LSP 120: Quantitative Reasoning and Technological Literacy Section 903

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LSP 120 Quantitative Reasoning and Technological
Literacy Section 903

• Özlem Elgün

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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

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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 wont 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

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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 (B3-B2)/(A3-A2)
4 7 21  
5 9 26  
6 11 31  
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. 
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• 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.

  A B C
1 x y Rate of Change
2 3 11  
3 5 16 2.5
4 7 21 2.5
5 9 26 2.5
6 11 31 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
  x0)
• We know x, y, and m, we need to calculate b
• Using the first set of values (x3 and y11) and
  2.5 for "m (slope)
• 112.53 b. 
• Solving  117.5 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
  Excels intercept function. 

  A B C
1 x y Rate of Change
2 3 11  
3 5 16 2.5
4 7 21 2.5
5 9 26 2.5
6 11 31 2.5
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PracticeFor the following, determine whether the
function is linear and if so, write the equation
for the function. 
x y
5 -4
10 -1
15 2
20 5
x y
1 1
2 3
5 9
7 18
x y
2 20
4 13
6 6
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?

t P
1980 67.38
1981 69.13
1982 70.93
1983 72.77
1984 74.67
1985 76.61
1986 78.60
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• 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
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"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 BAC
0 0.018
1 0.010
   
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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.
Time BAC Rate of change
0 0.018
1 0.010 -0.008
 
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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?

t P
1980 67.38
1981 69.13
1982 70.93
1983 72.77
1984 74.67
1985 76.61
1986 78.60
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• 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