Statistics for Business and Economics (13e)
Statistics for
Business and Economics (13e)
Anderson, Sweeney, Williams, Camm, Cochran
© 2017 Cengage Learning
Slides by John Loucks
St. Edwards University
© 2017 Cengage Learning. May not be scanned, copied or duplicated, or posted to a publicly accessible website, in whole or in part, except for use as permitted in a license distributed with a certain product or service or
otherwise on a password-protected website or school-approved learning management system for classroom use.
1
Statistics for Business and Economics (13e)
Chapter 1 - Data and Statistics
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Statistics
Applications in Business and Economics
Data
Data Sources
Descriptive Statistics
Statistical Inference
Analytics
Big Data and Data Mining
Computers and Statistical Analysis
Ethical Guidelines for Statistical Practice
© 2017 Cengage Learning. May not be scanned, copied or duplicated, or posted to a publicly accessible website, in whole or in part, except for use as permitted in a license distributed with a certain product or service or
otherwise on a password-protected website or school-approved learning management system for classroom use.
2
Statistics for Business and Economics (13e)
What is Statistics?
• The term statistics can refer to numerical facts such as averages, medians,
percentages, and maximums that help us understand a variety of business and
economic situations.
• Statistics can also refer to the art and science of collecting, analyzing, presenting,
and interpreting data.
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3
Statistics for Business and Economics (13e)
Applications in Business and Economics
Accounting
• Public accounting firms use statistical sampling procedures when conducting audits for
their clients.
Economics
• Economists use statistical information in making forecasts about the future of the
economy or some aspect of it.
Finance
• Financial advisors use price-earnings ratios and dividend yields to guide their investment
advice.
p/e ratio: the price an investor is paying for $1 of a company's earnings or profit. In other words, if a
company is reporting basic earnings per share of $2 and the stock is selling for $20 per share, the p/e
ratio is 10 ($20 per share divided by $2 earnings per share = 10 p/e)
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4
Statistics for Business and Economics (13e)
Applications in Business and Economics
Marketing
• Electronic point-of-sale scanners at retail checkout counters are used to collect
data for a variety of marketing research applications.
Production
• A variety of statistical quality control charts are used to monitor the output of a
production process.
Information Systems
• A variety of statistical information helps administrators assess the performance of
computer networks.
© 2017 Cengage Learning. May not be scanned, copied or duplicated, or posted to a publicly accessible website, in whole or in part, except for use as permitted in a license distributed with a certain product or service or
otherwise on a password-protected website or school-approved learning management system for classroom use.
5
Statistics for Business and Economics (13e)
Data and Data Sets
• Data are the facts and figures collected, analyzed, and summarized for
presentation and interpretation.
• All the data collected in a particular study are referred to as the data set for the
study.
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otherwise on a password-protected website or school-approved learning management system for classroom use.
6
Statistics for Business and Economics (13e)
Elements, Variables, and Observations
• Elements are the entities on which data are collected.
• A variable is a characteristic of interest for the elements.
• The set of measurements obtained for a particular element is called an
observation.
• A data set with n elements contains n observations.
• The total number of data values in a complete data set is the number of elements
multiplied by the number of variables.
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otherwise on a password-protected website or school-approved learning management system for classroom use.
7
Statistics for Business and Economics (13e)
Data, Data Sets, Elements, Variables, and Observations
Variables
Element Names
Company
Stock Exchange
Annual Sales ($M)
Earnings per share ($)
Dataram
NQ
73.10
0.86
EnergySouth
N
74.00
1.67
Keystone
N
365.70
0.86
LandCare
NQ
111.40
0.33
N
17.60
0.13
Psychemedics
Observation
Data Set
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otherwise on a password-protected website or school-approved learning management system for classroom use.
8
Statistics for Business and Economics (13e)
Scales of Measurement
• Scales of measurement include
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•
•
•
Nominal
Ordinal
Interval
Ratio
• The scale determines the amount of information contained in the data.
• The scale indicates the data summarization and statistical analyses that are most
appropriate.
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otherwise on a password-protected website or school-approved learning management system for classroom use.
9
Statistics for Business and Economics (13e)
Scales of Measurement
Nominal scale
• Data are labels or names used to identify an attribute of the element.
• A nonnumeric label or numeric code may be used.
Example
Students of a university are classified by the school in which they are enrolled using
a nonnumeric label such as Business, Humanities, Education, and so on.
Alternatively, a numeric code could be used for the school variable (e.g. 1 denotes
Business, 2 denotes Humanities, 3 denotes Education, and so on).
© 2017 Cengage Learning. May not be scanned, copied or duplicated, or posted to a publicly accessible website, in whole or in part, except for use as permitted in a license distributed with a certain product or service or
otherwise on a password-protected website or school-approved learning management system for classroom use.
10
Statistics for Business and Economics (13e)
Scales of Measurement
Ordinal scale
• The data have the properties of nominal data and the order or rank of the data is
meaningful.
• A nonnumeric label or numeric code may be used.
Example
Students of a university are classified by their class standing using a nonnumeric
label such as Freshman, Sophomore, Junior, or Senior.
Alternatively, a numeric code could be used for the class standing variable (e.g. 1
denotes Freshman, 2 denotes Sophomore, and so on).
© 2017 Cengage Learning. May not be scanned, copied or duplicated, or posted to a publicly accessible website, in whole or in part, except for use as permitted in a license distributed with a certain product or service or
otherwise on a password-protected website or school-approved learning management system for classroom use.
11
Statistics for Business and Economics (13e)
Scales of Measurement
Interval scale
• The data have the properties of ordinal data, and the interval between
observations is expressed in terms of a fixed unit of measure.
• Interval data are always numeric.
Example
Melissa has an SAT score of 1985, while Kevin has an SAT score of 1880. Melissa
scored 105 points more than Kevin.
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otherwise on a password-protected website or school-approved learning management system for classroom use.
12
Statistics for Business and Economics (13e)
Scales of Measurement
Ratio scale
• Data have all the properties of interval data and the ratio of two values is
meaningful.
• Ratio data are always numerical.
Example:
Price of a book at a retail store is $ 200, while the price of the same book sold
online is $100. The ratio property shows that retail stores charge twice the
online price.
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otherwise on a password-protected website or school-approved learning management system for classroom use.
13
Statistics for Business and Economics (13e)
Categorical and Quantitative Data
• Data can be further classified as being categorical or quantitative.
• The statistical analysis that is appropriate depends on whether the data for the
variable are categorical or quantitative.
• In general, there are more alternatives for statistical analysis when the data are
quantitative.
© 2017 Cengage Learning. May not be scanned, copied or duplicated, or posted to a publicly accessible website, in whole or in part, except for use as permitted in a license distributed with a certain product or service or
otherwise on a password-protected website or school-approved learning management system for classroom use.
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Statistics for Business and Economics (13e)
Categorical Data
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•
•
•
•
Labels or names are used to identify an attribute of each element
Often referred to as qualitative data
Use either the nominal or ordinal scale of measurement
Can be either numeric or nonnumeric
Appropriate statistical analyses are rather limited
© 2017 Cengage Learning. May not be scanned, copied or duplicated, or posted to a publicly accessible website, in whole or in part, except for use as permitted in a license distributed with a certain product or service or
otherwise on a password-protected website or school-approved learning management system for classroom use.
15
Statistics for Business and Economics (13e)
Quantitative Data
• Quantitative data indicate how many or how much.
• Quantitative data are always numeric.
• Ordinary arithmetic operations are meaningful for quantitative data.
© 2017 Cengage Learning. May not be scanned, copied or duplicated, or posted to a publicly accessible website, in whole or in part, except for use as permitted in a license distributed with a certain product or service or
otherwise on a password-protected website or school-approved learning management system for classroom use.
16
Statistics for Business and Economics (13e)
Cross-Sectional Data
Cross-sectional data are collected at the same or approximately the same point in
time.
Example
Data detailing the number of building permits issued in November 2013 in each of
the counties of Ohio.
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otherwise on a password-protected website or school-approved learning management system for classroom use.
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Statistics for Business and Economics (13e)
Time Series Data
Time series data are collected over several time periods.
Example
Data detailing the number of building permits issued in Lucas County, Ohio in each
of the last 36 months.
Graphs of time series data help analysts understand
• what happened in the past
• identify any trends over time, and
• project future levels for the time series
© 2017 Cengage Learning. May not be scanned, copied or duplicated, or posted to a publicly accessible website, in whole or in part, except for use as permitted in a license distributed with a certain product or service or
otherwise on a password-protected website or school-approved learning management system for classroom use.
