Chapter FourDay One

Transforming Data

P. 265 2,4 P. 276 5,7,9

Homework

Make a scatterplot of data

Note non-linear form

Think of a “common-sense” relationship

First Steps to Achieving Linearity

Average length and weight of Atlantic Ocean rockfish

Example

Age(years)

Length(cm)

Weight(g)

Age Length

Weight

1 5.2 2 11 28.2 3182 8.5 8 12 29.6 3713 11.5 21 13 30.8 4554 14.3 38 14 32.0 5045 16.8 69 15 33.0 5186 19.2 117 16 34.0 5377 21.3 148 17 34.9 6518 23.3 190 18 36.4 7199 25.0 264 19 37.1 72610 26.7 293 20 37.7 810

Achieving Linearity Now lets graph length3 vs. weight

L1 = length L2 = weight L3 = length3

Plot L3 vs. L2 Perform linreg on L3 vs. L2

weight = 4.066 + 0.0147(length)3

Now report r and r2

Make residual plot

Model

Types of Nonlinear Models Exponential Models y = bx

Use an exponential model if there is a linear relationship between x and log y.

Power Models y = xb

Use a power Model if there is a linear relationship between log x and log y.

Y = logb x if and only if by = x

Evaluate

Log10100 =

Log 2 8 =

Log 3 1/9 =

Review of Logartithms

logx 125 = 3

Log 4 x = 4

Solve each equation

Logb (MN) = logb M + logb N

Logb (M/N) = logb M – logb N

Logb Mp = p logb M

Laws of Logarthims

Exponential Models Show that if y = abx, then there is a linear

relationship between x and log y

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 (x versus log y) Write log(y) = bx + a Untransform to get final exponential model

Review of Exponential Models

Log y = a + bx

Untransform

Date(years since 1970)

Number of Transistors

Date Number of Transistors

1 2,250 19 1,180,0002 2,500 23 3,100,0004 5,000 27 7,500,0008 29,000 29 24,000,00012 120,000 30 42,000,00015 275,000

Building an Exponential Model