Lesson Quiz: Part I 1. Change 6 4 = 1296 to logarithmic form. log = 4 2. Change log 27 9 = to exponential form = log 100, log log 3 Calculate the following using mental math –3

6. Use the x-values {–2, –1, 0, 1, 2, 3} to graph f(x) =( ) X. Then graph its inverse. Describe the domain and range of the inverse function. 5 4 Lesson Quiz: Part II D: {x > 0}; R: all real numbers

More on relationships between two variables Unit 5-2 Transforming to achieve linearity

Body and brain weight of 96 species of mammals For this data, r = 0.86, but why might we not trust the given correlation? If we remove the elephant, the correlation changes to r = 0.5!

Body and brain weight of 96 species of mammals Here is a close up of the blob in the lower-left corner. Is the data linear? The data is not exactly linear- notice the data bends to the right as body weight increases.

Biologists know that data on sizes often behave better if we take logarithms before doing more analysis. This plot graphs the logarithm of brain weight against the logarithm of body weight for all 96 species. How does our data look now?

Applying a function such as the logarithm or square root to a quantitative variable is called transforming or re-expressing the data.

Why transform? And so you ask, why would we transform our data?

Why transform? To make the distribution of a single variable (as seen in a histogram, for example) more symmetric. To make the spread of several groups (as seen in side-by-side boxplots) more alike. To make the form of a scatterplot more nearly linear (as seen in the previous example). Make the scatter in a scatterplot spread out evenly rather than following a fan shape. In this chapter, we'll focus on the third reason.

Common transformations Transformations we may use include raising our data to a power (like squared or cubed), square rooting our data, taking the logarithm of our data, or taking the reciprocal of our data.

Common transformations The situation may help us know which transformations will best achieve linearity. For example... A problem dealing with area might benefit from squaring the data (power of 2) since area involves square units. A problem dealing with weight or volume might benefit from cubing or cube-rooting (a power of 3 or one-third) the data since volume involves cubic units. Data involving a ratio (like miles per gallon) might benefit from a reciprocal transformation (power of -1).

Example 4.2 This example has data comparing the lengths and weights of fish, and asks us to find a model that helps us predict the weight of a fish given its length.

Weight versus length of fish Here's a graph of the data. Describe the form of the data. Since the data is not linear, we want to try a transformation that will make it linear.

Common transformations Which transformation should we try? A problem dealing with area might benefit from squaring the data (power of 2) since area involves square units. A problem dealing with weight or volume might benefit from cubing or cube-rooting (a power of 3 or one-third) the data since volume involves cubic units. Data involving a ratio (like miles per gallon) might benefit from a reciprocal transformation (power of -1).

Weight versus length 3 Notice what happens to our graph when we cube all our lengths. Our form is now linear.

Weight versus length 3 The least-squares regression line is weight = length 3 with r 2 = Would you feel comfortable using this model for prediction? Notice our explanatory variable is length 3, because we cubed all our lengths.

Weight versus length 3 What can you say about the residual plot? Despite the slight pattern in the residual plot, the residuals themselves are quite small compared to the hundreds of grams we were measuring our fish in. We should be safe using our LSRL for prediction.

Prediction So to predict the weight of a fish with a length of 36 centimeters, plug 36 into our LSRL weight = length 3 weight = (36) 3 weight = grams

The ladder of powers A review of functions When transforming with powers (like in the last example), a general understanding of different power functions can sometimes help, since we could use any of these powers in transforming our data.

The ladder of powers A review of functions The power of 1 graph is a straight line

The ladder of powers A review of functions Powers greater than one (like 2 and 4) give graphs that bend upward.

The ladder of powers A review of functions Powers less than 1 but greater than 0 (like 0.5 or the square root) give graphs that bend downward.

The ladder of powers A review of functions Powers less than zero (like -1 or the reciprocal transformation) give graphs that decrease as x increases.

The ladder of powers A review of functions The zero power in the ladder is replaced by the graph of logx.