18
Statistics for Business and Economics (13e)
Time Series Data
Graph of Time Series Data
© 2017 Cengage Learning. May not be scanned, copied or duplicated, or posted to a publicly accessible website, in whole or in part, except for use as permitted in a license distributed with a certain product or service or
otherwise on a password-protected website or school-approved learning management system for classroom use.
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Statistics for Business and Economics (13e)
Data Sources
Existing Sources
• Internal company records – almost any department
• Business database services – Dow Jones & Co.
• Government agencies - U.S. Department of Labor
• Industry associations – Travel Industry Association of America
• Special-interest organizations – Graduate Management Admission Council
(GMAT)
• Internet – more and more firms
© 2017 Cengage Learning. May not be scanned, copied or duplicated, or posted to a publicly accessible website, in whole or in part, except for use as permitted in a license distributed with a certain product or service or
otherwise on a password-protected website or school-approved learning management system for classroom use.
20
Statistics for Business and Economics (13e)
Data Sources
Data Available From Internal Company Records
Record
Some of the Data Available
Employee records
Name, address, social security number
Production records
Part number, quantity produced, direct labor cost, material cost
Inventory records
Part number, quantity in stock, reorder level, economic order quantity
Sales records
Product number, sales volume, sales volume by region
Credit records
Customer name, credit limit, accounts receivable balance
Customer profile
Age, gender, income, household size
© 2017 Cengage Learning. May not be scanned, copied or duplicated, or posted to a publicly accessible website, in whole or in part, except for use as permitted in a license distributed with a certain product or service or
otherwise on a password-protected website or school-approved learning management system for classroom use.
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Statistics for Business and Economics (13e)
Data Sources
Data Available From Selected Government Agencies
Government Agency
Web address
Some of the Data Available
Census Bureau
www.census.gov
Population data, number of households, household income
Federal Reserve Board
www.federalreserve.gov
Data on money supply, exchange rates, discount rates
Office of Mgmt. & Budget
www.whitehouse.gov/omb Data on revenue, expenditures, debt of federal government
Department of Commerce www.doc.gov
Data on business activity, value of shipments, profit by industry
Bureau of Labor Statistics
Customer spending, unemployment rate, hourly earnings, safety
record
www.bls.gov
© 2017 Cengage Learning. May not be scanned, copied or duplicated, or posted to a publicly accessible website, in whole or in part, except for use as permitted in a license distributed with a certain product or service or
otherwise on a password-protected website or school-approved learning management system for classroom use.
22
Statistics for Business and Economics (13e)
Data Sources
Statistical Studies – Observational
• In observational (nonexperimental) studies no attempt is made to control or
influence the variables of interest.
• Example - Survey
• Studies of smokers and nonsmokers are observational studies because
researchers do not determine or control who will smoke and who will not smoke.
© 2017 Cengage Learning. May not be scanned, copied or duplicated, or posted to a publicly accessible website, in whole or in part, except for use as permitted in a license distributed with a certain product or service or
otherwise on a password-protected website or school-approved learning management system for classroom use.
23
Statistics for Business and Economics (13e)
Data Sources
Statistical Studies – Experimental
• In experimental studies the variable of interest is first identified. Then one or
more other variables are identified and controlled so that data can be obtained
about how they influence the variable of interest.
• The largest experimental study ever conducted is believed to be the 1954 Public
Health Service experiment for the Salk polio vaccine. Nearly two million U.S.
children (grades 1- 3) were selected.
• Across the US, 623,972 schoolchildren were injected with vaccine or placebo, and
more than a million others participated as “observed” controls. The results,
announced in 1955, showed good statistical evidence that Salk polio vaccine was 8090% effective in preventing paralytic poliomyelitis.
© 2017 Cengage Learning. May not be scanned, copied or duplicated, or posted to a publicly accessible website, in whole or in part, except for use as permitted in a license distributed with a certain product or service or
otherwise on a password-protected website or school-approved learning management system for classroom use.
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Statistics for Business and Economics (13e)
Data Acquisition Considerations
Time Requirement
• Searching for information can be time consuming.
• Information may no longer be useful by the time it is available.
Cost of Acquisition
• Organizations often charge for information even when it is not their primary
business activity.
Data Errors
• Using any data that happen to be available or were acquired with little care can
lead to misleading information.
© 2017 Cengage Learning. May not be scanned, copied or duplicated, or posted to a publicly accessible website, in whole or in part, except for use as permitted in a license distributed with a certain product or service or
otherwise on a password-protected website or school-approved learning management system for classroom use.
25
Statistics for Business and Economics (13e)
Descriptive Statistics
• Most of the statistical information in newspapers, magazines, company reports,
and other publications consists of data that are summarized and presented in a
form that is easy to understand.
• Such summaries of data, which may be tabular, graphical, or numerical, are
referred to as descriptive statistics.
Example
The manager of Hudson Auto would like to have a better understanding of the cost
of parts used in the engine tune-ups performed in her shop. She examines 50
customer invoices for tune-ups. The costs of parts, rounded to the nearest dollar,
are listed on the next slide.
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otherwise on a password-protected website or school-approved learning management system for classroom use.
26
Statistics for Business and Economics (13e)
Example: Hudson Auto Repair
Sample of Parts Cost ($) for 50 Tune-ups
91
78
93
57
75
52
99
80
97
62
71
69
72
89
66
75
79
75
72
76
104
74
62
68
97
105
77
65
80
109
85
97
88
68
83
68
71
69
67
74
62
82
98
101
79
105
79
69
62
73
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otherwise on a password-protected website or school-approved learning management system for classroom use.
27
Statistics for Business and Economics (13e)
Tabular Summary: Frequency and Percent Frequency
Parts Cost ($)
Frequency
Percent Frequency
50-59
2
4%
60-69
13
26%
70-79
16
32%
80-89
7
14%
90-99
7
14%
100-109
5
10%
50
100%
TOTAL
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otherwise on a password-protected website or school-approved learning management system for classroom use.
28
Statistics for Business and Economics (13e)
Graphical Summary: Histogram
Example: Hudson Auto
Tune-up Parts Cost
18
16
14
Frequency
12
10
8
6
4
2
0
50-59
60-69
70-79
80-89
90-99
Parts Cost ($)
© 2017 Cengage Learning. May not be scanned, copied or duplicated, or posted to a publicly accessible website, in whole or in part, except for use as permitted in a license distributed with a certain product or service or
otherwise on a password-protected website or school-approved learning management system for classroom use.
29
Statistics for Business and Economics (13e)
Numerical Descriptive Statistics
• The most common numerical descriptive statistic is the mean (or average).
• The mean demonstrates a measure of the central tendency, or central location of
the data for a variable.
• Hudson’s mean cost of parts, based on the 50 tune-ups studied is $79 (found by
summing up the 50 cost values and then dividing by 50).
© 2017 Cengage Learning. May not be scanned, copied or duplicated, or posted to a publicly accessible website, in whole or in part, except for use as permitted in a license distributed with a certain product or service or
otherwise on a password-protected website or school-approved learning management system for classroom use.
30
Statistics for Business and Economics (13e)
Statistical Inference
Population: The set of all elements of interest in a particular study.
Sample: A subset of the population.
Statistical inference: The process of using data obtained from a sample to make
estimates and test hypotheses about the characteristics of a population.
Census: Collecting data for the entire population.
Sample survey: Collecting data for a sample.
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otherwise on a password-protected website or school-approved learning management system for classroom use.
31
Statistics for Business and Economics (13e)
Process of Statistical Inference
Example: Hudson Auto
Step 1
Step 2
Step 3
Step 4
• Population consists
of all tune ups.
Average cost of
parts is unknown.
• A sample of 50
engine tune-ups is
examined.
• The sample data
provides a sample
average parts cost
of $79 per tune-up.
• The sample average
is used to estimate
the population
average.
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otherwise on a password-protected website or school-approved learning management system for classroom use.
32
Statistics for Business and Economics (13e)
Analytics
Analytics is the scientific process of transforming data into insight for making better
decisions.
Techniques:
• Descriptive analytics: This describes what has happened in the past.
• Predictive analytics: Use models constructed from past data to predict the future
or to assess the impact of one variable on another.
• Prescriptive analytics: The set of analytical techniques that yield a best course of
action.
© 2017 Cengage Learning. May not be scanned, copied or duplicated, or posted to a publicly accessible website, in whole or in part, except for use as permitted in a license distributed with a certain product or service or
otherwise on a password-protected website or school-approved learning management system for classroom use.
33
Statistics for Business and Economics (13e)
Big data and Data Mining:
Big data: Large and complex data set.
