Introduction and Linear Models

Report
LSP 120: Quantitative Reasoning and
Technological Literacy
Topic 1: Introduction to Quantitative
Reasoning and Linear Models
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Basic Definitions
• Data: numbers with a context
• Cell: each data point is recorded in a cell
• Observation: each row of cells form an
observation for a subject/individual
• Variable: any characteristic of an
individual
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Why Data?
1) Data beat anecdotes
“Belief is no substitute for arithmetic.”
Henry Spencer
Data are more reliable than anecdotes, because
they systematically describe an overall picture
rather than focus on a few incidents .
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Why Data?
2. Where the data come from is important.
“Figures won’t lie, but liars will figure.”
Gen. Charles H. Grosvenor
(1833-1917), Ohio Rep.
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Familiarizing with Data
• Open Excel
• Collect data:
– Ask 5 classmates the approximate # of text messages
they send per day
– Record the data on Excel spreadsheet
• Calculate average using the Average function on
Excel. (There are many functions such as sum,
count, slope, intercept etc. that we will use in this
class)
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What is a linear function?
• Most people would say it is a straight line or that it fits
the equation y = mx + b.
• They are correct, but what is true about a function that
when graphed yields a straight line?
• What is the relationship between the variables in a
linear function?
• A linear function indicates a relationship between x
and y that has a fixed or constant rate of change.
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Is the relationship between x and y is linear?
The first thing we want to do is be able to determine
whether a table of values for 2 variables represents a
linear function. In order to do that we use the
formula below:
x
y
3
11
5
16
7
21
9
26
11
31
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To determine if a relationship is linear in Excel, add a column in which you calculate
the rate of change. You must translate the definition of “change in y over change is
x” to a formula using cell references. Entering a formula using cell references allows
you to repeat a certain calculation down a column or across a row. Once you enter
the formula, you can drag it down to apply it to subsequent cells.
A
B
C
1
x
y
Rate of Change
2
3
11
3
5
16
4
7
21
5
9
26
6
11
31
=(B3-B2)/(A3-A2)
This is a cell reference
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• Note that we entered the formula for rate of change not next to the first set of
values but next to the second. This is because we are finding the change from the
first to the second. Then fill the column and check whether the values are
constant. To fill a column, either put the cursor on the corner of the cell with the
formula and double click or (if the column is not unbroken) put the cursor on the
corner and click and drag down. If the rate of change values are constant then the
relationship is a linear function.
1
2
3
4
5
6
A
x
3
5
7
9
11
B
y
11
16
21
26
31
C
Rate of Change
2.5
2.5
2.5
2.5
• So this example does represent a linear function. Rate of change is 2.5 and it is
constant. This means that that when the x value increases by 1, the y value
increases by 2.5.
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How to Write a Linear Equation
Next step is to write the equation for this function.
y = mx + b.
y and x are the variables
m is the slope (rate of change)
b is the y-intercept (the initial value when x=0)
1
2
3
4
5
6
A
B
x
3
5
7
9
11
y
11
16
21
26
31
C
Rate of
Change
2.5
2.5
2.5
2.5
We know x, y, and m, we need to calculate b:
Using the first set of values (x=3 and y=11) and 2.5 for "m“ (slope):
11=2.5*3 + b.
Solving: 11=7.5 + b
3.5 = b.
The equation for this function is : y = 2.5 x + 3.5
Another way to find the equation is to use Excel’s intercept function.
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Practice
For the following, determine whether the function is
linear and if so, write the equation for the function.
x
y
x
y
x
y
5
-4
1
1
2
20
10
-1
2
3
4
13
15
2
5
9
6
6
20
5
7
18
8
-1
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Warning:
Not all graphs that look like lines represent linear functions
The graph of a linear function is a line. However, a graph of a
function can look like a line even thought the function is not
linear. Graph the following data where t is years and P is the
population of Mexico (in millions):
• What does the graph look like?
