HAWKES LEARNING SYSTEMS
math courseware specialists
Copyright © 2010 by Hawkes Learning
Systems/Quant Systems, Inc.
All rights reserved.
Chapter 2
Data, Reality, and Problem Solving
HAWKES LEARNING SYSTEMS
Data, Reality, and Problem Solving
math courseware specialists
Sections 2.1-2.4 The Reality of Conducting a Study
Objectives:
• Learn the ethical concerns of conducting a study.
• Determine the practical concerns of conducting a study.
HAWKES LEARNING SYSTEMS
Data, Reality, and Problem Solving
math courseware specialists
Section 2.1 The Lords of Data
Measurement:
To develop suitable measurements we will be concerned with the
following:
• What should be measured?
• How should the concept be measured?
Now we need to know how good the measurements are. Ask yourself
the following:
• Is the concept under study adequately reflected by the proposed
measurements?
• Is the data measured accurately?
• Is there a sufficient quantity of the data to draw a reasonable
conclusion?
If you answered yes to all three of these questions, the data possesses
good properties.
HAWKES LEARNING SYSTEMS
Data, Reality, and Problem Solving
math courseware specialists
Section 2.2 Science and Data
The Scientific Method:
The scientific method is a collection of methods that have become
standard for exploring research problems.
The Scientific Method:
• Gather information about the phenomenon being studied.
• On the basis of that data, formulate a preliminary generalization
or hypothesis.
• Collect the data to test the hypothesis.
• If the data and other subsequent experiments support the
hypothesis, it becomes a law.
HAWKES LEARNING SYSTEMS
Data, Reality, and Problem Solving
math courseware specialists
Section 2.3 Decision Making and Data
Decision Making:
We make decisions every day and collecting data is a natural part of
our lives. One decision we make every day is:
“What will I have for dinner?”
• Although we may not apply the scientific method, most people perform
experiments and collect data (by eating). This leads to generalizations
such as “I like everything except liver, sweet peas, and beets.”
• After sufficient experimentation, these generalizations become
personal preference laws.
Do you have any personal preference laws?
HAWKES LEARNING SYSTEMS
Data, Reality, and Problem Solving
math courseware specialists
Section 2.3 Decision Making and Data
Decision Making Method:
• Clearly define the problem and any influential variables.
• Decide upon objectives and decision criteria for choosing a
solution.
• Create alternative solutions.
• Compare alternatives using the criteria established in the second
step.
• Implement the chosen alternative.
• Check the results to make sure the desired results are achieved.
HAWKES LEARNING SYSTEMS
Data, Reality, and Problem Solving
math courseware specialists
Section 2.4 Collecting Data
Ways to Collect Data:
The two ways to collect data are:
• Controlled experiments
• Observation
The data collection method is related to the nature of the problem to be
solved and the ethical and practical constraints of collecting data in
some particular environment.
HAWKES LEARNING SYSTEMS
Data, Reality, and Problem Solving
math courseware specialists
Section 2.4 Collecting Data
Controlled Experiments:
The purpose of a controlled experiment is to reveal the response of one
variable to the changes of another variable.
• In a controlled experiment there are two groups: the control group
and the experimental group.
• During the experiment a treatment is applied to the experimental
group.
• The exact treatment will depend on the particular experiment.
• The treatment changes the level of the explanatory variable in the
experiment. The effect of the treatment can be measured by comparing
the response variable in the control and experimental groups.
HAWKES LEARNING SYSTEMS
Data, Reality, and Problem Solving
math courseware specialists
Section 2.4 Collecting Data
Definitions:
•The response variable measures the outcome of interest in a study.
•An explanatory variable causes or explains changes in the response
variable.
HAWKES LEARNING SYSTEMS
Data, Reality, and Problem Solving
math courseware specialists
Section 2.4 Collecting Data
Calculus before Physics Experiment:
Control group
Students with the same
mathematical ability
Experimental group
Randomly assigning students to take
calculus before physics
Treatment
Taking calculus before physics
Response variable
Students’ grades in physics
Explanatory variable
Whether students take calculus
before physics
HAWKES LEARNING SYSTEMS
Data, Reality, and Problem Solving
math courseware specialists
Section 2.4 Collecting Data
Flowchart for the Calculus before Physics Experiment:
Treatment
Group
Treatment
(Take Calculus
before Physics)
Observe
Randomly
Select
Students
Compare
Performance
Control
Group
Observe
HAWKES LEARNING SYSTEMS
Data, Reality, and Problem Solving
math courseware specialists
Section 2.4 Collecting Data
Does an SAT Course Improve an SAT Score?