Three V’s of Big data:
Volume : Amount of available data
Velocity: Speed at which data is collected and processed
Variety: Different data types
© 2017 Cengage Learning. May not be scanned, copied or duplicated, or posted to a publicly accessible website, in whole or in part, except for use as permitted in a license distributed with a certain product or service or
otherwise on a password-protected website or school-approved learning management system for classroom use.
34
Statistics for Business and Economics (13e)
Data warehousing
Data warehousing is the process of capturing, storing, and maintaining the data.
• Organizations obtain large amounts of data on a daily basis by means of magnetic
card readers, bar code scanners, point of sale terminals, and touch screen
monitors.
• Wal-Mart captures data on 20-30 million transactions per day.
• Visa processes 6,800 payment transactions per second.
© 2017 Cengage Learning. May not be scanned, copied or duplicated, or posted to a publicly accessible website, in whole or in part, except for use as permitted in a license distributed with a certain product or service or
otherwise on a password-protected website or school-approved learning management system for classroom use.
35
Statistics for Business and Economics (13e)
Data Mining
• Methods for developing useful decision-making information from large
databases.
• Using a combination of procedures from statistics, mathematics, and computer
science, analysts “mine the data” to convert it into useful information.
• The most effective data mining systems use automated procedures to discover
relationships in the data and predict future outcomes prompted by general and
even vague queries by the user.
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otherwise on a password-protected website or school-approved learning management system for classroom use.
36
Statistics for Business and Economics (13e)
Data Mining Applications
• The major applications of data mining have been made by companies with a
strong consumer focus such as retail, financial, and communication firms.
• Data mining is used to identify related products that customers who have already
purchased a specific product are also likely to purchase (and then pop-ups are
used to draw attention to those related products).
• Data mining is also used to identify customers who should receive special
discount offers based on their past purchasing volumes.
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otherwise on a password-protected website or school-approved learning management system for classroom use.
37
Statistics for Business and Economics (13e)
Data Mining Requirements
• Statistical methodology such as multiple regression, logistic regression, and
correlation are heavily used.
• Also needed are computer science technologies involving artificial intelligence
and machine learning.
• A significant investment in time and money is required as well.
© 2017 Cengage Learning. May not be scanned, copied or duplicated, or posted to a publicly accessible website, in whole or in part, except for use as permitted in a license distributed with a certain product or service or
otherwise on a password-protected website or school-approved learning management system for classroom use.
38
Statistics for Business and Economics (13e)
Data Mining Model Reliability
• Finding a statistical model that works well for a particular sample of data does not
necessarily mean that it can be reliably applied to other data.
• With the enormous amount of data available, the data set can be partitioned into
a training set (for model development) and a test set (for validating the model).
• There is, however, a danger of overfitting the model to the point that misleading
associations and conclusions appear to exist.
• Careful interpretation of results and extensive testing is important.
© 2017 Cengage Learning. May not be scanned, copied or duplicated, or posted to a publicly accessible website, in whole or in part, except for use as permitted in a license distributed with a certain product or service or
otherwise on a password-protected website or school-approved learning management system for classroom use.
39
Statistics for Business and Economics (13e)
Ethical Guidelines for Statistical Practice
• In a statistical study, unethical behavior can take a variety of forms including:
•
•
•
•
•
Improper sampling
Inappropriate analysis of the data
Development of misleading graphs
Use of inappropriate summary statistics
Biased interpretation of the statistical results
• One should strive to be fair, thorough, objective, and neutral as you collect,
analyze, and present data.
• As a consumer of statistics, one should also be aware of the possibility of
unethical behavior by others.
© 2017 Cengage Learning. May not be scanned, copied or duplicated, or posted to a publicly accessible website, in whole or in part, except for use as permitted in a license distributed with a certain product or service or
otherwise on a password-protected website or school-approved learning management system for classroom use.
40
Statistics for Business and Economics (13e)
Ethical Guidelines for Statistical Practice
• The American Statistical Association developed the report “Ethical Guidelines for
Statistical Practice”.
• It contains 67 guidelines organized into 8 topic areas:
•
•
•
•
•
•
•
•
Professionalism
Responsibilities to Funders, Clients, Employers
Responsibilities in Publications and Testimony
Responsibilities to Research Subjects
Responsibilities to Research Team Colleagues
Responsibilities to Other Statisticians/Practitioners
Responsibilities Regarding Allegations of Misconduct
Responsibilities of Employers Including Organizations, Individuals, Attorneys, or
Other Clients
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otherwise on a password-protected website or school-approved learning management system for classroom use.
41
Statistics for Business and Economics (13e)
End of Chapter 1
© 2017 Cengage Learning. May not be scanned, copied or duplicated, or posted to a publicly accessible website, in whole or in part, except for use as permitted in a license distributed with a certain product or service or
otherwise on a password-protected website or school-approved learning management system for classroom use.
42
Statistics for Business and Economics (13e)
Statistics for
Business and Economics (13e)
Anderson, Sweeney, Williams, Camm, Cochran
© 2017 Cengage Learning
Slides by John Loucks
St. Edwards University
© 2017 Cengage Learning. May not be scanned, copied or duplicated, or posted to a publicly accessible website, in whole or in part, except for use as permitted in a license distributed with a certain product or service or
otherwise on a password-protected website or school-approved learning management system for classroom use.
1
Statistics for Business and Economics (13e)
Chapter 2
Part A - Descriptive Statistics: Tabular and Graphical Displays
• Summarizing Data for a Categorical Variable
• Categorical data use labels or names to identify categories of like items.
• Summarizing Data for a Quantitative Variable
• Quantitative data are numerical values that indicate how much or how many.
© 2017 Cengage Learning. May not be scanned, copied or duplicated, or posted to a publicly accessible website, in whole or in part, except for use as permitted in a license distributed with a certain product or service or
otherwise on a password-protected website or school-approved learning management system for classroom use.
2
Statistics for Business and Economics (13e)
Summarizing Categorical Data
•
•
•
•
•
Frequency Distribution
Relative Frequency Distribution
Percent Frequency Distribution
Bar Chart
Pie Chart
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3
Statistics for Business and Economics (13e)
Frequency Distribution
• A frequency distribution is a tabular summary of data showing the number
(frequency) of observations in each of several non-overlapping categories or
classes.
• The objective is to provide insights about the data that cannot be quickly
obtained by looking only at the original data.
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otherwise on a password-protected website or school-approved learning management system for classroom use.
4
Statistics for Business and Economics (13e)
Frequency Distribution
Example: Marada Inn
• Guests staying at Marada Inn were asked to rate the quality of their
accommodations as being excellent, above average, average, below average, or
poor.
• The ratings provided by a sample of 20 guests are:
Below Average
Average
Above Average
Above Average
Above Average
Above Average
Above Average
Below Average
Below Average
Average
Poor
Poor
Above Average
Excellent
Above Average
Average
Above Average
Average
Above Average
Average
© 2017 Cengage Learning. May not be scanned, copied or duplicated, or posted to a publicly accessible website, in whole or in part, except for use as permitted in a license distributed with a certain product or service or
otherwise on a password-protected website or school-approved learning management system for classroom use.
5
Statistics for Business and Economics (13e)
Frequency Distribution
• Example: Marada Inn
Rating
Frequency
Poor
2
Below Average
3
Average
5
Above Average
9
Excellent
1
Total
20
© 2017 Cengage Learning. May not be scanned, copied or duplicated, or posted to a publicly accessible website, in whole or in part, except for use as permitted in a license distributed with a certain product or service or
otherwise on a password-protected website or school-approved learning management system for classroom use.
6
Statistics for Business and Economics (13e)
Relative Frequency Distribution
• The relative frequency of a class is the fraction or proportion of the total
number of data items belonging to the class.
Relative frequency of a class =
Frequency of the class
𝑛
• A relative frequency distribution is a tabular summary of data showing the
relative frequency for each class.
© 2017 Cengage Learning. May not be scanned, copied or duplicated, or posted to a publicly accessible website, in whole or in part, except for use as permitted in a license distributed with a certain product or service or
otherwise on a password-protected website or school-approved learning management system for classroom use.
7
Statistics for Business and Economics (13e)
Percent Frequency Distribution
• The percent frequency of a class is the relative frequency multiplied by
100.
• A percent frequency distribution is a tabular summary of a set of data
showing the percent frequency for each class.
© 2017 Cengage Learning. May not be scanned, copied or duplicated, or posted to a publicly accessible website, in whole or in part, except for use as permitted in a license distributed with a certain product or service or
otherwise on a password-protected website or school-approved learning management system for classroom use.
