Focus Fox handout Q’s 3, 6, 7 ,8, 10

Transforming DataTransformations can straighten a nonlinear pattern. Then, we can use a least-square regression line to make inferences and test a claim about the slope of the population.

Pg. 766-767

Transforming data is changing the scale of measurement that was used when data was collected.

Linear transformations cannot straighten curved relationship between two variables

Functions that are not linear can transform curved relationships to a linear model

Transforming DataWe will use relationships that we know have a power or root propertyPizza – 10’’, 12’’, 14’’ area of a pizza πr2

Distance an object dropped from different heights and time since release distance = a(time)2

Time it takes a pendulum to complete 1 swing and length of pendulum period = a = a(length)½

Intensity of light bulb and distance from the bulb intensity = = a(distance)-2

This is a power model y = axp with p being the power.

Transforming DataImagine that you have been put in charge of organizing a fishing tournament in which prizes will be given for the heaviest Atlantic Ocean rockfish caught. You know that many of the fish caught during the tournament will be measured and released. You are also aware that using delicate scales to try to weigh a fish that is flopping around in a moving boat will probably not yield very accurate results. It would be much easier to measure the length of the fish while on the boat. What you need is a way to convert the length of the fish to its weight.

You contact the nearby marine research laboratory, and they provide reference data on the length (in centimeters) and weight (in grams) for Atlantic Ocean rockfish of several sizes.

Transforming DataBecause length is one-dimensional and weight (like volume) is three-dimensional, a power model of the form weight = a(length)3 should describe the relationship. What happens if we cube the lengths in the data table and then graph weight versus length3?

Pg. 769

Length: 5.2 8.5 11.5 14.3 16.8 19.2 21.3 23.3 25.0 26.7

Weight: 2 8 21 38 69 117 148 190 264 293

Length: 28.2 29.6 30.8 32 33 34 34.9 36.4 37.1 37.7

Weight: 318 371 455 504 518 537 651 719 726 810

Transforming DataSimilarly, we could use a cube root to transform the data.

= length

This is the same relationship as weight = length3

Transforming DataGive the equation of least-squares regression line. Define any variables.

Transforming DataSuppose a contestant in the fishing tournament catches an Atlantic Ocean rockfish that’s 36 cm long. Use each model to predict the fish’s weight.

Transforming DataInterpret the value of s in context.

Transforming DataWhen experience on theory suggest that the relationship between two variables is described by a power model of the form y = axp, you have two strategies to transform the data:

1. Raise the values of the explanatory variable x to the p power and plot the points (xp, y)

2. Take the pth root of the values of the response variable y and plot the points ( x,