• Now, calculate the rate of change
for each set of data points
(as we learned under
Does the data represent a
linear function?) Is it constant?
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t
P
1980
67.38
1981
69.13
1982
70.93
1983
72.77
1984
74.67
1985
76.61
1986
78.60
14
• What if you were given the population for every ten years? Would
the graph no longer appear to be linear? Graph the following data.
• Does this data (derived from
the same equation as the table
above) appear to be linear?
Both of these tables represent
an exponential model (which we
will be discussing shortly).
The important thing to note is that
exponential data can appear to be
linear depending on how many data
points are graphed. The only way to
determine if a data set is linear is to
calculate the rate of change (slope)
and verify that it is constant.
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t
P
1980
67.38
1990
87.10
2000
112.58
2010
145.53
2020
188.12
2030
243.16
2040
314.32
15
"Real world" example of a linear function:
• Studies of the metabolism of alcohol consistently show that blood
alcohol content (BAC), after rising rapidly after ingesting alcohol,
declines linearly. For example, in one study, BAC in a fasting person
rose to about 0.018 % after a single drink. After an hour the level
had dropped to 0.010 %. Assuming that BAC continues to decline
linearly (meaning at a constant rate of change), approximately
when will BAC drop to 0.002%?
• In order to answer the question, you must express the relationship
as an equation and then use to equation. First, define the variables
in the function and create a table in excel.
• The two variables are time and BAC.
• Calculate the rate of change.
Time
0
1
BAC
0.018%
0.010%
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Time
BAC
0
0.018%
1
0.010%
Rate of
change
-0.008%
This rate of change means when the time
increases by 1, the BAC decreases (since rate
of change is negative) by .008. In other words,
the BAC % is decreasing .008 every hour. Since
we are told that BAC declines linearly, we can
assume that figure stays constant. Now write
the equation with Y representing BAC and X
the time in hours. Y = -.008x + .018.
This equation can be used to make
predictions. The question is "when will the
BAC reach .002%?" Plug in .002 for Y and solve
for X.
.002 = -.008x + .018
-.016 = -.008x
x=2
Therefore the BAC will reach .002% after 2
hours.
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Warning:
Not all graphs that look like lines represent linear functions
The graph of a linear function is a line. However, a graph of a
function can look like a line even thought the function is not
linear. Graph the following data where t is years and P is the
population of Mexico (in millions):
• What does the graph look like?
• Now, calculate the rate of change
for each set of data points
(as we learned under
Does the data represent a
linear function?) Is it constant?
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t
P
1980
67.38
1981
69.13
1982
70.93
1983
72.77
1984
74.67
1985
76.61
1986
78.60
18
• What if you were given the population for every ten years? Would
the graph no longer appear to be linear? Graph the following data.
• Does this data (derived from
the same equation as the table
above) appear to be linear?
Both of these tables represent
an exponential model (which we
will be discussing shortly).
The important thing to note is that
exponential data can appear to be
linear depending on how many data
points are graphed. The only way to
determine if a data set is linear is to
calculate the rate of change (slope)
and verify that it is constant.
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t
P
1980
67.38
1990
87.10
2000
112.58
2010
145.53
2020
188.12
2030
243.16
2040
314.32
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Linear Modeling-Trendlines
• The Problem - To date, we have studied linear equations (models)
where the data is perfectly linear. By using the slope-intercept
formula, we derived linear equation/models. In the “real world”
most data is not perfectly linear. How do we handle this type of
data?
• The Solution - We use trendlines (also known as line of best fit and
least squares line).
• Why - If we find a trendline that is a good fit, we can use the
equation to make predictions. Generally we predict into the future
(and occasionally into the past) which is called extrapolation.
Constructing points between existing points is referred to as
interpolation.
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Is the trendline a good fit for the data?