8
Statistics for Business and Economics (13e)
Relative Frequency and Percent Frequency Distributions
• Example: Marada Inn
Rating
Relative Frequency
Percent Frequency
Poor
.10
10
Below Average
.15
15
Average
.25
25
Above Average
.45
45
Excellent
.05
5
1.00
100
Total
© 2017 Cengage Learning. May not be scanned, copied or duplicated, or posted to a publicly accessible website, in whole or in part, except for use as permitted in a license distributed with a certain product or service or
otherwise on a password-protected website or school-approved learning management system for classroom use.
9
Statistics for Business and Economics (13e)
Bar Chart
• A bar chart is a graphical display for depicting qualitative data.
• On one axis (usually the horizontal axis), we specify the labels that are used
for each of the classes.
• A frequency, relative frequency, or percent frequency scale can be used for
the other axis (usually the vertical axis).
• Using a bar of fixed width drawn above each class label, we extend the height
appropriately.
• The bars are separated to emphasize the fact that each class is a separate
category.
© 2017 Cengage Learning. May not be scanned, copied or duplicated, or posted to a publicly accessible website, in whole or in part, except for use as permitted in a license distributed with a certain product or service or
otherwise on a password-protected website or school-approved learning management system for classroom use.
10
Statistics for Business and Economics (13e)
Bar Chart
Marada Inn Quality Ratings
10
9
Frequency
8
7
6
5
4
3
2
1
Poor
Below
Average
Average
Above
Average
Excellent
Quality
Rating
© 2017 Cengage Learning. May not be scanned, copied or duplicated, or posted to a publicly accessible website, in whole or in part, except for use as permitted in a license distributed with a certain product or service or
otherwise on a password-protected website or school-approved learning management system for classroom use.
11
Statistics for Business and Economics (13e)
Pareto Diagram
• In quality control, bar charts are used to identify the most important causes
of problems.
• When the bars are arranged in descending order of height from left to right
(with the most frequently occurring cause appearing first) the bar chart is
called a Pareto diagram.
• This diagram is named for its founder, Vilfredo Pareto, an Italian economist.
© 2017 Cengage Learning. May not be scanned, copied or duplicated, or posted to a publicly accessible website, in whole or in part, except for use as permitted in a license distributed with a certain product or service or
otherwise on a password-protected website or school-approved learning management system for classroom use.
12
Statistics for Business and Economics (13e)
Pie Chart
• The pie chart is a commonly used graphical display for presenting relative
frequency and percent frequency distributions for categorical data.
• First draw a circle; then use the relative frequencies to subdivide the circle into
sectors that correspond to the relative frequency for each class.
• Since there are 360 degrees in a circle, a class with a relative frequency of .25
would consume .25(360) = 90 degrees of the circle.
© 2017 Cengage Learning. May not be scanned, copied or duplicated, or posted to a publicly accessible website, in whole or in part, except for use as permitted in a license distributed with a certain product or service or
otherwise on a password-protected website or school-approved learning management system for classroom use.
13
Statistics for Business and Economics (13e)
Pie Chart
Marada Inn Quality Ratings
Excellent
5%
Poor
10%
Above
Average
45%
Below
Average
15%
Average
25%
Average
25%
© 2017 Cengage Learning. May not be scanned, copied or duplicated, or posted to a publicly accessible website, in whole or in part, except for use as permitted in a license distributed with a certain product or service or
otherwise on a password-protected website or school-approved learning management system for classroom use.
14
Statistics for Business and Economics (13e)
Example: Marada Inn
from thesurveyed
Preceding
Pie
Chart a quality rating of “above
• Insights
One-half Gained
of the customers
gave
Marada
average” or “excellent” (looking at the left side of the pie). This might
please the manager.
• For each customer who gave an “excellent” rating, there were two customers
who gave a “poor” rating (looking at the top of the pie). This should
displease the manager.
© 2017 Cengage Learning. May not be scanned, copied or duplicated, or posted to a publicly accessible website, in whole or in part, except for use as permitted in a license distributed with a certain product or service or
otherwise on a password-protected website or school-approved learning management system for classroom use.
15
Statistics for Business and Economics (13e)
Summarizing Quantitative Data
• Frequency Distribution
• Relative Frequency and Percent Frequency Distributions
• Dot Plot
• Histogram
• Cumulative Distributions
• Stem-and-Leaf Display
© 2017 Cengage Learning. May not be scanned, copied or duplicated, or posted to a publicly accessible website, in whole or in part, except for use as permitted in a license distributed with a certain product or service or
otherwise on a password-protected website or school-approved learning management system for classroom use.
16
Statistics for Business and Economics (13e)
Frequency Distribution
• Example: Hudson Auto Repair
The manager of Hudson Auto would like to gain a better understanding of
the cost of parts used in the engine tune-ups performed in the shop. She
examines 50 customer invoices for tune-ups. The costs of parts, rounded to
the nearest dollar, are listed on the next slide.
© 2017 Cengage Learning. May not be scanned, copied or duplicated, or posted to a publicly accessible website, in whole or in part, except for use as permitted in a license distributed with a certain product or service or
otherwise on a password-protected website or school-approved learning management system for classroom use.
17
Statistics for Business and Economics (13e)
Frequency Distribution
• Example: Hudson Auto Repair
Sample of Parts Cost($) for 50 Tune-ups
91
71
104
85
62
78
69
74
97
82
93
72
62
88
98
57
89
68
68
101
75
66
97
83
79
52
75
105
68
105
99
79
77
71
79
80
75
65
69
69
97
72
80
67
62
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otherwise on a password-protected website or school-approved learning management system for classroom use.
62
76
109
74
73
18
Statistics for Business and Economics (13e)
Frequency Distribution
The three steps necessary to define the classes for a frequency distribution
with quantitative data are:
1. Determine the number of non-overlapping classes.
2. Determine the width of each class.
3. Determine the class limits.
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otherwise on a password-protected website or school-approved learning management system for classroom use.
19
Statistics for Business and Economics (13e)
Frequency Distribution
• Guidelines for Determining the Number of Classes
• Use between 5 and 20 classes.
• Data sets with a larger number of elements usually require a larger
number of classes.
• Smaller data sets usually require fewer classes.
• The goal is to use enough classes to show the variation in the data, but
not so many classes that some contain only a few data items.
© 2017 Cengage Learning. May not be scanned, copied or duplicated, or posted to a publicly accessible website, in whole or in part, except for use as permitted in a license distributed with a certain product or service or
otherwise on a password-protected website or school-approved learning management system for classroom use.
20
Statistics for Business and Economics (13e)
Frequency Distribution
• Guidelines for Determining the Width of Each Class
• Use classes of equal width.
• Approximate Class Width =
Largest data value − Smallest data value
Number of classes
• Making the classes the same width reduces the chance of inappropriate
interpretations.
© 2017 Cengage Learning. May not be scanned, copied or duplicated, or posted to a publicly accessible website, in whole or in part, except for use as permitted in a license distributed with a certain product or service or
otherwise on a password-protected website or school-approved learning management system for classroom use.
21
Statistics for Business and Economics (13e)
Frequency Distribution
• Note on Number of Classes and Class Width
• In practice, the number of classes and the appropriate class width are
determined by trial and error.
• Once a possible number of classes is chosen, the appropriate class width
is found.
• The process can be repeated for a different number of classes.
• Ultimately, the analyst uses judgment to determine the combination of
the number of classes and class width that provides the best frequency
distribution for summarizing the data.
© 2017 Cengage Learning. May not be scanned, copied or duplicated, or posted to a publicly accessible website, in whole or in part, except for use as permitted in a license distributed with a certain product or service or
otherwise on a password-protected website or school-approved learning management system for classroom use.
22
Statistics for Business and Economics (13e)
Frequency Distribution
• Guidelines for Determining the Class Limits
• Class limits must be chosen so that each data item belongs to one and
only one class.
• The lower class limit identifies the smallest possible data value assigned
to the class.
• The upper class limit identifies the largest possible data value assigned to
the class.
• An open-end class requires only a lower class limit or an upper class
limit.
© 2017 Cengage Learning. May not be scanned, copied or duplicated, or posted to a publicly accessible website, in whole or in part, except for use as permitted in a license distributed with a certain product or service or
otherwise on a password-protected website or school-approved learning management system for classroom use.
23
Statistics for Business and Economics (13e)
Frequency Distribution
• Class Midpoint
• In some cases, we want to know the midpoints of the classes in a frequency
distribution for quantitative data.
• The class midpoint is the value halfway between the lower and upper
class limits.
© 2017 Cengage Learning. May not be scanned, copied or duplicated, or posted to a publicly accessible website, in whole or in part, except for use as permitted in a license distributed with a certain product or service or
otherwise on a password-protected website or school-approved learning management system for classroom use.
