 0.150
    0.100                                                        0.100
                                                                 0.050
    0.000                                                        0.000
             0   2   4   6    8 10 12 14 16 18 20 22 24                  0       1   2       3   4    5   6   7    8   9       10
                             Number of Statistics                                            Number of Successes                    33
Doing Statistics for Business
          Binomial Distribution for n = 10 and  = 0.50                                                                      Binomial Distribution for n = 25 and  = 0.50
       0.25                                                                                                                 1.60E-01

                                                                                                                            1.40E-01
        0.2
                                                                                                                            1.20E-01

                                                                                                                            1.00E-01
       0.15
                                                                                                                            8.00E-02

p(x)    0.1                                                                                                            p(x) 6.00E-02
                                                                                                                            4.00E-02
       0.05
                                                                                                                            2.00E-02

                                                                                                                            0.00E+00
         0
                                                                                                                                       0       2        4        6     8         10   12        14    16        18    20   22        24
                0        1       2            3         4        5         6         7        8         9         10
                                                                                                                                                                           Number of Successes
                                                  Number of Successes


                                                                                                                                Binomial Distribution for n = 100 and  = 0.50
                             Binomial Distribution for n = 50 and  = 0.50                                                      8.00E-02
          1.20E-01
                                                                                                                                7.00E-02

          1.00E-01                                                                                                              6.00E-02

          8.00E-02                                                                                                              5.00E-02

                                                                                                                         p(x)   4.00E-02
          6.00E-02
  p(x)
                                                                                                                                3.00E-02
          4.00E-02
                                                                                                                                2.00E-02
          2.00E-02                                                                                                              1.00E-02

         0.00E+00                                                                                                               0.00E+00
                     0   3   6       9                                                                                                     0   6
                                         12       15   18   21   24   27       30   33   36   39   42   45   48                                    12       18   24   30    36   42   48   54    60   66   72    78   84   90   96
                                              Number of Successes                                                                                                     Number of Successes




       Figures 6.5 Binomial Distribution for large values
                   of n                                                                                                                                                                                                                   34
Doing Statistics for Business

  A Probability Density Function, f(x), is
  a smooth curve that represents the
  probability distribution of a continuous
  random variable.




                                             35
Doing Statistics for Business
Figure 6.6. Probability
Distribution for a
Continuous Random                Probability Density Function for a Continuous Random Variable



Variable.


                          f(x)




                                                           Values of X




                                                                                                 36
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Figure 6.7. Probability
represented by an area
under the curve           P(x1 < X < x2)




                                       x1   x2




                                                 37
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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.




                                          38
Doing Statistics for Business
Figure 6.8. Normal
Probability Curve
                            Normal Curve




                     f(x)




                             Values of X




                                           39
Doing Statistics for Business
          Normal Curves with Equal Means and Different Standard
                                                                                                         Normal Curves with Different Means and Equal Standard
                               Deviations                                                                                      Deviations




                                                                                                                                                                                    
                                                                                     
                                                                                                                                                                                    
                                                                                     




15   17   19   21   23   25   27   29   31   33   35   37   39   41   43   45
                               Values of X                                                     15   17   19   21   23   25   27   29   31   33   35   37   39   41   43   45
                                                                                                                              Values of X




                    Figure 6.9 Probability represented by an
                               area under the curve
                                                                                                                                                                                         40
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.
                                                                        41
Doing Statistics for Business

  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.



                                           42
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.
                                                                       43
Doing Statistics for Business
Figure 6.10 Probability
given by the standard
normal table




