Statistics Demystified, 2nd edition

Stan Gibilisco

Explanations for Quiz Answers in Chapter 7

1. According to the Malthusian theory, the supply of available resources can increase arithmetically -- that is, steadily, according to a linear function -- as the years go by. The correct choice is C. (Today's scientists might debate Thomas Malthus's view, suggesting that resources can only increase up to a certain maximum with the passage of time.)

2. If two variables correlate strongly, either in the positive sense or the negative sense, the points in a scatter plot tend to cluster near a straight line. With positive correlation, the line has positive slope (it ramps upward as we move toward the right); in the case of negative correlation, the line has negative slope (it ramps downward as we move toward the right). The correct choice is A.

3. Point P lies precisely on the least-squares line as shown in Fig. 7-13 (on page 255). If we add P as a new data point, the slope of the least-squares line won't change. The answer is A.

4. Point Q lies above the least-squares line as shown in Fig. 7-13, and also lies to the left of the center of the X (independent-variable) axis. If we add Q as a new data point, its presence will cause the upper-left-hand end of the least-squares line to move slightly higher in the coordinate grid, while having little or no effect on the lower-right-hand end of the least-squares line. The line's slope will therefore get a little steeper while remaining negative. The correct choice is B.

5. Point R lies below the least-squares line as shown in Fig. 7-13, and also lies to the right of the center of the independent-variable axis. If we add R as a new data point, its presence will cause the lower-right-hand end of the least-squares line to move slightly downward in the coordinate grid, while having less effect on the upper-left-hand end of the least-squares line. The line's slope will therefore become slightly steeper while remaining negative. The correct choice is B.

6. If we see a correlation factor of +50% between two parameters X and Y, then we know that in general, an increase in X will accompany an increase in Y (and vice-versa). We can expect a few exceptions for individual data points, but in the majority of cases, the foregoing rule will hold true. The correct choice is D.

7. Isolated, small events can sometimes create chain reactions resulting in gigantic changes in the outcomes of experiments (or real-world events). Consider, for example, a snow avalanche on a mountainside. The snow stays in place until someone sets off a small firecracker, creating a little shock wave in the air. As a result of the "acoustic bump," the snow starts to move, only a little bit at first, but picking up more and more, until tons of snow cascade down the slope, crushing trees and buildings! Statisticians have given this phenomenon the name butterfly effect, based on the notion of a butterfly taking off someplace, creating just enough of an atmospheric disturbance to eventually trigger a storm somewhere else (as described on page 245). The correct choice is C.

8. The half-parabola defined by the equation Y = X 2, when we confine both X and Y to the set of positive real numbers, slopes generally upward as we move to the right. But it's not a straight line. If we encounter a scatter plot between two variables X and Y in which the points all lie precisely along the half-parabola Y = X 2, we can conclude that the correlation is positive but less than +1. The correct choice is B.

9. In Fig. 7-14 (on page 256), the graph for Phenomenon Y appears to constitute an exact "upside-down mirror image" of the graph for Phenomenon X. Whenever X increases, Y decreases. Whenever X decreases, Y increases. The variables' trends appear to perfectly oppose each other at every point along the time axis, suggesting a perfect negative correlation of -1 or -100%. The correct choice is D.

10. Figure 7-15 (on page 257) shows a graph for two variables called X and Y (the same variable names as in the previous problem, but obviously different phenomena). In this case, the two curves trend more or less together. We can see that X usually increases when Y increases, and X usually decreases when Y decreases, so we can surmise that the correlation is positive. However, in some intervals along the time axis, the foregoing rule doesn't apply. A positive correlation exists, but it isn't perfect. We can conclude that the correlation factor must lie between 0 at 1 (or between 0 and +100%). The correct choice is B.