A country's GDP & life expectancy So let's say we were looking at a graph such as this, which compares a country's gross domestic product and life expectancy, and we wanted to linearize the data.

A country's GDP & life expectancy There isn't an obvious relationship between GDP and life expectancy like there was between length & weight, so just start somewhere on the ladder and move down.

A country's GDP & life expectancy Here's our data to the power of 0.5, or in other words square rooted. Compare our new r value to the old. How linear is the data?

A country's GDP & life expectancy We could do better, so let's go down the ladder another step to see what happens.

A country's GDP & life expectancy Here's the log of our data (which takes the power of 0 on the ladder). Compare our new r value to the old. How linear is the data? Let's go one more step on the ladder.

A country's GDP & life expectancy Here's our data to the power of -0.5, or in other words the reciprocal square rooted. Compare our new r value to the old. How linear is the data?

A country's GDP & life expectancy I'm sure you noticed that as we moved down the ladder of powers, the scatterplots became straighter. This final plot has a fairly linear form apart from the outliers.

NOTE Although this guess and check method ultimately accomplished the goal of achieving linearity, the ladder of powers is rarely used in practice. It is much more satisfactory to begin with a theory or mathematical model that we expect to describe a relationship, (as in the length and weight of fish example.)‏ Also note that not all data will become linear with a transformation.

Transformations on the TI We will use the next example to show YOU how to perform your own transformations on your calculator, as well as to make a very important point about a particular type of model.

More on Moore's law You may recall last time talking about Moore's law, which predicted in 1965 that the number of transistors on an integrated circuit chip would double every 18 months

More on Moore's law Enter the data into L1 and L2 on your calculator (stat, edit) and construct a scatterplot of the data (stat plot). You may need to adjust your window to show all the data.

Your plot should have looked like this We will answer two questions 1.What transformation will linearize the data? 2.Is the data exponential?

The answer to question 1: Taking the log of the response variable will linearize the data The answer to questions 2: the data is exponential Verifying exponential growth In fact, logs can be used to verify exponential growth, because the log of exponential data will always produce a linear relationship!

Verifying exponential growth In other words, if our data are growing exponentially and we plot the logarithm (base 10 or base e) or y against x, we should observe a straight line for the transformed data.

Verifying exponential growth Go back to our calculator data on Moore's law. To perform the transformation, go back to your lists (stat, edit). Highlight L3 by scrolling all the way up in that column. With L3 highlighted, you can type in a formula. Type “ln (L2)” and hit enter. This takes the natural log of our response variable. (Take a quick note of the range of your values).

Verifying exponential growth Now graph the points (x, lny) by going to stat plot and changing your lists to x list: L1 y list: L3 Remember you will need to adjust your window again.

Verifying exponential growth Your graph should look like this. It's fairly linear. Let's perform a regression to see how linear.

Verifying exponential growth Perform a linear regression (stat, calc, LinReg, then type L1, L3) and record your regression equation, correlation, and r 2 values. Not only was our data linear, confirming the data is expontial, but our regression line explains 99.5% of our data. As the book states “That's impressive!”

Verifying exponential growth It’s also a good idea to check our residual plot. On the calculator, go back to your lists (stat, edit). Highlight L4 by scrolling all the way up in that column. Now insert a blank list by pressing INS. Our calculator actually already has our residuals stored. To access them, press 2 nd, LIST, then find RESID in your names menu and press enter. Now go back to stat plot, select plot1, and enter the following: Xlist: L1 Ylist: once again find RESID from 2 nd, LIST Remember to change your window!

Verifying Exponential Growth The residual plot is shown on page 274 of your text Once again, there is a slight pattern to our residuals, but they are so small that we can justify using our model to make predictions.

Predictions using our LSRL With our regression equation, we can now use it to make predictions. To predict the number of transistors on Intel’s Itanium 2 chip, which was released in 2003, we substitute 33 for “years since 1970” in the regression equation. Ln(transistors) = years since 1970 Ln(transistors) = (33) = Then change to exponential form (remember ln is base e)