24
Statistics for Business and Economics (13e)
Frequency Distribution
• Example: Hudson Auto Repair
If we choose six classes:
Approximate Class Width = (109 - 52)/6 = 9.5 10
Parts Cost ($)
50-59
60-69
70-79
80-89
90-99
100-109
Frequency
2
13
16
7
7
5
Total 50
© 2017 Cengage Learning. May not be scanned, copied or duplicated, or posted to a publicly accessible website, in whole or in part, except for use as permitted in a license distributed with a certain product or service or
otherwise on a password-protected website or school-approved learning management system for classroom use.
25
Statistics for Business and Economics (13e)
Relative Frequency and Percent Frequency Distributions
• Example: Hudson Auto Repair
Parts
Cost ($)
50-59
60-69
70-79
80-89
90-99
100-109
Relative
Frequency
.04 = 2/50
.26
.32
.14
.14
.10
Total 1.00
Percent
Frequency
4 = .04(100)
26
32
14
14
10
100
© 2017 Cengage Learning. May not be scanned, copied or duplicated, or posted to a publicly accessible website, in whole or in part, except for use as permitted in a license distributed with a certain product or service or
otherwise on a password-protected website or school-approved learning management system for classroom use.
26
Statistics for Business and Economics (13e)
Relative Frequency and Percent Frequency Distributions
• Example: Hudson Auto Repair
Insights Gained from the Percent Frequency Distribution:
• Only 4% of the parts costs are in the $50-59 class.
• 30% of the parts costs are under $70.
• The greatest percentage (32% or almost one-third) of the parts costs are
in the $70-79 class.
• 10% of the parts costs are $100 or more.
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otherwise on a password-protected website or school-approved learning management system for classroom use.
27
Statistics for Business and Economics (13e)
Dot Plot
• One of the simplest graphical summaries of data is a dot plot.
• A horizontal axis shows the range of data values.
• Then each data value is represented by a dot placed above the axis.
© 2017 Cengage Learning. May not be scanned, copied or duplicated, or posted to a publicly accessible website, in whole or in part, except for use as permitted in a license distributed with a certain product or service or
otherwise on a password-protected website or school-approved learning management system for classroom use.
28
Statistics for Business and Economics (13e)
Dot Plot
• Example: Hudson Auto Repair
50
60
70
80
90
100
110
Tune-up Parts Cost ($)
© 2017 Cengage Learning. May not be scanned, copied or duplicated, or posted to a publicly accessible website, in whole or in part, except for use as permitted in a license distributed with a certain product or service or
otherwise on a password-protected website or school-approved learning management system for classroom use.
29
Statistics for Business and Economics (13e)
Histogram
• Another common graphical display of quantitative data is a histogram.
• The variable of interest is placed on the horizontal axis.
• A rectangle is drawn above each class interval with its height corresponding
to the interval’s frequency, relative frequency, or percent frequency.
• Unlike a bar graph, a histogram has no natural separation between rectangles
of adjacent classes.
© 2017 Cengage Learning. May not be scanned, copied or duplicated, or posted to a publicly accessible website, in whole or in part, except for use as permitted in a license distributed with a certain product or service or
otherwise on a password-protected website or school-approved learning management system for classroom use.
30
Statistics for Business and Economics (13e)
Histogram
• Example: Hudson Auto Repair
18
Tune-up Parts Cost
16
Frequency
14
12
10
8
6
4
2
50-59
60-69
70-79
80-89
90-99 100-110
Parts
Cost ($)
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otherwise on a password-protected website or school-approved learning management system for classroom use.
31
Statistics for Business and Economics (13e)
Histograms Showing Skewness
• Symmetric
• Left tail is the mirror image of the right tail
• Example: Heights of People
Relative Frequency
.35
.30
.25
.20
.15
.10
.05
0
© 2017 Cengage Learning. May not be scanned, copied or duplicated, or posted to a publicly accessible website, in whole or in part, except for use as permitted in a license distributed with a certain product or service or
otherwise on a password-protected website or school-approved learning management system for classroom use.
32
Statistics for Business and Economics (13e)
Histograms Showing Skewness
• Moderately Skewed Left
• A longer tail to the left
• Example: Exam Scores
Relative Frequency
.35
.30
.25
.20
.15
.10
.05
0
© 2017 Cengage Learning. May not be scanned, copied or duplicated, or posted to a publicly accessible website, in whole or in part, except for use as permitted in a license distributed with a certain product or service or
otherwise on a password-protected website or school-approved learning management system for classroom use.
33
Statistics for Business and Economics (13e)
Histograms Showing Skewness
• Moderately Right Skewed
• A Longer tail to the right
• Example: Housing Values
Relative Frequency
.35
.30
.25
.20
.15
.10
.05
0
© 2017 Cengage Learning. May not be scanned, copied or duplicated, or posted to a publicly accessible website, in whole or in part, except for use as permitted in a license distributed with a certain product or service or
otherwise on a password-protected website or school-approved learning management system for classroom use.
34
Statistics for Business and Economics (13e)
Histograms Showing Skewness
• Highly Skewed Right
• A very long tail to the right
• Example: Executive Salaries
Relative Frequency
.35
.30
.25
.20
.15
.10
.05
0
© 2017 Cengage Learning. May not be scanned, copied or duplicated, or posted to a publicly accessible website, in whole or in part, except for use as permitted in a license distributed with a certain product or service or
otherwise on a password-protected website or school-approved learning management system for classroom use.
35
Statistics for Business and Economics (13e)
Skewness – Left and Right
© 2017 Cengage Learning. May not be scanned, copied or duplicated, or posted to a publicly accessible website, in whole or in part, except for use as permitted in a license distributed with a certain product or service or
otherwise on a password-protected website or school-approved learning management system for classroom use.
36
Statistics for Business and Economics (13e)
Cumulative Distributions
• Cumulative frequency distribution - shows the number of items with values
less than or equal to the upper limit of each class.
• Cumulative relative frequency distribution – shows the proportion of items
with values less than or equal to the upper limit of each class.
• Cumulative percent frequency distribution – shows the percentage of items
with values less than or equal to the upper limit of each class.
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otherwise on a password-protected website or school-approved learning management system for classroom use.
37
Statistics for Business and Economics (13e)
Cumulative Distributions
• The last entry in a cumulative frequency distribution always equals the total
number of observations.
• The last entry in a cumulative relative frequency distribution always equals
1.00.
• The last entry in a cumulative percent frequency distribution always equals
100.
© 2017 Cengage Learning. May not be scanned, copied or duplicated, or posted to a publicly accessible website, in whole or in part, except for use as permitted in a license distributed with a certain product or service or
otherwise on a password-protected website or school-approved learning management system for classroom use.
38
Statistics for Business and Economics (13e)
Frequency Distribution
• Example: Hudson Auto Repair
If we choose six classes:
Approximate Class Width = (109 - 52)/6 = 9.5 10
Parts Cost ($)
50-59
60-69
70-79
80-89
90-99
100-109
Frequency
2
13
16
7
7
5
Total 50
© 2017 Cengage Learning. May not be scanned, copied or duplicated, or posted to a publicly accessible website, in whole or in part, except for use as permitted in a license distributed with a certain product or service or
otherwise on a password-protected website or school-approved learning management system for classroom use.
39
Statistics for Business and Economics (13e)
Relative Frequency and Percent Frequency Distributions
• Example: Hudson Auto Repair
Parts
Cost ($)
50-59
60-69
70-79
80-89
90-99
100-109
Relative
Frequency
.04 = 2/50
.26
.32
.14
.14
.10
Total 1.00
Percent
Frequency
4 = .04(100)
26
32
14
14
10
100
© 2017 Cengage Learning. May not be scanned, copied or duplicated, or posted to a publicly accessible website, in whole or in part, except for use as permitted in a license distributed with a certain product or service or
otherwise on a password-protected website or school-approved learning management system for classroom use.
40
Statistics for Business and Economics (13e)
Cumulative Distributions
• Hudson Auto Repair
Cost ($)
< 59
< 69
< 79
< 89
< 99
< 109
Cumulative
Frequency
2
15 = 2+13
31
38
45
50
Cumulative
Relative
Frequency
.04
.30 = 15/50
.62
.76
.90
1.00
Cumulative
Percent
Frequency
4
30 = .30(100)
62
76
90
100
© 2017 Cengage Learning. May not be scanned, copied or duplicated, or posted to a publicly accessible website, in whole or in part, except for use as permitted in a license distributed with a certain product or service or
otherwise on a password-protected website or school-approved learning management system for classroom use.
