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Chapter 14 Answers to Chapter Questions: The Statistical Im agination, 2 Editionnd

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Answers to End-of-Chapter Questions for Chapter 14

Solutions Manual to Accompany The Statistical Imagination:

Elementary Statistics for the Social Sciences by Ferris J. Ritchey

2 edition, McGraw-Hill, 2008/2009nd

by

Brian P. Hinote, Jason Wasserman and Ferris J. Ritchey

Questions for Chapter 14 are on page 538 of the text.

1. With the help of a hand-sketched scatterplot and the regression equation explain the central idea behind regression analysis.

The central idea behind regression analysis is to use the formula of a straight line to improve best estimates of an interval/ratio dependent variable (Y ) for all values of an interval/ratio independent variable (X ). Without knowledge that X is correlated with Y, the best estimate of Y for any value of X is the mean of Y. With knowledge of a relationship between variables, however, we can predict to the regression line rather than the mean line.

This is illustrated with a scatterplot Figure CQ14-1 below. If we had no idea that horsepower was related to engine size, to estimate horsepower all we could do is predict the value of the mean, which is represented by the flat line just above a horsepower of 100. But if we know that horsepower is related to engine size, we can improve our estimates by predicting to the regression line, the line that slopes upward. For example, we can estimate that engines with over 400 cubic inches tend to have horsepowers around 200. In fact, with the regression formula we can make rather precise estimates of Y for any given value of X.

Chapter 14 Answers to Chapter Questions: The Statistical Im agination, 2 Editionnd

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Figure CQ14-1. Scatterplot of Automobile Engine Horsepower Regressed on Size of Engine

2. With the help of hand-sketched scatterplots, illustrate why linear correlation and regression analysis applies only when there is a cigar-shaped, linear pattern of coordinates.

With linear correlation and regression, we use the formula for a straight line, which is: No matter the pattern of coordinates of a scatterplot, if this formula is used, it produces a straight line. If we are to use this line to make good estimates, then the coordinates must fit around the line. Figure 14-8 on page 532 of the text illustrates the futility of using a linear formula for a nonlinear plot.

3. With the help of hand-sketched scatterplots, illustrate the pattern of coordinates for positive, negative, and no relationships between X and Y.

(Provide sketches similar to Figures 14-1, 14-2 and 14-3 on pages 514-515 of the text.)

4. With the help of hand-sketched scatterplots, illustrate why it sometimes is necessary to truncate the axis of a scatterplot.

(Provide a sketch similar to that in Figure 14-7, page 529 of the text. Provide a second sketch with truncated axes.)

Chapter 14 Answers to Chapter Questions: The Statistical Im agination, 2 Editionnd

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5. What does Pearson’s r correlation coefficient measure?

Pearson’s r measures the tightness of fit of X,Y-coordinates around a regression line of a scatterplot. If the points fit closely to the line, the Pearson’s r will have a rather large absolute value. This means that X will provide a good estimate of Y. For example, in the following illustration (Figure CQ14-2), Pearson’s r calculates to a high value of .93:

Figure CQ14-2. Scatterplot of Vehicle Weight Regressed on Size of Engine (n = 406 automobiles), r = .93.

If the points fit rather loosely, however, then X will be of limited help in improving estimates of Y. In the following scatterplot (Figure CQ14-3) Pearson’s r calculates to the rather low value of .13:

Chapter 14 Answers to Chapter Questions: The Statistical Im agination, 2 Editionnd

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Figure CQ14-3. Scatterplot of Education of a Family Caregiver and How Many Weeks Assistance Has Been Provided to a Relative (n = 104 caregivers)

6. What does the regression coefficient b measure?

In the regression equation, , b is the slope of the regression line and is called the regression coefficient. The regression coefficient tells us how many units to add to an estimate of Y for each additional one-unit increase in X. The larger the value of b, the steeper the slope of the regression line. (See, for example, Figure 14-6, page 526 in the text.) In addition, the sign of b tells us the direction of a relationship.

7. On the regression line, the Y-intercept, a, is the value of Ý where X = .

Zero. (The function of a in the regression formula is to distinguish among regression lines with the same slope.)

Sometimes the value of a is an abstraction that cannot exist in reality. For example, in the illustration in this chapter, when predicting weight from height, a = -159.31 pounds, an impossible weight. Also, the value of a will not be apparent on a scatterplot with truncated axes, such as Figure 14-4.

Chapter 14 Answers to Chapter Questions: The Statistical Im agination, 2 Editionnd

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8. With a linear relationship between two interval/ratio variables, what is the X,Y-coordinate that will always fall on the regression line?

The coordinate This is due to the fact that correlation and regression techniques are based on deviations from the means of the two variables. For example, a high positive correlation occurs when subjects who deviate on the high end of X (e.g., especially tall subjects) also deviate on the high end of Y (e.g., especially heavy subjects). Figure 14-5, page 522 in the text, illustrates linear patterning of X,Y-coordinates relative to the means of X and Y.

9. Match the following regarding the direction of correlations:

a. A positive correlation The regression line has no slope; r = 0 and b = 0

b. No correlation The regression line slopes downward; r and b have a negative sign; an increase in X is related to a decrease in Y

c. A negative correlation The regression line slopes upward; r and b have a positive sign; an increase in X is related to an increase in Y

Answers from top to bottom: b, c, a

10. Attenuation or inflation of the calculation of correlation and regression coefficients can result from the presence of in the scatterplot.

outlier coordinates

END OF ANSWERS TO CHAPTER 13 QUESTIONS