Regression Analysis

Key Terms and Concepts Before taking the quiz, you need to be able to explain the meanings (and recognize symbols in cases where there is an associated symbol) of each of these terms or concepts. You should also know when and how to use them in statistics problems.

Unless otherwise noted (by a note such as "see lesson 2") these terms and concepts are defined in the glossary.

categorical variables cause and effect conditional distributions data transformation explanatory variable finding residuals on the graphing calculator (see lessons 3 and 4 and your calculator manual) influential points interpreting MINITAB for regression (see lesson 3) joint frequencies least-squares regression line (line of best fit) LinReg (a + bx) on the graphing calculator (On the TI83/84 it is STAT CALC 8. On the TI-89 it

is [F4] 3:Regressions) marginal frequencies negative association non-linear bivariate data numerical variables outliers (in bivariate data) positive association relation between r and the slope of the regression line resid on the graphing calculator residual response variable row and column percents scatterplot Simpson's Paradox sum of squared residuals the coefficient of determination (r2) the correlation coefficient (r) two-way table use the graphing calculator to transform data to achieve linearity (see lesson 5) using residuals to test a linear model (see lessons 3, 4, and 5)

______________________________ Copyright 2011 Apex Learning Inc. (See Terms of Use at www.apexvs.com/TermsOfUse) TI-83 screens are used with the permission of the publisher. Copyright 1996, Texas Instruments, Incorporated. TI-89 screens are used with the permission of the publisher. Copyright 2011, Texas Instruments, Incorporated.

AP Statistics Review: Bivariate Data: Regression Analysis and Two-Way Tables

Page 1 of 22

Objectives, Example Problems, and Study Tips

Introduction to Bivariate Data

Objective 1 Distinguish between quantitative and categorical data. Examples 1. Which of the following statistics or variables are derived from quantitative data and

which are derived from categorical data? A. Your G.P.A B. The political party your father belongs to C. The cities of residence of 300 people D. The populations of 10 different cities

Tips

Categorical data are also called qualitative data. Quantitative data are also called numerical data.

A categorical variable (which holds categorical data) tells which of several groups an individual belongs to. A quantitative variable has a numerical value that can be manipulated mathematically.

Categorical variables can be thought of as labels, or names; quantitative variables can be thought of as numerical values, or quantities.

Some categorical data appear as numbers, but they are really just names, or labels, for categories. For example, the variable "favorite radio station" may have the value 102.3 or 104.1, but these numbers are meant as labels and not as quantities. It would be meaningless to add 102.3 and 104.1 to get the average radio station.

Answers 1.

A. Quantitative. Your G.P.A, is an arithmetic mean of grades you received from many classes.

B. Categorical. "Political party" is a label, not a quantity. C. Categorical D. Quantitative

Objective 2 Distinguish between explanatory and response variables. Examples 1. You want to be able to predict class rank from number of hours a student spends on

homework. Which is the explanatory variable, and which is the response variable?

2. True or False: There is always a cause-and-effect relationship between the explanatory and response variables.

Tips

Sometimes things can be associated without knowing which, if either, variable caused the other.

The explanatory variable is sometimes called the independent variable, and the response variable is sometimes called the dependent variable.



Page 2 of 22

Answers 1. The variable you're using to predict from is the explanatory variable and the variable you're

trying to predict is the response variable. Therefore, hours spent on homework is the explanatory variable, and class rank is the response variable.

2. False. We might suspect a cause-an-effect relationship between two variables if they're strongly related, but association alone does not prove cause and effect. For example, it's well known that success on the SAT predicts success in college (that's one reason why many colleges use SAT scores to help decide on admissions). In no way, however, does it mean that a high score on the SAT causes success in college.

Objective 3 Construct a scatterplot when given a set of paired data. Examples 1. The number of calories and the number of grams of fat in 25 common fast foods

(hamburgers, pie, french fries, onion rings, etc.) are given in the following table. Construct a scatterplot of the data where grams of fat is the explanatory (independent) variable: (Data taken from Landwehr & Watkins, Exploring Data, Dale Seymour Publications, Palo Alto, 1995, pg. 21)

Grams of Calories Fat 31 570 38 660 48 800 55 890 14 300 19 350 10 260 15 300 28 470 33 530 25 450 10 280 32 620 28 450 13 236 9 178 4 142 5 95 0 25 20 372 19 339 14 320 13 360 8 290 14 220



Page 3 of 22

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Page 4 of 22

Tip Associations are considered to be positive if the response variable increases as the

explanatory variable decreases, and negative if the response variable decreases as the explanatory variable increases.