41
Statistics for Business and Economics (13e)
Stem-and-Leaf Display
• A stem-and-leaf display shows both the rank order and shape of a distribution
of data.
• It is similar to a histogram on its side, but it has the advantage of showing the
actual data values.
• The leading digits of each data item are arranged to the left of a vertical line.
• To the right of the vertical line we record the last digit for each item in rank
order.
• Each line (row) in the display is referred to as a stem.
• Each digit on a stem is a leaf.
© 2017 Cengage Learning. May not be scanned, copied or duplicated, or posted to a publicly accessible website, in whole or in part, except for use as permitted in a license distributed with a certain product or service or
otherwise on a password-protected website or school-approved learning management system for classroom use.
42
Statistics for Business and Economics (13e)
Example: Hudson Auto Repair
The manager of Hudson Auto would like to gain a better understanding of the
cost of parts used in the engine tune-ups performed in the shop. She examines
50 customer invoices for tune-ups. The costs of parts, rounded to the nearest
dollar, are listed on the next slide.
© 2017 Cengage Learning. May not be scanned, copied or duplicated, or posted to a publicly accessible website, in whole or in part, except for use as permitted in a license distributed with a certain product or service or
otherwise on a password-protected website or school-approved learning management system for classroom use.
43
Statistics for Business and Economics (13e)
Stem-and-Leaf Display
• Example: Hudson Auto Repair
Sample of Parts Cost ($) for 50 Tune-ups
91
71
104
85
62
78
69
74
97
82
93
72
62
88
98
57
89
68
68
101
75
66
97
83
79
52
75
105
68
105
99
79
77
71
79
80
75
65
69
69
97
72
80
67
62
© 2017 Cengage Learning. May not be scanned, copied or duplicated, or posted to a publicly accessible website, in whole or in part, except for use as permitted in a license distributed with a certain product or service or
otherwise on a password-protected website or school-approved learning management system for classroom use.
62
76
109
74
73
44
Statistics for Business and Economics (13e)
Stem-and-Leaf Display
• Example: Hudson Auto Repair
5
6
7
8
9
10
Stems
2
2
1
0
1
1
7
2
1
0
3
4
2
2
2
7
5
2
2
3
7
5
5
3
5
7
9
6
4
8
8
7 8 8 8 9 9 9
4 5 5 5 6 7 8 9 9 9
9
9
Leaves
© 2017 Cengage Learning. May not be scanned, copied or duplicated, or posted to a publicly accessible website, in whole or in part, except for use as permitted in a license distributed with a certain product or service or
otherwise on a password-protected website or school-approved learning management system for classroom use.
45
Statistics for Business and Economics (13e)
Stretched Stem-and-Leaf Display
• Whenever a stem value is stated twice, the first value corresponds to leaf
values of 0 - 4, and the second value corresponds to leaf values of 5 - 9.
© 2017 Cengage Learning. May not be scanned, copied or duplicated, or posted to a publicly accessible website, in whole or in part, except for use as permitted in a license distributed with a certain product or service or
otherwise on a password-protected website or school-approved learning management system for classroom use.
46
Statistics for Business and Economics (13e)
Stretched Stem-and-Leaf Display
• Example: Hudson Auto Repair
5
5
6
6
7
7
8
8
9
9
10
10
2
7
2
5
1
5
0
5
1
7
1
5
2
6
1
5
0
8
3
7
4
5
2
7
2
5
2
9
2
8 8 8 9 9 9
2 3 4 4
6 7 8 9 9 9
3
7 8 9
9
© 2017 Cengage Learning. May not be scanned, copied or duplicated, or posted to a publicly accessible website, in whole or in part, except for use as permitted in a license distributed with a certain product or service or
otherwise on a password-protected website or school-approved learning management system for classroom use.
47
Statistics for Business and Economics (13e)
Stem-and-Leaf Display
• Leaf Units
• A single digit is used to define each leaf.
• In the preceding example, the leaf unit was 1.
• Leaf units may be 100, 10, 1, 0.1, and so on.
• Where the leaf unit is not shown, it is assumed to equal 1.
• The leaf unit indicates how to multiply the stem-and-leaf numbers in order
to approximate the original data.
© 2017 Cengage Learning. May not be scanned, copied or duplicated, or posted to a publicly accessible website, in whole or in part, except for use as permitted in a license distributed with a certain product or service or
otherwise on a password-protected website or school-approved learning management system for classroom use.
48
Statistics for Business and Economics (13e)
Stem-and-Leaf Display
• Example: Leaf Unit = 0.1
8.6
If we have data with values such as
11.7 9.4
9.1
10.2 11.0
8.8
Leaf Unit = 0.1
8
9
10
11
6 8
1 4
2
0 7
© 2017 Cengage Learning. May not be scanned, copied or duplicated, or posted to a publicly accessible website, in whole or in part, except for use as permitted in a license distributed with a certain product or service or
otherwise on a password-protected website or school-approved learning management system for classroom use.
49
Statistics for Business and Economics (13e)
End of Chapter 2, Part A
© 2017 Cengage Learning. May not be scanned, copied or duplicated, or posted to a publicly accessible website, in whole or in part, except for use as permitted in a license distributed with a certain product or service or
otherwise on a password-protected website or school-approved learning management system for classroom use.
50
.
.
.
.
.
.
.
.
.
.
.
.
SLIDES BY
John Loucks
St. Edward’s Univ.
© 2013 Cengage Learning. All Rights Reserved. May not be scanned, copied
or duplicated, or posted to a publicly accessible website, in whole or in part.
Slide 1
Chapter 3, Part A
Discrete Probability Distributions
Introduction to probability
Random Variables
Discrete Probability Distributions
Binomial Probability Distribution
Poisson Probability Distribution
.40
.30
.20
.10
0
1
© 2013 Cengage Learning. All Rights Reserved. May not be scanned, copied
or duplicated, or posted to a publicly accessible website, in whole or in part.
2
3
4
Slide 2
Uncertainties
Managers often base their decisions on an
analysis of uncertainties such as the following:
What are the chances that sales will decrease
if we increase prices?
What is the likelihood a new assembly method
will increase productivity?
What are the odds that a new investment will
be profitable?
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or duplicated, or posted to a publicly accessible website, in whole or in part.
Slide 3
Probability
Probability is a numerical measure of the likelihood
that an event will occur.
Probability values are always assigned on a scale
from 0 to 1.
A probability near zero indicates an event is quite
unlikely to occur.
A probability near one indicates an event is almost
certain to occur.
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or duplicated, or posted to a publicly accessible website, in whole or in part.
Slide 4
Probability as a Numerical Measure
of the Likelihood of Occurrence
Increasing Likelihood of Occurrence
Probability:
0
The event
is very
unlikely
to occur.
.5
The occurrence
of the event is
just as likely as
it is unlikely.
1
The event
is almost
certain
to occur.
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or duplicated, or posted to a publicly accessible website, in whole or in part.
Slide 5
Statistical Experiments
In statistics, the notion of an experiment differs
somewhat from that of an experiment in the
physical sciences.
In statistical experiments, probability determines
outcomes.
Even though the experiment is repeated in exactly
the same way, an entirely different outcome may
occur.
For this reason, statistical experiments are sometimes called random experiments.
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or duplicated, or posted to a publicly accessible website, in whole or in part.
Slide 6
Assigning Probabilities
◼ Basic Requirements for Assigning Probabilities
1. The probability assigned to each experimental
outcome must be between 0 and 1, inclusively.
0 < P(Ei) < 1 for all i
Where: Ei is the ith experimental outcome and P(Ei) is its
probability
2. The sum of the probabilities for all experimental
outcomes must equal 1.
P(E1) + P(E2) + . . . + P(En) = 1
Where: n is the number of experimental outcomes
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or duplicated, or posted to a publicly accessible website, in whole or in part.
Slide 7
1. Random Variables
A random variable is a numerical description of the
outcome of an experiment.
A discrete random variable may assume either a
finite number of values or an infinite sequence of
values.
A continuous random variable may assume any
numerical value in an interval or collection of
intervals.
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or duplicated, or posted to a publicly accessible website, in whole or in part.
Slide 8
Random Variables
Examples of Random Variables
The first, second, and fourth variables above are discrete,
while the third one is continuous.
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or duplicated, or posted to a publicly accessible website, in whole or in part.
Slide 9
Random Variables
Examples of Random Variables
Question
Family
size
Type
Random Variable x
x = Number of dependents in
family reported on tax return
Discrete
Distance from x = Distance in miles from
home to store home to the store site
Continuous
Own dog
or cat
Discrete
x = 1 if own no pet;
= 2 if own dog(s) only;
= 3 if own cat(s) only;
= 4 if own dog(s) and cat(s)
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or duplicated, or posted to a publicly accessible website, in whole or in part.
