Transforming to Achieve Linearity Lesson 3: Section 12.2 (part 1)

objectives Use transformation involving powers and roots to achieve linearity for a relationship between two variables.

What if the data is nonlinear?? In Chapter 2, we discussed transforming the data by changing the scale of measurement that was used when the data were collected. Examples: Changing temperature to Fahrenheit to Celsius Changing inches to centimeters In this section, we are going to apply functions, such as square root or logarithmic functions, to straighten out nonlinear patterns.

Transforming with Powers and Roots Some situations in real life do not follow a linear pattern - some follow a power model with the form y = axp . Examples: 1.) The distance that an object dropped from a given height falls is related to time since release by the model: distance = a(time)2 2.) The time it takes a pendulum to complete one back-and-forth swing (period) is related to its length by the model: period = or 3.) The intensity of a light bulb is related to distance from the bulb by the model: intensity = a/distance2 or a(distance)-2 In this case (x, y) does NOT have a linear relationship, but (xp, y) does! Examples 1.) (x2, y) or (x, 𝒚 ) 2.) ( 𝑥 , y) or (x, y2) 3.) (x-2, y) or (1/x2, y) or (x2, 1/y) or (x, 1/ 𝒚 )

EXAMPLE 1: Go Fish! Imagine 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.

EXAMPLE 1: Go Fish! 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. (a) Look at the scatterplot of the data. Describe what you see. Answer: The relationship between length (cm) and weight (g) is positive and curved.

EXAMPLE 1: Go Fish! (b) Because length is one-dimensional and weight (like volume) is 3-dimensional, a power model of the form weight = a(length)3 What happened to the graph below to transform it to a linear pattern? Answer: Graphed (x3, y) or weight vs. length3

EXAMPLE 1: Go Fish! (c) Besides graphing (x3, y) or weight vs. length3 , how else could we have achieved linearity? What happened to the graph below to transform it to a linear pattern? Answer: Graphed(x, ) or vs. length.

EXAMPLE 2: Child Mortality and income What does a country’s mortality rate for children under five years of age (per 1000 births) tell us about the income per person (measured in gross domestic product per person adjusted for difference in purchasing power) for residents of that country? Here are the data and a scatterplot for a random sample of 14 countries in 2009 (data from www.gapminder.org)

EXAMPLE 2: Child Mortality and income The scatterplot shows a strong negative association that is clearly nonlinear. Because the horizontal and vertical axes look like asymptotes, perhaps there is a reciprocal relationship between the variables, such as: income per person = a(childmortalityrate)-1 or income per person = a/(childmortalityrate) How can we find linearity in this situation? (explain the 2 graphs below) Graph of 1/income per person vs. child mortality rate Graph of income vs. 1/child mortality rate

EXAMPLE 3: Child Mortality and income Here is the Minitab output from separate regression analyses of the two sets of transformed data. Transformation 1: (1/under 5, income) Transformation 2: (under 5, 1/income)

EXAMPLE 3: Child Mortality and income Do the following for both transformations: (a)Give the equation of least-squares regression line. Define any variables you use. Transformation 1: 𝑖𝑛𝑐𝑜𝑚𝑒 =869+ 104868 𝑢𝑛𝑑𝑒𝑟 5 Transformation 2: 1 𝑖𝑛𝑐𝑜𝑚𝑒 = −0.00000585+0.00000829(𝑢𝑛𝑑𝑒𝑟 5) (b) Predict the income per person for Turkey, which had a child mortality rate of 20.3. Transformation 1: 𝑖𝑛𝑐𝑜𝑚𝑒 =869+ 104868 20.3 =$6034.91 Transformation 2: 1 𝑖𝑛𝑐𝑜𝑚𝑒 = −0.00000585+0.00000829(20.3) =$6156.23

(c) Interpret the value of r2 in context. Transformation 1: 84.3% of the variation in income is accounted for by the least-squares regression line using x=1/under5 Transformation 2: 92.1% of the variation in 1/income is accounted for by the least-squares regression line using x = under 5 mortality rate.

homework Assigned reading: p. 765-771 Complete HW problems: Check answers to odd problems. Study for a QUIZ on Section 12.1 next class