AP Statistics – 12.2 Notes Name: ____________________ Transforming to Achieve Linearity

Transforming with Powers and Roots 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

Transforming with Powers 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.

Transforming with Roots Another way to transform the data to achieve linearity is to take the cube root of the weight values and graph the cube root of weight versus length. Transforming with Powers and Roots Summary When experience or theory suggests that the relationship between two variables is described by a power model of the form y = axp, you now have two strategies for transforming the data to achieve linearity.

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, √𝑦𝑝 )

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AP Statistics – 12.2 Notes Name: ____________________ Transforming to Achieve Linearity

Example: Power Models and Logarithmic Transformations On July 31, 2005, a team of astronomers announced that they had discovered what appeared to be a new planet in our solar system. They had first observed this object almost two years earlier using a telescope at Caltech’s Palomar Observatory in California. Originally named UB313, the potential planet is bigger than Pluto and has an average distance of about 9.5 billion miles from the sun. (For reference, Earth is about 93 million miles from the sun.) Could this new astronomical body, now called Eris, be a new planet? At the time of the discovery, there were nine known planets in our solar system. Here are data on the distance from the sun and period of revolution of those planets. Note that distance is measured in astronomical units (AU), the number of earth distances the object is from the sun.

The graphs below show the results of two different transformations of the data.

a. Explain why a power model would provide a more appropriate description of the relationship between period of revolution and distance from the sun than an exponential model.

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The seaHerplot of lnperiod vs distance is clearly curved so anexponentialmodelwouldnotbe appropriate

However the graphof lnperiod vs Indistance has a stronglinearPattern indicating that apowermodelwould be moreappropriate

AP Statistics – 12.2 Notes Name: ____________________ Transforming to Achieve Linearity

b. Minitab output from a linear regression analysis on the transformed data is shown below. Give the equation of the least-squares regression line. Be sure to define any variables you use.

c. Use your model from part (b) to predict the period of revolution for Eris, which is 9,500,000,000/93,000,000 = 102.15 AU from the sun. Show your work.

d. A residual plot for the linear regression in part (b) is shown below. Do you expect your prediction in part (c) to be too high, too low, or just right? Justify your answer.

Power Models vs Exponential Models • Power Model is when you take the logarithm of both the explanatory and response variable:

log(𝑦 𝑣𝑎𝑟𝑖𝑎𝑏𝑙𝑒) = log(𝑥 𝑣𝑎𝑟𝑖𝑎𝑏𝑙𝑒)

• Exponential Model is when you take the logarithm of the response variable only: log(𝑦 𝑣𝑎𝑟𝑖𝑎𝑏𝑙𝑒) = 𝑥

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n p 0.0002544 t l49986Indistance

Review

logax bn p d 0.0002544 t l49986In 102.15 abIn period 6.939

6.939 InX b logex bperiod e

e xperiod 1032years

Eri's valuefor In distance isIn 102.15 4.626

Since4.626would fall to the rightoftheresidualplot we mayexpecttheresidual to bepositive Thismeansthepredictionwould be toolow

AP Statistics – 12.2 Notes Name: ____________________ Transforming to Achieve Linearity

Transformations on the Calculator How is the braking distance for a motorcycle related to the speed the motorcycle was going when the brake was applied? Statistics teacher Aaron Waggoner gathered data to answer this question. The table and scatterplot below shows the speed (in miles per hour, or mph) and the distance (in feet) needed to come to a complete stop when the brake was applied.

Speed (mph) Distance (feet) 6 1.42 9 4.92

19 18.00 30 44.75 32 52.08 40 84.00 48 110.33

1. Does a power model or an exponential model provide a more appropriate description of the relationships between braking distance and speed? Compare scatterplots of the appropriate transformations.

2. Give the equation of the least-squares regression line. Be sure to define any variables you use. 3. Use your model from question 2 to predict the braking distance if the motorcycle was going 55 miles

per hour. Show your work. 4. A residual plot for the linear regression from question 2 is shown below. Do you expect your

prediction from question 3 to be too high, too low, or just right? Justify your answer.

ThescalterplotofLogdistance vs speed is clearlycurved so anexponential model wouldnot be appropriate However thegraphoflogdistance vs togspeed is roughlylinearindicatingthat apowermodelwouldbe moreappropriate seegraphsfromCaleabove

logdistance 1.35063 t 2.036101logspeed

Review

logX Y logfdistance 1.35063 t 2.036101log 55logoX y logdistance _2.193

102.19310K distancedistance 155.96

log55 1.74Sincethereis no obvious pattern in thethe prediction should be justaboutright In otherwords we cannotestimatewhether the prediction will be toohighor too low