Transcript LSP 120: Quantitative Reasoning and Technological Literacy

LSP 120: Quantitative Reasoning and
Technological Literacy
Section 202
Özlem Elgün
Why are we here?
• 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
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 .
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.
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)
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.
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
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
• 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.
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
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.
C
Rate of
Change
2.5
2.5
2.5
2.5
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
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?
t
P
1980
67.38
1981
69.13
1982
70.93
1983
72.77
1984
74.67
1985
76.61
1986
78.60
• 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.
t
P
1980
67.38
1990
87.10
2000
112.58
2010
145.53
2020
188.12
2030
243.16
2040
314.32
"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%
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.
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?
t
P
1980
67.38
1981
69.13
1982
70.93
1983
72.77
1984
74.67
1985
76.61
1986
78.60
• 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.
t
P
1980
67.38
1990
87.10
2000
112.58
2010
145.53
2020
188.12
2030
243.16
2040
314.32