• There are five guidelines to answer this question:
1. Guideline 1: Do you have at least 7 data points?
2. Guideline 2: Does the R-squared value indicate a
relationship?
3. Guideline 3: Verify that your trendline fits the shape of
your graph.
4. Guideline 4: Look for outliers.
5. Guideline 5: Practical Knowledge, Common Sense
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Guideline 1: Do you have at least 7 data points?
• For the datasets that we use in this class, you
should use at least 7 of the most recent data
points available.
• If there are more data points, you will also
want to include them (unless your data fails
one of the guidelines below).
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Guideline 2: Does the R-squared value indicate
a relationship?
• R2 is a standard measure of how well the line fits the data.
(Tells us how linear the relationship between x and y is)
• In statistical terms, R2 is the percentage of variance of y
that is explained by our trendline.
• It is more useful in the negative sense: if R2 is very low, it
tells us the model is not very good and probably shouldn't
be used.
• If R2 is high, we should also look at other guidelines to
determine whether our trendline is a good fit for the data,
and whether we can have confidence in our predictions.
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What does the R2 value mean?
• R2 = 1 indicates a perfect match between the trendline and the data.
• R2 = 0, indicates no linear relationship between the x and y variables.
• 0.7 < R2 < 1.0 indicates a possible strong linear relationship between
x and y variables, conditional on other guidelines.
• 0.4 < R2 < 0.7 indicates a possible moderate linear relationship and x
and y variables, conditional on other guidelines.
• If the R2 value is below .4, the linear relationship between x and y is
weak and you cannot use the trendline to make predictions.
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Even more on R-squared…
 The coefficient of determination, r 2, is
 The coefficient of determination
useful because it gives the proportion of
represents the percent of the data that is
the variance (fluctuation) of one variable
the closest to the line of best fit. For
that is predictable from the other
example, if r = 0.922, then r 2 = 0.850,
variable.
which means that 85% of the total
variation in y can be explained by the
linear relationship between x and y (as
 It is a measure that allows us to
described by the regression
determine how certain one can be, in
equation). The other 15% of the total
making predictions from a certain
variation in y remains unexplained.
model/graph.
 The coefficient of determination is a
measure of how well the regression line
represents the data. If the regression line
passes exactly through every point on the
scatter plot, it would be able to explain all
 The coefficient of determination is such
of the variation. The further the line is
2
that 0 < r < 1, and denotes the strength
away from the points, the less it is able to
of the linear association between x and
explain.
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y.
 The coefficient of determination is the
ratio of the explained variation to the
total variation.
Calculating the coefficient of determination
The mathematical formula for computing r is:
where n is the number of pairs of data.
To compute r2, just square the result from the above formula.
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NOW BACK TO OUR GUIDELINES FOR
DETERMINING WHETHER A
TRENDLINE IS A GOOD FIT FOR THE
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DATA...
Guideline 3: Verify that your trendline fits
the shape of your graph.
• For example, if your trendline continues
upward, but the data makes a downward turn
during the last few years, verify that the
“higher” prediction makes sense (see practical
knowledge).
• In some cases it is obvious that you have a
localized trend. Localized trends will be
discussed at a later date.
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Guideline 4: Look for outliers
• Outliers should be investigated carefully. Often they
contain valuable information about the process under
investigation or the data gathering and recording
process. Before considering the possible elimination of
these points from the data, try to understand why they
appeared and whether it is likely similar values will
continue to appear. Of course, outliers are often bad
data points. If the data was entered incorrectly, it is
important to find the right information and update it.
• In some cases, the data is correct and an anomaly
occurred that partial year. The outlier can be removed
if it is justified. It must also be documented.
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Guideline 5: Practical Knowledge,
Common Sense
• How many years out can we predict?
• Based on what you know about the topic,
does it make sense to go ahead with the
prediction?
• Use your subject knowledge, not your
mathematical knowledge to address this
guideline.