Answers 1. Negative. As the explanatory variable (age) increases, the response variable (number of

years to live) decreases.

2. Negative

3. Negative, since typically people who exercise weigh less.

The Least-Squares Regression Line

Objective 1 Calculate the linear regression line from a bivariate data set, interpret the correlation coefficient, and use the line to predict values of the response variable when given values for the explanatory variable. Examples 1. In what sense is the linear regression line also the "line of best fit?"

2. Define the least-squares regression line.

3. Consider again the fat vs. calories data you saw earlier:

Fat 31 38 48 55 14 19 10 15 28 33 25 10 32 Calories 570 660 800 890 300 350 260 300 470 530 450 280 620

Fat 28 13 9 4 5 0 20 19 14 13 8 14 Calories 450 236 178 142 95 25 372 339 320 360 290 220

Calculate the line of best fit of calories on fat (that is, use fat as the explanatory variable and use calories as the response variable) and interpret the regression coefficient. (Do this on your calculator, not by hand!)

Tips

The line of "best fit" for any set of points is the line that comes closest to containing all the ordered pairs in the data.

When you interpret a correlation coefficient, you simply make a statement that tells the amount of increase in the response variable for every unit increase (an increase of 1) in the explanatory variable. The amount of increase is simply the regression coefficient, or the slope of the regression line.

On the TI-83/TI-84 there is a LinReg(a+bx), which you can get to by Press STAT Choose [CALC] Choose [LinReg(a+bx)] (or just press 8)

On the TI-89 there is a LinReg(a+bx), which you can get to when you are in the Stats/List Editor press [F4] 3:Regressions [ENTER].



Page 5 of 22

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Page 6 of 22

B. The sum of the squares of the residuals 2 .y y Here's the data again:

Fat 31 38 48 55 14 19 10 15 28 33 25 10 32 Calories 570 660 800 890 300 350 260 300 470 530 450 280 620

Fat 28 13 9 4 5 0 20 19 14 13 8 14 Calories 450 236 178 142 95 25 372 339 320 360 290 220

Tip

You should be able to calculate the residuals without using the built-in function in your calculator; it teaches you a lot about just what residuals are. However, you should remember that when you do a linear regression on the graphing calculator, a set of residuals is created and stored under the list name [RESID] on the TI-83/TI-84 which can be found under the [NAMES] menu; and [statvars\resid] on the TI-89.

Answer on the TI-83/TI-84: 1. Enter the fat data in L1 and the calorie data in L2.

2. Press STAT CALC 8 (to get LinReg(a+bx)).

3. Press 2nd [L1], [L2], VARS Y-VARS 1 1 (to get LinReg(a+bx) L1,L2,Y1).

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5. Press STAT 1 and move the cursor on top of L3 (clear L3 if it has numbers in it).

6. Press CLEAR 2nd [L2] - VARS Y-VARS 1 1 ( 2nd [L1] ) ENTER.

7. This will place the residuals in L3. The expression L2-Y1(L1) is equivalent to subtracting the predicted y-value from the actual y-value.

8. Press STAT CALC 1 2nd [L3].

9. The sum of the residuals is y y and should be very close to 0. The sum of the residuals squared is 2y y and is something like 44942.81.

10. Just to be sure you've done it right in L3, do the following:

2nd [LIST], select MATH, press 5, press 2nd [LIST], select RESID and press ENTER, press x2. (Your screen should end up with this expression: sum( LRESID)2).

11. Press ENTER again, and your answer should be 44942.81, which is the sum of all of the residuals squared.

Note: Save the data for fat, calories, and residuals for the next part of this review. Answer on the TI-89: 1. Go to the Stats/List Editor and clear list1. Enter the Fat data there. Clear list 2 and put the

Calorie data there.

2. Compute the least-squares regression line for the data while you are in the lists, by pressing [F4] 3:Regressions->1:LineReg(a+bx) [ENTER] (your list names of list1 and



Page 7 of 22

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Page 8 of 22

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Page 9 of 22

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Page 10 of 22

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Page 11 of 22

Tips sx is the standard deviation of the values in the variable x, and sy is the standard

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s

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where b = the regression coefficient r = the correlation coefficient sx = the standard deviation of the values in the variable X sy = the standard deviation of the values in the variable Y

If you standardize the data in x and y (if you convert the numbers to z-scores), sx and

sy will both be one, and the formula is simplified to: 11

b r b = r

Therefore, the correlation coefficient and the regression coefficient will have the same value.