                                44
Doing Statistics for Business
                                        Second Decimal Place
    z      0.00     0.01     0.02     0.03     0.04    0.05      0.06     0.07     0.08     0.09
    0.0   0.5000   0.5040   0.5080   0.5120   0.5160   0.5199   0.5239   0.5279   0.5319   0.5359
    0.1   0.5398   0.5438   0.5478   0.5517   0.5557   0.5596   0.5636   0.5675   0.5714   0.5753
    0.2   0.5793   0.5832   0.5871   0.5910   0.5948   0.5987   0.6026   0.6064   0.6103   0.6141
    0.3   0.6179   0.6217   0.6255   0.6293   0.6331   0.6368   0.6406   0.6443   0.6480   0.6517
    0.4   0.6554   0.6591   0.6628   0.6664   0.6700   0.6736   0.6772   0.6808   0.6844   0.6879
    0.5   0.6915   0.6950   0.6985   0.7019   0.7054   0.7088   0.7123   0.7157   0.7190   0.7224
    0.6   0.7257   0.7291   0.7324   0.7357   0.7389   0.7422   0.7454   0.7486   0.7517   0.7549
    0.7   0.7580   0.7611   0.7642   0.7673   0.7704   0.7734   0.7764   0.7794   0.7823   0.7852
    0.8   0.7881   0.7910   0.7939   0.7967   0.7995   0.8023   0.8051   0.8078   0.8106   0.8133
    0.9   0.8159   0.8186   0.8212   0.8238   0.8264   0.8289   0.8315   0.8340   0.8365   0.8389
    1.0   0.8413   0.8438   0.8461   0.8485   0.8508   0.8531   0.8554   0.8577   0.8599   0.8621
    1.1   0.8643   0.8665   0.8686   0.8708   0.8729   0.8749   0.8770   0.8790   0.8810   0.8830
    1.2   0.8849   0.8869   0.8888   0.8907   0.8925   0.8944   0.8962   0.8980   0.8997   0.9015




  Figure 6.11 The Standard Normal Table

                                                                                                    45
Doing Statistics for Business
Figure 6.12 Comparison
of Upper and Lower
Probabilities


                     P(Z < -1)
                                 P( Z > 1)




                                             46
Doing Statistics for Business
 Figure 6.13
 Finding the area
 between two Z values
                                 0.9772




                        0.0228




                                          47
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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.

                                                                              48
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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.




                                                         49
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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.


                                                                       50
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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?



                                                                         51
Doing Statistics for Business



                                   Top p%
                                   (p known)


                       X




    Figure 6.14 The “inverse” normal probability
                problem
                                                   52
Doing Statistics for Business



      10% or 0.1000




    Figure 6.15 Bottom 10% of the normal
                distribution
                                           53
Doing Statistics for Business

     z      0.00     0.01     0.02     0.03     0.04     0.05     0.06     0.07     0.08     0.09
    -1.9   0.0287   0.0281   0.0274   0.0268   0.0262   0.0256   0.0250   0.0244   0.0239   0.0233
    -1.8   0.0359   0.0351   0.0344   0.0336   0.0329   0.0322   0.0314   0.0307   0.0301   0.0294
    -1.7   0.0446   0.0436   0.0427   0.0418   0.0409   0.0401   0.0392   0.0384   0.0375   0.0367
    -1.6   0.0548   0.0537   0.0526   0.0516   0.0505   0.0495   0.0485   0.0475   0.0465   0.0455
    -1.5   0.0668   0.0655   0.0643   0.0630   0.0618   0.0606   0.0594   0.0582   0.0571   0.0559
    -1.4   0.0808   0.0793   0.0778   0.0764   0.0749   0.0735   0.0721   0.0708   0.0694   0.0681
    -1.3   0.0968   0.0951   0.0934   0.0918   0.0901   0.0885   0.0869   0.0853   0.0838   0.0823
    -1.2   0.1151   0.1131   0.1112   0.1093   0.1075   0.1056   0.1038   0.1020   0.1003   0.0985
    -1.1   0.1357   0.1335   0.1314   0.1292   0.1271   0.1251   0.1230   0.1210   0.1190   0.1170
    -1.0   0.1587   0.1562   0.1539   0.1515   0.1492   0.1469   0.1446   0.1423   0.1401   0.1379
    -0.9   0.1841   0.1814   0.1788   0.1762   0.1736   0.1711   0.1685   0.1660   0.1635   0.1611
    -0.8   0.2119   0.2090   0.2061   0.2033   0.2005   0.1977   0.1949   0.1922   0.1894   0.1867




    Figure 6.16 Normal Probability Table
                Containing the Probability 0.1000
                                                                                                     54
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.
                                                                            55
Doing Statistics for Business
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.”
                                                          56
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.                       57
Doing Statistics for Business
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.
                                                            58
Doing Statistics for Business
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.


                                                             59
Doing Statistics for Business
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.


                                                             60
Doing Statistics for Business
Figure 6.25 The Binomial Random Variable
           Dialog Box




                                           61
Doing Statistics for Business
Figure 6.26 Random Data from Binomial
           Distribution




                                        62
Doing Statistics for Business
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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Doing Statistics for Business
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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