Section 12.2 Transforming to Achieve Linearity

Transforming Nonlinear relationships between two quantitative variables can sometimes be changed into linear relationships by transforming one or both of the variables. Transformation is particularly effective when there is a reason to think that the data are governed by some nonlinear mathematical model. Once we transform the data to achieve linearity, we can fit a least-squares regression line to the transformed data and use this linear model to make predictions.

Power Models When theory or experience suggests that the relationship between two variables follows a power model of the form 𝑦=𝑎 𝑥 𝑝 , there are two options… Option 1: Raise the values of the explanatory variable x to the power p, the look at a graph of ( 𝑥 𝑝 ,𝑦) Option 2: Take the pth root of the values of the response variable y, then look at a graph of (𝑥, 𝑝 𝑦 )

Example Imagine that you have been put in charge of organizing a fishing tournament in which prizes will be given for the heaviest fish caught. It would be easiest to measure the length of the fish on the boat, but you need a way to convert the length of the fish to its weight. You contact the nearby marine research lab and they provide reference data on several sizes of fish:

Transform power to linear Because 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.

Option 1: ( 𝑙𝑒𝑛𝑔𝑡ℎ 3 , 𝑤𝑒𝑖𝑔ℎ𝑡) Write the equation of the least-squares regression line: Suppose someone caught a fish that is 36 cm long. Predict their weight: Interpret s:

Option 2: (𝑙𝑒𝑛𝑔𝑡ℎ, 3 𝑤𝑒𝑖𝑔ℎ𝑡 ) Write the equation of the least-squares regression line: Suppose someone caught a fish that is 36 cm long. Predict their weight: Interpret s:

Exponential and Log Models A useful strategy for straightening a curved pattern in a scatterplot is to take the logarithm of one or both variables. First – try to take the log or ln of y…if that doesn’t give you a linear model…do both sides. If you log both – you are saying there is a power relationship between the two variables.

Example Suppose you invest $100 in a savings account that pays 6% interest compounded annually. The table shows your balance in the account after each of the first six years.

𝑦=100(1.06 ) 𝑥

Example One of the founders of Intel Corporation predicted in 1965 that the number of transistors on an integrated circuit chip would double every 18 months.

When I take the ln(y)

Minitab output from transformed data Linear eq: Predict the number of transistors in 2020.

Example On July 31, 2005 a team of astronauts announced they had discovered what might be a new planet. At the time of discovery, there were nine planets in our solar system. Here are the data on the distance from the sun and period of revolution of those planets.

If I take the ln(y)

So I have to take the ln of both sides

Find the least-squares regression line Eq: Predict the period of revolution for Eris (the “new planet”), which is 102.15 AU from the sun and show your work.

Homework Pg. 787 (35, 37-39, 42, 45-48)