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 28.2 318

2 8.5 8 12 29.6 371

3 11.5 21 13 30.8 455

4 14.3 38 14 32.0 504

5 16.8 69 15 33.0 518

6 19.2 117 16 34.0 537

7 21.3 148 17 34.9 651

8 23.3 190 18 36.4 719

9 25.0 264 19 37.1 726

10 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 1,180,000

2 2,500 23 3,100,000

4 5,000 27 7,500,000

8 29,000 29 24,000,000

12 120,000 30 42,000,000

15 275,000

Building an Exponential Model