please put the following data in your calculator....
TRANSCRIPT
Please put the following data
in your calculator. Thanks!Volume
(cm3)Pressure
(atmospheres)
6 2.9589
8 2.4073
10 1.9905
12 1.7249
14 1.5288
16 1.349
18 1.2223
20 1.1201
Transforming to
Achieve
Linearity12.2!
Gapminder.orgMission: “Fighting devastating ignorance with
fact-based worldviews everyone can
understand” Gapminder was founded in Stockholm on February 25, 2005, by Ola Rosling,
Anna Rosling Rönnlund and Hans Rosling. In 2006, Hans Rosling held his first TED talk on “The best statistics you’ve never seen”. It became one of the most seen TED talks ever, thanks to its unique combination of knowledge-testing, animating bubble charts and storytelling about global development.
The animated bubble graph is a software called Trendalyzer, which Gapminder had developed to make global public data understandable. In 2007 Google acquired Trendalyzer and the team of developers moved to Google’s headquarters in California. Over three years the team improved the user experience for search and exploration of global public data. In 2010 Anna and Ola decided to leave Google and return to Gapminder to develop free teaching material. In order to prioritize what content to include in the teaching material, they started measuring public knowledge (or rather the lack of knowledge) with the Ignorance project. They soon realized that spreading facts was not enough since the problem was bigger. People had an incorrect dramatic worldview.
The Factfulness project was born to help people get a fact-based worldview. Since the world is better than most people think, a fact-based worldview actually reduces stress and anxiety. Using Factfulness you can dismantle the misconceptions that shape the overdramatic worldview.
How would you describe the relationship
between wealth and life expectancy?
As the years go on, in general, what is
happening?
Why do we need a regression
line?
Regression lines are used to predict linear
data, by using an equation to model the
relationship between two variables.
Other models that could fit data:
Power models
Exponential Models
Trigonometric Models
But if data’s not linear, how can we
apply a regression line?
Straightening Curved Data
To straighten curved patterns on a
scatterplot, quantitative variables can be
transformed.
This is when the scale of measurement is
changed using a function such as a square
root, reciprocal, or logarithm to one or both
quantitative variables.
Example 1:
Boyle’s Law (p785, #32)If you have taken a chemistry or physics class, then you are probably familiar with Boyle’s law: for gas in a confined space kept at a constant temperature, pressure times volume is a constant (in symbols, PV = k). Students collected the following data on pressure and volume using a syringe and a pressure probe.
Make a scatterplot of this data.
Volume (cm3)
Pressure (atmospheres)
6 2.9589
8 2.4073
10 1.9905
12 1.7249
14 1.5288
16 1.349
18 1.2223
20 1.1201
Transforming Method #1: using powers/roots
Using PV = k, algebraically isolate…
P V
Transformation 1: ( 1/V, P )
Find reciprocal of every
explanatory variable, keep response the same
Transformation 2: ( V, 1/ P )
Find reciprocal of every response
variable, keep explanatory the same
Using your calculators, keep L1, L2 as is, and then…
L3: type in as 1/L1
Your scatterplot will be for
(L3, L2)
Label the axes below for the
scatterplot, make a rough
sketch of the scatterplot and
comment what you see.
• L4: type in as 1/L2
• Your scatterplot will be for (L1, L4)
• Label the axes below for the
scatterplot, make a rough sketch
of the scatterplot and comment
what you see.
Continued…Do a residual plot for each to determine if a line is the best
model for the transformed data.
Using your calculators, determine the equation for the least-
squares regression line for these
transformations…remember your units are no longer the
same! Define any variables used. L3: type in as 1/L1
Your scatterplot will be for
(L3, L2)
Label the axes below for the
scatterplot, make a rough
sketch of the scatterplot and
comment what you see.
• L4: type in as 1/L2
• Your scatterplot will be for (L1,
L4)
• Label the axes below for the
scatterplot, make a rough
sketch of the scatterplot and
comment what you see.
Use the model to predict the pressure in the syringe when
the volume is 17 cm3.
1.303 atmospheres 1.287 atmospheres
We perform specific transformations on
specific types of models that two variables
follow. The most popular are …
Power Models: (note, x is the
base)
Exponential Models: (note, x is
the exponent)
Many biological models are
described by using power
models
Abundance of species follows
a power model, with x = body
weights
With x as the explanatory,
the same relationship is
seen with pulse rate, life
length, number of eggs a
bird lays, etc.
Populations grow
exponentially (uninterrupted)
Compounded interest
Power vs. Exponential Model?
But what if we are unsure of the type of relationship between our
variables?
Transforming Method #2:
To transform either of these models, the best option is to take
logarithms.
If a power model describes the
relationship between two
variables, transform the
relationship by taking the
logarithm of both sides.
A scatterplot of logs of both
variables should produce a linear
pattern.
If an exponential model
describes the relationship
between two variable,
transform by taking the
logarithm of response variable
ONLY.
A scatterplot of the
explanatory variable and the
logarithm of the response
variable should produce a
linear pattern.
Back to Boyle
g. So…do you think when we transform the
equation for Boyle’s low, is it a Power model
or Exponential?
h. Give the equation of the least-squares
regression line, defining any variables you
use.
Which to choose?
The type of model that best represents a set
of nonlinear data can be deducted if we
transform it as if it was a power model and
then as an exponential: whichever
transformation creates a linear pattern
indicates which type of model it is.
Which to choose?If a power model best describes a linear relationship, plot log x and log y.
If an exponential model best describes a linear relationship, then plot x to log y.
For both, ask yourself:
is there a line in the scatterplot?
What does the residual plot of the explanatory to resididual supposed to look like?
Which to choose?
If both (log x, log y) and (x, log y) look
linear…
Fit least-squares regression line to both sets
of transformed data
Compare residual plots: look for most
random scatter
Use s (which is smaller?) and r2 (which is
bigger?)