Answer

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x

yb rs

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Objective 3 Calculate the coefficient of determination (r2) for a set of paired data and explain its meaning. Examples

1. Given the following data set:

X 114 87 93 74 50 Y 14 12 10 9 7

A. Find the line of best fit, and the correlation coefficient, of Y on X.

B. Find the value of r2.

C. Interpret r2 in the context of the data set.

2. Describe what's meant by the phrase, "r2 is the amount of variation in Y explained by X."

Tip

The "amount of variation in Y" refers to the total variation in Y measured from the



Page 12 of 22

average y-value, that is, from y (as opposed to y ). The total variation is often referred

to as the "total sum of squares" (SST) and equals 21 .y y Answers 1.

A. y = 1.51 + .11x,r = .93 B. r2 = .87 C. 87% of the variation in Y (as measured from y ) is attributable to variation in X.

2. The short explanation:

Some of the variation in Y is tied to the trend shown by the least-squares line; as the x-variable increases, the y-variable increases or decreases by a certain amount. That's the variation explained by X. But other components of the variation in Y aren't related to changes in X. That's the variation not explained by X. The coefficient of determination (r2) is simply the proportion of the variation in Y that is explained by X. The longer but more precise explanation: Imagine that we had a set of y-values with no knowledge of x-values they might be paired with. With no ability to predict a value of y, our best guess at any value of Y would have to be y , the average value. The distance from any point to y can be considered an "error" in our prediction. You can compute the sum of the squares of all such "errors" in the data set of y-values and call it total sum of squares. Now find the line of best fit, which is simply the horizontal line y =y . Now, consider an "error" to be the distance from the actual y-value to the predicted y-value (that's right, it's a residual). Find the sum of the squares of those residuals. The (total sum of squares)-(sum of squares of residuals) is the amount of error eliminated because of basing our predictions on the regression line rather than the average y-value. The fraction of the total error from y that this represents is the amount of variation "explained" by the variable X.

Objective 4 Interpret MINITAB output for regression and correlation. Examples Consider this printout:



Page 13 of 22

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Page 14 of 22

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Objective 3 Use the graphing calculator to do transformation of data and analyze the results to determine if the transformation has resulted in data that can be appropriately modeled with a straight line.

Examples 1. Consider the following set of observations: Obs. 1 2 3 4 5 6 7 8 9 10 11 12 13 14

x 13.5 13.5 14 15 17.5 19 20 21 22 23 25 25 26 27 y 5 15 35 25 25 70 80 140 75 125 190 300 240 315

A. Enter the data in L1 and L2 in your TI-83/84 or list1 and list2 in your TI-89; find the

regression line, and construct a scatterplot with the regression line included. Does a line appear to be a good model for these data? Why not?

B. What is r2 for this model? C. Find the natural logarithms (ln) of the y-data. Put these values in L3 (L3=ln(L2)) or list3

(list3 = ln(list2)). D. Draw a scatterplot of x vs. ln y. Find the regression equation of ln y on x and include it

on the graph. Does this appear to be a better linear fit? What is r2 for this model? E. Use the regression equation you found in #4 to predict the value of y when x = 16

(remember to "back transform" to the original data!)

Tips To transform a data set (for instance, to find ln y), follow these steps:

1. Go to your lists. 2. Place your cursor over the top of L3 and clear the data in L3 (assuming that your

data are in L1 and L2) or list3 and clear the data in list3 on the TI-89 with your data in list1 and list3.

3. With your cursor over the top of L3 (or list3), press [ENTER] so that your cursor is blinking at the bottom of the screen, and you see the expression L3= (or list3= on the TI-89).

4. Press LN 2nd L2 ENTER (or [2nd] LN list2 [ENTER] on the TI-89). 5. You can now do bivariate statistics and make plots of L1 vs. L2 (list1 vs. list2). Be

sure to recalculate the linear regression equation for L1 vs. L3 (list1 vs. list3) and change your STAT PLOT to L1 vs. L3 (list1 vs. list3 for plots on the TI-89).

Important note: Your graphing calculator has options for quadratic regressions, cubic regressions, and other nonlinear regressions. Do not use these, as they will yield incorrect residuals. Instead, transform the data first as taught in the tutorial.



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Regression Analysis

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