Chapter Four Day Two

Power Models

P. 285 11,12,13

Homework

Show that if y = a*bx taking then there is a linear relationship between x and log(y).

Review of Exponential Models

Make scatterplot and note very strong non-linear form.

Take the log of the y-values and put the results in L3.

Do a linreg on L1 vs. L3 Write log(y) = bx + a Untransform to get final exponential model

Review of Exponential Models

Untransform log(y) = ax + b

Example

Hierarchy of Powers

y = ax linear Y = ax2 quadratic Y = ax3 cubic Y = ax4 quadratic Y = ax5 5th degree

For large x axb < abx for any b

Power Models y = a* xb

Show that if y = abx then there is a linear relationship between log x and log y.

Example

Untransform log y = alog(x) + b

Example

Plot data and note nonlinear form Put log of the x-values in L3 Put log of y –values in L4 Do linreg on log x vs log y Write log(y) = a(log x) +b Untransform to get final power model

Steps to Making a Power Model

Find a model that predicts a planet’s period of revolution using the distance from the sun as an explanatory variable.

Example - Planets

Planet Distance from Sun (AU)

Period of Revolution (Earth years)

Mercury .387 .241Venus .723 .615Earth 1.000 1.000Mars 1.524 1.881Jupiter 5.203 11.862Saturn 9.539 29.456Uranus 19.191 84.070Neptune 30.061 164.810Pluto 39.529 248.530

Try an exponential model – linear relationship between x and log y

Try a Power Model – Linear Relationship between log(x) and log(y)

ln(period) = .000254 + 1.50 ln(distance)

Untransform linear log-log Model to get final power model

Evaluating a Model Comment on r2

Comment on residual Plot