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Adding a Trendline
Using Excel
• Open the file: MileRecordsUpdate.xls and
calculate the slope (rate of change) in column
C.
• Is this women’s data perfectly linear?
• No, there is not a constant rate of change.
(See table below.)
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Calculating rate of change
Date
1967
1969
1971
1973
1979
1981
1985
1989
1996
Women's
Record
seconds
Change
277
276
275
269
262
261
257
255
253
-0.50
-0.50
-3.00
-1.17
-0.50
-1.00
-0.50
-0.29
Graphing the data produces the following graph which confirms that the
data is not perfectly linear. To graph data, highlight the data you want to
graph (not headers or empty cells). Choose a chart type: Under the Insert
tab click on Scatter located under the Charts group. Under Scatter, choose
Scatter with only Markers (the first option). A simple graph is created.
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280
275
270
265
Series1
260
255
250
1965
1970
1975
1980
1985
1990
1995
2000
We can clearly see that the data is not linear but we can use a linear model to
approximate the data. You will need to add a title, axis labels and trendline
(including the equation and r-squared value). First click on the graph to activate the
Chart Tools menu and then choose the Design tab. Under the Charts Layout group,
select #9. (Click on the "more" arrow to display all eleven layouts. Slide over each
layout until you locate #9.) Your Prepared
graphbyshould
look like this:
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Chart Title
280
275
Axis Title
270
265
Series1
260
Linear (Series1)
255
250
y = -0.929x + 2103.4
R² = 0.9342
245
1965
1970
1975
1980
1985
1990
1995
2000
Axis Title
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• Click on the Chart Title and add a descriptive title (consider who,
what, where and when). Click on each Axis Title and label both your
x-axis (horizontal axis) and your y-axis (vertical axis). If you are
graphing only one series of data, always be certain to remove the
legend (just click on the legend and use either the delete or
backspace button). To move the equation/r-squared value slide on
the text box containing both the equation/r-squared value. Once
your cursor changes to "cross-hairs" press on the left mouse button
and slide the text box to a location on the graph where it is easier to
read.
• It is suggested that you remove the minor axis gridline by changing
them to the same color as your background. Right-click on the y-axis
(vertical axis), choose Format Minor Gridlines then Solid Line. Change
the color of the line to match your background (currently your
background is white).
• It is important to add a text box stating the data source used to create
the graph. Under the Insert tab choose text box under the Text
group. Draw a text box on your chart and then type in "Source:"
followed by the data source. If no data source is listed, type
"Unknown".
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Women's World Records in the Mile Run from 1967 through
1996
280
275
y = -0.929x + 2103.4
R² = 0.9342
Record in seconds
270
265
260
255
250
245
1965
1970
Source: USA Track and Field
1975
1980
1985
1990
1995
2000
Year
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In the preceding graph…
• The black trendline is the line that “best fits” the data. It is a line
that comes as close the all the data points as possible.
• The R2 value indicates how linear the data actually is. The R2 value
will be a decimal between 0 and 1. The closer it is to one, the
closer the data is to linear. The smaller the R2 value, the less linear
the data. We can see here that the R2 value for the women’s mile
record is .9342 which is very close to one, so the data is very close
to linear.
• The equation is the equation of the trendline in y = mx + b form.
We can see that the slope or the rate of change of the trendline is .929 which means that according to the trendline, the mile record is
decreasing by just under 1 second every year.
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Always use excel functions (slope, intercept) and
cell references, never use the equation given in
the graph, to calculate future predictions
You learned in class to use the =slope() and
+intercept() functions. You should use the slope and
intercept functions when you are modeling and
calculating predictions because the equation that
Excel puts on the graph is often rounded to only a
few decimal places. Using the equation that Excel
puts on the graph can lead to aberrant results
because of this rounding.
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Why do we add a trendline and how do we use it?
• Since the trendline is an approximation what is
happening with data, we can use it to make
predictions about the data.