Slide 10
Example: JSL Appliances
Discrete random variable with a finite number of
values
Let x = number of TVs sold at the store in one day,
where x can take on 5 values (0, 1, 2, 3, 4)
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or duplicated, or posted to a publicly accessible website, in whole or in part.
Slide 11
Example: JSL Appliances
Discrete random variable with an infinite sequence
of values
Let x = number of customers arriving in one day,
where x can take on the values 0, 1, 2, . . .
We can count the customers arriving, but there is no
finite upper limit on the number that might arrive.
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or duplicated, or posted to a publicly accessible website, in whole or in part.
Slide 12
2. Discrete Probability Distributions
The probability distribution for a random variable
describes how probabilities are distributed over
the values of the random variable.
We can describe a discrete probability distribution
with a table, graph, or equation.
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or duplicated, or posted to a publicly accessible website, in whole or in part.
Slide 13
Discrete Probability Distributions
The probability distribution is defined by a
probability function, denoted by f(x), which provides
the probability for each value of the random variable.
The required conditions for a discrete probability
function are:
f(x) > 0
f(x) = 1
For a discrete probability distribution we calculate
the probability of being less than some value x, i.e.
P(X < x), by simply summing up the probabilities of
the values less than x.
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or duplicated, or posted to a publicly accessible website, in whole or in part.
Slide 14
Example: DiCarlo Motors, Inc.
Using past data on daily car sales, …
a tabular representation of the probability
distribution for car sales was developed.
x
0
1
2
3
4
5
f(x)
.18
.39
.24
.14
.04
.01
1.00
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or duplicated, or posted to a publicly accessible website, in whole or in part.
Slide 15
Example: DiCarlo Motors, Inc.
Graphical Representation of the Probability
Distribution
Probability
.50
.40
.30
.20
.10
0
1
2
3
4
5
Values of Random Variable x (car sales)
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or duplicated, or posted to a publicly accessible website, in whole or in part.
Slide 16
Example: DiCarlo Motors, Inc.
The probability distribution provides the following
information.
• There is a 0.18 probability that no cars will be
sold during a day.
• The most probable sales volume is 1, with f(1) =
0.39.
• There is a 0.05 probability of an outstanding sales
day with four or five cars being sold.
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or duplicated, or posted to a publicly accessible website, in whole or in part.
Slide 17
Discrete Uniform Probability Distribution
The discrete uniform probability distribution is the
simplest example of a discrete probability
distribution given by a formula.
The discrete uniform probability function is
f(x) = 1/n
the values of the
random variable
are equally likely
where:
n = the number of values the random
variable may assume
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or duplicated, or posted to a publicly accessible website, in whole or in part.
Slide 18
Expected Value and Variance
The expected value, or mean, of a random variable
is a measure of its central location.
E(x) = = xf(x)
The variance summarizes the variability in the
values of a random variable.
Var(x) = 2 = (x - )2f(x)
The standard deviation, , is defined as the positive
square root of the variance.
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or duplicated, or posted to a publicly accessible website, in whole or in part.
Slide 19
Example: DiCarlo Motors, Inc.
Expected Value of a Discrete Random Variable
x
0
1
2
3
4
5
f(x)
xf(x)
.18
.00
.39
.39
.24
.48
.14
.42
.04
.16
.01
.05
E(x) = 1.50
expected number of
cars sold in a day
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or duplicated, or posted to a publicly accessible website, in whole or in part.
Slide 20
Example: DiCarlo Motors, Inc.
Variance and Standard Deviation
of a Discrete Random Variable
x
x-
(x - )2
0
1
2
3
4
5
.18
-1.5
2.25
.4050
.39
-0.5
0.25
.0975
.24
0.5
0.25
.0600
.14
1.5
2.25
.3150
.04
2.5
6.25
.2500
.01
3.5
12.25
.1225
Variance of daily sales = 2 = 1.2500
f(x)
(x - )2f(x)
cars
squared
Standard deviation of daily sales = 1.118 cars
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or duplicated, or posted to a publicly accessible website, in whole or in part.
Slide 21
3. Binomial Probability Distribution
Four Properties of a Binomial Experiment
1. The experiment consists of a sequence of n
identical trials.
2. Two outcomes, success and failure, are possible
on each trial.
3. The probability of a success, denoted by p, does
not change from trial to trial.
stationarity
assumption
4. The trials are independent.
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or duplicated, or posted to a publicly accessible website, in whole or in part.
Slide 22
Binomial Probability Distribution
Our interest is in the number of successes
occurring in the n trials.
We let x denote the number of successes
occurring in the n trials.
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or duplicated, or posted to a publicly accessible website, in whole or in part.
Slide 23
Binomial Probability Distribution
Binomial Probability Function
n!
f (x) =
p x (1 − p)( n − x )
x !(n − x )!
where:
f(x) = the probability of x successes in n trials
p = the probability of success on any one trial
n = the number of trials
x = number of successes in n trials
n! =n( n - 1)( n - 2) . . . (2)(1)
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or duplicated, or posted to a publicly accessible website, in whole or in part.
Slide 24
Binomial Probability Distribution
Binomial Probability Function
n!
f (x) =
p x (1 − p)( n − x )
x !(n − x )!
Number of experimental
outcomes providing exactly
x successes in n trials
Probability of a particular
sequence of trial outcomes
with x successes in n trials
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or duplicated, or posted to a publicly accessible website, in whole or in part.
Slide 25
Example: Nastke Clothing Store
Binomial Probability Distribution
The store manager estimates that the probability
of a customer making a purchase is 0.30. What is the
probability that 2 of the next 3 customers entering the
store make a purchase?
Let: p = .30 (success), n = 3, x = 2
n!
f ( x) =
p x (1 − p ) (n − x )
x !( n − x )!
3!
f (2) =
(0.3)2 (0.7)1 = 3(.09)(.7) = .189
2!(3 − 2)!
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or duplicated, or posted to a publicly accessible website, in whole or in part.
Slide 26
Binomial Probability Distribution
TABLE 3.6 PROBABILITY DISTRIBUTION FOR THE
NUMBER OF CUSTOMERS
MAKING A PURCHASE
X
f (x)
0
0.343
1
0.441
2
0.189
3
0.027
Total 1.000
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or duplicated, or posted to a publicly accessible website, in whole or in part.
Slide 27
Binomial Probability Distribution
Expected Value
E(x) = = np
Variance
Var(x) = 2 = np(1 − p)
Standard Deviation
s = np(1- p)
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or duplicated, or posted to a publicly accessible website, in whole or in part.
Slide 28
Example: Nastke Clothing Store
Binomial Probability Distribution
• Expected Value
E(x) = = 3(.3) = .9 customers out of 3
• Variance
Var(x) = 2 = 3(.3)(.7) = .63
• Standard Deviation
= 3(.3)(.7) = .794 customers
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or duplicated, or posted to a publicly accessible website, in whole or in part.
Slide 29
Poisson
Probability
Distribution
Poisson
Probability Distribution
A Poisson distributed random variable is often
useful in estimating the number of occurrences
over a specified interval of time or space
It is a discrete random variable that may assume
an infinite sequence of values (x = 0, 1, 2, . . . ).
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or duplicated, or posted to a publicly accessible website, in whole or in part.
Slide 30
Poisson
Probability
Distribution
Poisson
Probability Distribution
Examples of Poisson distributed random variables:
the number of knotholes in 14 linear feet of
pine board
the number of vehicles arriving at a toll
booth in one hour
Bell Labs used the Poisson distribution to model
the arrival of phone calls.
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or duplicated, or posted to a publicly accessible website, in whole or in part.
Slide 31
Poisson
Probability
Distribution
Poisson
Probability Distribution
Two Properties of a Poisson Experiment
1. The probability of an occurrence is the same
for any two intervals of equal length.
2. The occurrence or nonoccurrence in any
interval is independent of the occurrence or
nonoccurrence in any other interval.
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or duplicated, or posted to a publicly accessible website, in whole or in part.
Slide 32
Poisson Probability Distribution
Poisson Probability Function
𝜇 𝑥 𝑒 −𝜇
𝑓 𝑥 =
𝑥!
where:
x = the number of occurrences in an interval
f(x) = the probability of x occurrences in an interval
= mean number of occurrences in an interval
e = 2.71828
x! = x(x – 1)(x – 2) . . . (2)(1)
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or duplicated, or posted to a publicly accessible website, in whole or in part.
Slide 33
Poisson Probability Distribution
Poisson Probability Function
Since there is no stated upper limit for the number
of occurrences, the probability function f(x) is
applicable for values x = 0, 1, 2, … without limit.
