Lecture 1: Thu, Sept 5

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Lecture 1 Thu, Sept 5

• Introduction/Syllabus (web page)
• Todays material
• Key Statistical Concepts
• Types of Data
• Pie and Bar Charts
• Histograms, Stem-and-Leaf Plots
• Scatter Plots
• Intro to JMP-IN (Xr 2.94 2.95)
• Homework Assignment

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Key Definitions

• Statistics the art of data analysis. Involves
  classifying, summarizing, organizing, and
  interpreting numerical information.
• Population the set of all items of interest in a
  statistical problem.
• Sample a subset of items in the population.
• Descriptive Statistics a body of methods used to
  summarize and organize the characteristics of
  sample data.
• Inferential Statistics a body of methods used to
  draw inferences about characteristics of
  populations based on sample data.

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• Variable characteristic or property of an
  individual item of a population or sample.
• Observation the value assigned to a variable.
• Parameter descriptive measure of a population.
• Statistic descriptive measure of a sample.
• Statistical Inference process of making an
  estimate, prediction or decision about a
  population based on information contained in a
  sample.
• Measure of Reliability a statement about the
  degree of uncertainty.

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Example Cola Wars

• Cola wars is the popular term for the intense
  competition between Coca-Cola and Pepsi displayed
  in their marketing campaigns. Their campaigns
  have featured movie and television stars, rock
  videos, athletic endorsements, and claims of
  consumer preference based on taste tests.
  Suppose, as part of a Pepsi marketing campaign,
  1,000 cola consumers are given a blind taste test
  (ie, a taste test in which the two brand names
  are disguised). Each consumer is asked to state
  their gender, age and a preference for brand A or
  brand B.

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• a. Describe the population.
• b. Describe the variables of interest.
• c. Describe the sample.
• d. Describe the inference about the taste
  preference.
• e. Assume the cola preferences of 1,000 consumers
  were indicated in a taste test. Describe how the
  reliability of an inference concerning the
  preferences of all cola consumers in the Pepsi
  bottlers marketing region could be measured.

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Solutions

• a. Population of interest the collection or set
  of all cola consumers.
• b. Variables of interest gender, age and cola
  preference.
• c. Sample 1,000 cola consumers selected from the
  population of all cola consumers.
• d. Inference of interest generalization of the
  cola preferences of the 1,000 sampled consumers
  to the population of all cola consumers. In
  particular, the preferences of the consumers in
  the sample can be used to estimate the percentage
  of all cola consumers who prefer each brand.

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• e. When the preferences of 1,000 consumers who
  are used to estimate the preference of all
  consumers in the region, the estimate will not
  exactly mirror the preferences of the population.
  For example, if the taste test shows that 56 of
  the 1,000 consumers chose Pepsi, it does not
  follow (nor is it likely) that exactly 56 of all
  cola drinkers in the region prefer Pepsi.
• Nevertheless, we can use sound statistical
  reasoning (which is presented later in the
  course) to ensure that our sampling procedure
  will generate estimates that are almost certainly
  within a specified limit of the true percentage
  of all consumers who prefer Pepsi.
• For example, such reasoning might assure us
  that the estimate of the preference for Pepsi
  from the sample is almost certainly within 5 of
  the actual population preference. The implication
  is that the actual preference for Pepsi is
  between 51 ie, (56-5) and 61 ie, (565)-
  that is, (56 5) This interval represents a
  measure of reliability for the inference.

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Types of Data (Chapter 2)

• Quantitative Data are obtained when the variable
  being observed takes numerical values.
• Qualitative Data are obtained when the variable
  being observed can only be categorized into
  different groups (classes).
• Ranked Data variable is categorized into
  different groups, but the groups are ranked.

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Questions

• In the Cola Wars example, what type of data are
  the variables of interest?
• Gender
• Age
• Cola preference
• Give one example of each type of data numerical,
  categorical, ranked.

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Types of data - examples
Interval data
Nominal
Age - income 55 75000 42 68000 . . . .
Person Marital status 1 married 2 single 3 sin
gle . . . .
Weight gain 10 5 . .
Computer Brand 1 IBM 2 Dell 3 IBM . . . .
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Types of data - examples
Interval data
Nominal data
With nominal data, all we can do is, calculate
the proportion of data that falls into each
category.
Age - income 55 75000 42 68000 . . . .
Weight gain 10 5 . .
IBM Dell Compaq Other Total 25
11 8 6 50
50 22 16 12
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Types of data analysis

• Knowing the type of data is necessary to properly
  select the technique to be used when analyzing
  data.
• Type of analysis allowed for each type of data
• Interval data arithmetic calculations
• Nominal data counting the number of observation
  in each category
• Ordinal data - computations based on an ordering
  process

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Cross-Sectional/Time-Series Data

• Cross sectional data is collected at a certain
  point in time
• Marketing survey (observe preferences by gender,
  age)
• Test score in a statistics course
• Starting salaries of an MBA program graduates
• Time series data is collected over successive
  points in time
• Weekly closing price of gold
• Amount of crude oil imported monthly

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Graphical Techniques for Qualitative Data

• How to summarize? Count the number of times and
  compute the proportion of times of the occurrence
  of each value of the data.
• Pie Chart is a circle divided into a number of
  slices that represent the various categories such
  that the size of each slice is proportional to
  the percentage corresponding to that category.
• Bar Chart uses bars to represent the frequencies
  (or relative frequencies) such that the height of
  each bar equals the frequency or relative
  frequency of each of the categories.