Response variable
High school students that
have taken the SAT exactly
one time
The high school students are
given an SAT preparation
course.
The same high school students
take the SAT again
SAT scores
Explanatory variable
SAT preparation course
Control group
Treatment
Experimental group
HAWKES LEARNING SYSTEMS
Data, Reality, and Problem Solving
math courseware specialists
Section 2.4 Collecting Data
Flowchart for the SAT Experiment:
Before
The control group:
Is high school
students that have
taken the SAT
exactly one time.
The value of the
response variable is
the students’ scores
on their first SAT.
Treatment
The same
high school
students are
given an
SAT
preparation
course.
After
Now, the students
are the treatment
group and they
take the SAT again.
The value of the
response variable
is the score on the
second SAT.
Compare
Compare
students
performance
on the SAT
before and
after the
course
HAWKES LEARNING SYSTEMS
Data, Reality, and Problem Solving
math courseware specialists
Section 2.4 Collecting Data
Definitions:
• Sometimes in clinical trials people are given placebos. Placebos are
“fake” treatments.
• Double blind studies are used to counteract the placebo effect. In
these studies neither the subjects know if they are members of the
control or experimental group, nor do the evaluators know whether their
subjects are members of the control or experimental group.
HAWKES LEARNING SYSTEMS
Data, Reality, and Problem Solving
math courseware specialists
Section 2.4 Collecting Data
Observational Data:
Observational data comes by measuring “what is.”
For example, observational data is:
• What is happening in the marketplace at the time.
• The measure of how things are in a specific geographic area at a
given point in time.
Virtually all data we routinely encounter is observational.
HAWKES LEARNING SYSTEMS
Data, Reality, and Problem Solving
math courseware specialists
Section 2.4 Collecting Data
Examples of Observational Data:
• Stock, commodity, bond, option, and currency market data.
• Almost all federal data, including census, economic, and
educational data.
• Sports data.
Can you think of an example of observational data?
HAWKES LEARNING SYSTEMS
Data, Reality, and Problem Solving
math courseware specialists
Section 2.4 Collecting Data
Example:
8,442 men and 4,321 women applied to graduate school at Berkeley.
Subsequently, 3,714 men and 1,512 women accepted. Thus 44% of the
men and 35% of the women were accepted. Was the graduate school
discriminating against women?
College Acceptance Rates
Men
Women
Total Number Percent Total Number Percent
Major
of Applicants Admitted of Applicants Admitted
I
800
65
120
85
II
550
62
32
70
III
400
40
410
36
IV
350
36
347
37
V
200
24
387
27
VI
300
8
352
10
To the left is the
admissions data from
Berkeley.
Does this data show
discrimination?
HAWKES LEARNING SYSTEMS
Data, Reality, and Problem Solving
math courseware specialists
Section 2.4 Collecting Data
Solution:
College Acceptance Rates
Men
Women
Total Number Percent Total Number Percent
Major
of Applicants Admitted of Applicants Admitted
I
800
65
120
85
II
550
62
32
70
III
400
40
410
36
IV
350
36
347
37
V
200
24
387
27
VI
300
8
352
10
• In all but one of the majors, women were admitted more frequently than
men. Not only were women not being discriminated against, but there
appears to be potential discrimination against men in Major I.
• The majors that had very low acceptance rates had relatively very few
men and a large number of women applying.
•The variable, major field of study, was confounding the variable gender in
the original analysis and biasing the original conclusion.
HAWKES LEARNING SYSTEMS
math courseware specialists
Copyright
©
2010
Hawkes
Learning
Ch
2. Data,
Reality,
&byProblem
Solving
Data, Reality,
and
Problem
Solving
Systems/Quant Systems, Inc.
Data Classifications
Section 2.4 2.5
Collecting
Data
All
rights reserved.
Example:
Determine whether each of the following studies is observational
or experimental.
•
A medical researcher wants to examine the effects of
exercise on cardiovascular health.
Solution: Experimental
•
A recording company is interested in knowing the
percentage of teenagers that download music off of the
internet.
Solution: Observational
•
A chain of grocery stores wants to know how much the
average family spends on produce each month.
Solution: Observational
HAWKES LEARNING SYSTEMS
Data, Reality, and Problem Solving
math courseware specialists
Sections 2.5-2.6 Levels of Measurement
Objectives:
• Classify data as discrete or continuous.