In practical applications, x will eventually become
large enough so that f(x) is approximately zero
and the probability of any larger values of x
becomes negligible.
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or duplicated, or posted to a publicly accessible website, in whole or in part.
Slide 34
Poisson Probability Distribution
Example: Mercy Hospital
Patients arrive at the emergency room of Mercy
Hospital at the average rate of 6 per hour on
weekend evenings.
What is the probability of 4 arrivals in 30 minutes
on a weekend evening?
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or duplicated, or posted to a publicly accessible website, in whole or in part.
Slide 35
Poisson
Probability
Distribution
Poisson
Probability Distribution
Example: Mercy Hospital
= 6/hour = 3/half-hour, x = 4
𝑓 4 =
34 (2.71828)−3
4!
Using the
probability
function
= .1680
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or duplicated, or posted to a publicly accessible website, in whole or in part.
Slide 36
Chapter 3, Part B
Continuous Probability Distributions
Uniform Probability Distribution
Normal Probability Distribution
Exponential Probability Distribution
f (x)
f (x) Exponential
Uniform
f (x)
Normal
x
x
x
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or duplicated, or posted to a publicly accessible website, in whole or in part.
Slide 37
Continuous Random Variables
Examples of continuous random variables include the
following:
• The flight time of an airplane traveling from
Chicago to New York
• The lifetime of the picture tube in a new television
set
• The drilling depth required to reach oil in an
offshore drilling operation
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or duplicated, or posted to a publicly accessible website, in whole or in part.
Slide 38
Continuous Probability Distributions
A continuous random variable can assume any value
in an interval on the real line or in a collection of
intervals.
It is not possible to talk about the probability of the
random variable assuming a particular value.
Instead, we talk about the probability of the random
variable assuming a value within a given interval.
For a continuous probability distribution we calculate
the probability of being less than some value x, i.e.
P(X < x), by calculating the area under the curve to
the left of x.
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or duplicated, or posted to a publicly accessible website, in whole or in part.
Slide 39
Continuous Probability Distributions
The probability of the random variable assuming a
value within some given interval from x1 to x2 is
defined to be the area under the graph of the
probability density function between x1 and x2.
f (x)
f (x) Exponential
Uniform
f (x)
x1 x 2
Normal
x
x1
x1 x2
xx12 x2
x
x
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or duplicated, or posted to a publicly accessible website, in whole or in part.
Slide 40
Normal Probability Distribution
The normal probability distribution is the most
important distribution for describing a continuous
random variable.
It is widely used in statistical inference.
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or duplicated, or posted to a publicly accessible website, in whole or in part.
Slide 41
Normal Probability Distribution
It has been used in a wide variety of applications:
Heights
of people
Test
scores
Amounts
of rainfall
Scientific
measurements
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or duplicated, or posted to a publicly accessible website, in whole or in part.
Slide 42
Normal Probability Distribution
Normal Probability Density Function
f (x) =
1
s 2p
e
-(x-m )2 /2s 2
where:
= mean
= standard deviation
= 3.14159
e = 2.71828
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or duplicated, or posted to a publicly accessible website, in whole or in part.
Slide 43
Normal Probability Distribution
Characteristics
The distribution is symmetric, and is bell-shaped.
x
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or duplicated, or posted to a publicly accessible website, in whole or in part.
Slide 44
Normal Probability Distribution
Characteristics
The entire family of normal probability
distributions is defined by its mean and its
standard deviation .
Standard Deviation
Mean
x
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or duplicated, or posted to a publicly accessible website, in whole or in part.
Slide 45
Normal Probability Distribution
Characteristics
The highest point on the normal curve is at the
mean, which is also the median and mode.
x
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or duplicated, or posted to a publicly accessible website, in whole or in part.
Slide 46
Normal Probability Distribution
Characteristics
The mean can be any numerical value: negative,
zero, or positive.
x
-10
0
20
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or duplicated, or posted to a publicly accessible website, in whole or in part.
Slide 47
Normal Probability Distribution
Characteristics
The standard deviation determines the width of the
curve: larger values result in wider, flatter curves.
= 15
= 25
x
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or duplicated, or posted to a publicly accessible website, in whole or in part.
Slide 48
Normal Probability Distribution
Characteristics
Probabilities for the normal random variable are
given by areas under the curve. The total area
under the curve is 1 (.5 to the left of the mean and
.5 to the right).
.5
.5
x
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or duplicated, or posted to a publicly accessible website, in whole or in part.
Slide 49
Normal Probability Distribution
Characteristics
68.26% of values of a normal random variable
are within +/- 1 standard deviation of its mean.
95.44% of values of a normal random variable
are within +/- 2 standard deviations of its mean.
99.72% of values of a normal random variable
are within +/- 3 standard deviations of its mean.
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or duplicated, or posted to a publicly accessible website, in whole or in part.
Slide 50
Normal Probability Distribution
Characteristics
99.72%
95.44%
68.26%
– 3
– 1
– 2
+ 3
+ 1
+ 2
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or duplicated, or posted to a publicly accessible website, in whole or in part.
x
Slide 51
Standard Normal Probability Distribution
-Use probability tables that have been calculated on a
computer.
- Only one special Normal distribution, N(0, 1), has
been tabulated.
- If we want to calculate probabilities from different
Normal distributions we convert the probability to one
involving the standard Normal distribution.
→ standardization
A random variable having a normal distribution
with a mean of 0 and a standard deviation of 1 is
said to have a standard normal probability
distribution.
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or duplicated, or posted to a publicly accessible website, in whole or in part.
Slide 52
Standard Normal Probability Distribution
The letter z is used to designate the standard
normal random variable.
=1
z
0
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or duplicated, or posted to a publicly accessible website, in whole or in part.
Slide 53
Standard Normal Probability Distribution
Converting to the Standard Normal Distribution
z=
x−
We can think of z as a measure of the number of
standard deviations x is from .
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or duplicated, or posted to a publicly accessible website, in whole or in part.
Slide 54
Example: Pep Zone
Standard Normal Probability Distribution
Pep Zone sells auto parts and supplies including
a popular multi-grade motor oil. When the stock of
this oil drops to 20 gallons, a replenishment order is
placed.
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or duplicated, or posted to a publicly accessible website, in whole or in part.
Slide 55
Example: Pep Zone
Standard Normal Probability Distribution
The store manager is concerned that sales are
being lost due to stockouts while waiting for an
order. It has been determined that demand during
replenishment lead-time is normally distributed with
a mean of 15 gallons and a standard deviation of 6
gallons.
The manager would like to know the probability
of a stockout, P(x > 20).
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or duplicated, or posted to a publicly accessible website, in whole or in part.
Slide 56
Example: Pep Zone
Solving for the Stockout Probability
Step 1: Convert x to the standard normal distribution.
z = (x - )/
= (20 - 15)/6
= .83
Step 2: Find the area under the standard normal
curve between the mean and z = .83.
see next slide
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or duplicated, or posted to a publicly accessible website, in whole or in part.
Slide 57
Example: Pep Zone
Probability Table for the
Standard Normal Distribution
z
.00
.01
.02
.03
.04
.05
.06
.07
.08
.09
.
.
.
.
.
.
.
.
.
.
.
.5
.1915 .1695 .1985 .2019 .2054 .2088 .2123 .2157 .2190 .2224
.6
.2257 .2291 .2324 .2357 .2389 .2422 .2454 .2486 .2517 .2549
.7
.2580 .2611 .2642 .2673 .2704 .2734 .2764 .2794 .2823 .2852
.8
.2881 .2910 .2939 .2967 .2995 .3023 .3051 .3078 .3106 .3133
.9
.3159 .3186 .3212 .3238 .3264 .3289 .3315 .3340 .3365 .3389
.
.
.
.
.
.
.
.
.
.
.
P(0 < z < .83)
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or duplicated, or posted to a publicly accessible website, in whole or in part.
Slide 58
Example: Pep Zone
Solving for the Stockout Probability
Step 3: Compute the area under the standard normal
curve to the right of z = .83.
P(z > .83) = .5 – P(0 < z < .83)
= .5- .2967
= .2033
Probability
of a stockout
P(x > 20)
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or duplicated, or posted to a publicly accessible website, in whole or in part.
Slide 59
Example: Pep Zone
Solving for the Stockout Probability
Area = .5 - .2967
Area = .2967
= .2033
0
.83
z
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or duplicated, or posted to a publicly accessible website, in whole or in part.
Slide 60
End of Chapter 3
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or duplicated, or posted to a publicly accessible website, in whole or in part.
Slide 61
Purchase answer to see full
attachment