ISP 120

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ISP 120

• Linear Models
• and Trendlines

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Linear Relationships

• What makes a graph or function or table of values
  linear?
• For a fixed change in x, there is a fixed change
  in y, or
• The change in y per unit change in x is constant,
  or
• There is a constant rate of change

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Linear Relationships

• Look at a set of values. Is it a linear
  relationship? Apply (B3-B2)/(A3-A2)

A B
2 4 25 3 5 50 4 6 75 5 7
100 6 8 125 7 9 150
(B3-B2)/(A3-A2) (50-25)/(5-4)
25 (B4-B3)/(A4-A3) (75-50)/(6-5)
25 (B5-B4)/(A5-A4) (100-75)/(7-6)
25 (B6-B5)/(A6-A5) (125-100)/(8-7)
25 (B7-B6)/(A7-A6) (150-125)/(9-8) 25
All the results are the same (25), so this is a
linear set of values.
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Linear Relationships

• If it is linear, what is the function?
• Recall y mx b
• m is the slope, or the (change in y) / (change in
  x)
• b is the y intercept
• So calculate the slope
• Then plug in slope and first x and y values into
  ymxb and solve for b

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Linear Relationships
A B
2 4 25 3 5 50 4 6 75 5 7
100 6 8 125 7 9 150
(B3-B2)/(A3-A2) (50-25)/(5-4)
25 (B4-B3)/(A4-A3) (75-50)/(6-5)
25 (B5-B4)/(A5-A4) (100-75)/(7-6)
25 (B6-B5)/(A6-A5) (125-100)/(8-7)
25 (B7-B6)/(A7-A6) (150-125)/(9-8) 25
y mx b m change in y / change in x 25/1
25 25 25 4 b b -75 (this is the
y-intercept) y 25x 75 This is the
equation of the line
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Examples
x y 5 -4 10 -1 15 2 20 5
x y 0 1 2 4 4 16 6 36
x y 3 5 4 9 6 11 9 17
mchange y / change x 3/50.6 y mxb
-40.65b b-7 y 0.6x - 7
x y Rate of Change 3 5 6 9 (B3-B2)/(A3-A2) 9 13 1
2 17
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Linear Relationships

• Linear growth occurs when a quantity grows by
  the same absolute amount
• Exponential growth occurs when a quantity grows
  by the same relative amount that is by the same
  percentage in each unit of time
• There is also linear decay and exponential decay

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Examples

• The number of students at Wilson High School has
  increased by 50 in each of the past four years.
• The price of milk has been rising with inflation
  at 3 per year.
• Tax law allows you to depreciate the value of
  your equipment by 200 per year.
• The memory capacity of computer hard drives is
  doubling approximately every two years.
• The price of DVD records has been falling by
  about 25 per year.

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Trendlines

• Real data is seldom perfectly linear
• Suppose the data is reasonably linear how does
  one make a linear model of the data?
• The standard approach is to use a best-fit line

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Trendlines

• How was this line created?
• Least squares regression
• Create a line f(x)mx b, such that the sum of
  the squares of (vertical) errors is as small as
  possible.

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Trendlines

• If R2 1, then 100 of the variance in y is
  explained by the line, so we have a perfect fit.
• If R2 0, then we have a terrible fit.

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Trendlines

• Do we have to do this calculation?
• No, Excel can do it for you.
• Consider the dataset MileRecords.xls
• Graph the data using an XY Scatter. After the
  graph is done, right click on line, select Add
  Trendline, Options
• Can you make a prediction? Be careful! Ones
  confidence in predictions made far from the data
  must be tempered.