• Determine if data are qualitative or quantitative.
• Identify the level of measurement.
HAWKES LEARNING SYSTEMS
Data, Reality, and Problem Solving
math courseware specialists
Section 2.5 Data Classifications
Data Classifications:
Data or variables can be categorized in several ways:
• Discrete or continuous
• Level of measurement
• Quantitative or qualitative
HAWKES LEARNING SYSTEMS
Data, Reality, and Problem Solving
math courseware specialists
Section 2.5 Data Classifications
Discrete Data:
Discrete data is a restricted set of values.
Examples of discrete data:
• The number of people in a classroom
• A variable that only has the values 1, 1.5, 2, 2.5
• A variable that contains integer values
Can you think of some data that is discrete?
HAWKES LEARNING SYSTEMS
Data, Reality, and Problem Solving
math courseware specialists
Section 2.5 Data Classifications
Continuous Data:
Continuous data can take on any value in an interval.
Examples of continuous data:
• Height
• Age
• A variable that has any value between 0 and 2
Can you think of some data that is continuous?
HAWKES LEARNING SYSTEMS
Data, Reality, and Problem Solving
math courseware specialists
Section 2.5 Data Classifications
Level of Measurement:
The quality of data is referred to as its level of measurement.
Terms used to describe the level of measurement:
• Nominal
• Ordinal
• Interval
• Ratio
HAWKES LEARNING SYSTEMS
Data, Reality, and Problem Solving
math courseware specialists
Section 2.5 Data Classifications
Nominal Data:
Nominal measures offer names or labels for certain characteristics.
Examples of nominal measures:
• Gender
• Hair color
• Jersey number of a basketball player
Can you think of a measure that is nominal?
HAWKES LEARNING SYSTEMS
Data, Reality, and Problem Solving
math courseware specialists
Section 2.5 Data Classifications
Ordinal Data:
Ordinal data represents data in an associated order.
Examples of ordinal data:
• Ranking of sports teams
• Year in college
• Letter grades
Can you think of data that is ordinal?
HAWKES LEARNING SYSTEMS
Data, Reality, and Problem Solving
math courseware specialists
Section 2.5 Data Classifications
Interval Data:
If the data can be ordered and the arithmetic difference is meaningful, the
data is interval.
Examples of interval data:
• Temperature
• Dates
• Level of difficulty rated from 1 to 5
Can you think of data that is interval?
HAWKES LEARNING SYSTEMS
Data, Reality, and Problem Solving
math courseware specialists
Section 2.5 Data Classifications
Ratio Data:
Ratio data has a meaningful zero point and the ratio of two data points is
meaningful.
Examples of ratio data:
• Money
• Height
• Age
Can you think of data that is ratio?
HAWKES LEARNING SYSTEMS
Data, Reality, and Problem Solving
math courseware specialists
Section 2.5 Data Classifications
Example:
Determine the level of measurement.
• Today’s high temperature (in Fahrenheit) for varying cities across the
U.S.
Solution: Interval
• The colors contained in a box of crayons.
Solution: Nominal
• The boiling point (in Kelvin’s) for varying chemical compounds.
Solution: Ratio
• The individual page numbers at the bottom of each page in the
statistics book.
Solution: Ordinal
HAWKES LEARNING SYSTEMS
Data, Reality, and Problem Solving
math courseware specialists
Section 2.5 Data Classifications
Qualitative and Quantitative Data:
• Qualitative data is measured on a nominal or ordinal scale.
• Quantitative data is measured on an interval or ratio scale.
Qualitative
Quantitative
Descriptions and
labels
Counts and
measurements
HAWKES LEARNING SYSTEMS
Data, Reality, and Problem Solving
math courseware specialists
Section 2.5 Data Classifications
Example:
Classify each of the following as qualitative or quantitative data.
• The weights of members of the football team.
Solution: Quantitative
• The flavors of Ben and Jerry’s Ice Cream.
Solution: Qualitative
• The jersey numbers of a women’s basketball team.
Solution: Qualitative
• Student ID numbers.
Solution: Qualitative
HAWKES LEARNING SYSTEMS
math courseware specialists
Copyright
© 2010
by Hawkes Learning
Data, Reality,
and Problem
Solving
Systems/Quant Systems, Inc.
Section 2.5 All
Data
Classifications
rights
reserved.
Levels of Measurement:
Ratio
Interval
Ordinal
Order
Nominal
Names
0 is a
placeholder
0 means the
absence of
something