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Turboprop Airplanes

• In 1994, a spate of small aircraft crashes made
  the safety of turboprop airplanes an issue. As
  part of an analysis of different types of
  accidents, Airjet Ltd determined where accidents
  occurred for both turboprop airplanes and jets in
  the period 1984-1993. The data are stored using
  the following format

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• Results for turboprops are stored in column 1
  (n260) Results for jets are stored in column 2
  (n298).
• Identify the type of data stored in each column.
• Use two pie charts to summarize these data.
• Does it appear that turboprop airplanes and jets
  have similar accident patterns?

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Graphical Techniques for Quantitative Data

• Frequency Distribution a table that groups data
  in non-overlapping intervals called classes and
  records the number of observations (frequencies)
  in each class.

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• Frequency Histogram is created by drawing
  rectangles. The bases of the rectangles
  correspond to the class interval, and the height
  of each rectangle equals the number of
  observations in that class.
• Stem-and-Leaf Displays similar to histogram but
  with each observation represented by leafs. (see
  description next page)
• Ogive is the graphical representation of the
  cumulative relative frequency distribution.

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Shapes of histograms
Symmetry

• There are four typical shape characteristics

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Shapes of histograms
Skewness
Negatively skewed
Positively skewed
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Modal classes

• A modal class is the one with the largest number
  of observations.
• A unimodal histogram

The modal class
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Modal classes
A bimodal histogram
A modal class
A modal class
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Bell shaped histograms

• Many statistical techniques require that the
  population be bell shaped.
• Drawing the histogram helps verify the shape of
  the population in question

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Example MBA Salaries

• The table contains the top salary offer (in
  thousands of dollars) received by each member of
  a sample of 50 MBA students who recently
  graduated from the Graduate School of Management
  at Rutgers, the state university of New Jersey.

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MBA Salary Data
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Frequency Distribution
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Histogram of MBA Salaries
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Shapes of Histograms

• Symmetric histogram which if you draw a line
  down the middle looks identical on both sides
• Positively skewed histogram with a long tail
  extending to the right
• Negatively skewed histogram with a long tail
  extending to the left
• Bell-shaped histogram looks like a bell
• Number of modal classes the number of distinct
  peaks in a histogram

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Stem-and-Leaf Plot

• Split each datum into stem and leaf
• Stem the first part of the number
• Leaves last digit of number
• Examples
• ?

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Stem-and-Leaf Example 2

• ?

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Histogram Stem-and-Leaf
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Cumulative Frequency Distribution
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Histogram Ogive Plot
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Example Production

• In order to estimate how long it will take to
  produce a particular product, a manufacturer will
  study the relationship between production time
  per unit time and the number of units that have
  been produced. The line or curve characterizing
  this relationship is called a learning curve
  (Adler and Clark, Management Science, Mar 1991).
• Twenty-five employees, all of whom were
  performing the same production task for the 10th
  time, were observed. Each persons task
  completion time (in minutes) was recorded. The
  same 25 employees were observed again the 30th
  time they performed the same task and the 50th
  time they performed the task. The resulting
  completion times are shown in the table below.

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• Use a statistical software package to construct a
  frequency histogram for each of the three data
  sets.
• Compare the histograms. Does it appear that the
  relationship between task completion and the
  number of times the task is performed is in
  agreement with the observations note above about
  production processes in general? Explain.

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Graphical Techniques for 2 Quantitative Variables

• Scatter Plot
• Graphical method to describe the relationship
  between two quantitative variables
• Two-dimensional plot, with one variables values
  plotted along the vertical axis and the other
  along the horizontal axis.

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Typical Patterns of Scatter Diagrams
Negative linear relationship
Positive linear relationship
No relationship
Negative nonlinear relationship
Nonlinear (concave) relationship
This is a weak linear relationship.A non linear
relationship seems to fit the data better.
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House Sales and Mortgage Levels

• The economics department of a national investment
  banking firm is conducting a study to determine
  how house sales are related to mortgage rate
  levels. The number of house sales are related to
  mortgage rate levels. The number of houses sold
  and the average monthly mortgage rate for 36
  months recorded.

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• a. Draw a scatter diagram for these data with
  number of houses sold on the vertical axis.
• b. Describe the relationship between mortgage
  rates and number of homes sold.

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Graphing the Relationship Between Two Nominal
Variables

• We create a contingency table.
• This table lists the frequency for each
  combination of values of the two variables.
• We can create a bar chart that represent the
  frequency of occurrence of each combination of
  values.

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Contingency table

• Example 2.8
• To conduct an efficient advertisement campaign
  the relationship between occupation and
  newspapers readership is studied. The following
  table was created

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Contingency table

• Solution
• If there is no relationship between occupation
  and newspaper read, the bar charts describing the
  frequency of readership of newspapers should look
  similar across occupations.

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Bar charts for a contingency table
Blue-collar workers prefer the Star and the
Sun.
White-collar workers and professionals mostly
read the Post and the Globe and Mail
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2.6 Describing Time-Series Data

• Data can be classified according to the time it
  is collected.
• Cross-sectional data are all collected at the
  same time.
• Time-series data are collected at successive
  points in time.
• Time-series data is often depicted on a line
  chart (a plot of the variable over time).

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Line Chart

• Example 2.9
• The total amount of income tax paid by
  individuals in 1987 through 1999 are listed
  below.
• Draw a graph of this data and describe the
  information produced

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Line Chart
For the first five years total tax was
relatively flat From 1993 there was a rapid
increase in tax revenues.
Line charts can be used to describe nominal data
time series.
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Homework Assignment 1

• Due next Thursday, Sept 19, at the start of
  class.
• Full assignment will be posted on the Stat 101
  web page this Thu at 5pm.