• For example, to predict what the mile record was
in 1999, use the equation of the trendline. First
identify the variables. X is year and Y is record in
seconds. Calculate slope and intercept on Excel.
Then plug 1999 in for X in the linear equation and
solve for Y.
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Five guidelines to see if the trendline a good fit for the data
• Guideline 1: Do you have at least 7 data points?
• Guideline 2: Does the R2 value indicate a relationship?
Reminder: R2 is the percentage of variance of y that is explained
by our trendline. It is a standard measure of how well the
trendline fits the data.
• Guideline 3: Verify that your trendline fits the shape of your graph.
• Guideline 4: Look for outliers
• Guideline 5: Use practical knowledge/ common sense to evaluate
your findings
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Justifying your prediction in words
Once we calculate the answer to the question, we cannot simply report the numbers. We
need to present them in meaningful sentences that explain their meaning in their contexts.
SAMPLE LEAD SENTENCES
“If the trend established from 1967- 1996 persists, we expect the Women’s world record to be
----------- seconds in 1998. “
SUPPORTING SENTENCES
“We are confident in our prediction because the r-squared value of ---------- shows that the
data has a strong/ moderate/weak linear relationship.
Even though in the long term we expect the rate of change in women’s mile records to
decrease and not stay constant, we expect that in the very near future the linear trend should
continue, giving us confidence in our prediction.
ITEMS THAT MUST BE POINTED OUT WHEN APPLICABLE
Reason for using less than 7 data points.
Omitting any single data point.
Focusing on a localized linear trend.
Continuing to predict a higher amount when they trend actually decreases (or the opposite).
Olympic Record Evolution
forbyWomen’s
1500m Olympic race. 40
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Women’s World Record in 1500 m. since 1967
TIME
NAME ATHLETE
DATE
PLACE
4:17.3+
Anne Rosemary Smith (GBR)
1967-06-03
Chiswick, Great Britain
4:15.6
Maria Gommers (NED)
1967-10-24
Sittard, Netherlands
4:12.4
Paola Pigni (ITA)
1969-07-02
Milan, Italy
4:10.77*
Jaroslava Jehličková (CZE)
1969-09-20
Athens, Greece
4:09.62*
Karin Burneleit (GDR)
1971-08-15
Helsinki, Finland
4:06.9
Ludmila Bragina (URS)
1972-07-18
Moscow, Soviet Union
4:06.47*
Ludmila Bragina (URS)
1972-09-04
Munich, Germany
4:05.07*
Ludmila Bragina (URS)
1972-09-07
Munich, Germany
4:01.38*
Ludmila Bragina (URS)
1972-09-09
Munich, Germany
3:56.0
Tatyana Kazankina (URS)
1976-06-28
Podolsk, Soviet Union
3:55.0
Tatyana Kazankina (URS)
1980-07-06
Moscow, Soviet Union
3:52.47
Tatyana Kazankina (URS)
1980-08-03
Zurich, Switzerland
3:50.46
Qu Yunxia (CHN)
1993-09-11
Beijing, China
Adding a Trendline
(in Excel 2007)
• Open the file: MileRecordsUpdate.xls and calculate the slope (average
rate of change) in column H for Men’s World records in the Mile Run.
• Is this men’s data perfectly linear?
• Can you use a linear model to describe the data? (Hint: Graph the data in a
simple scatter plot)
• Create a graph with a trendline, title your graph appropriately.
• What would the men’s world record be in the year 2000? (Hint: in your
calculations you need to use the SLOPE and INTERCEPT Excel functions,
and use the linear equation.)
• Check you answer by extending the trendline to year 2000. (right click on
trendline, under forecast, increase it forward by number of units you need
to, to reach 2000). Does your trendline show a similar number as your
prediction.
• Once you calculate your answers write your answers our in meaningful
sentences, justifying your prediction in words. (Hint: report your
prediction, the R-squared value, and any possible caveats.)
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