1. Our brains don’t like losses. Most people dislike losses more than they like gains. In money terms, people are about as sensitive to a loss of $10 as to a gain of $20. To discover what parts of the brain are active in decisions about gain and loss, psychologists presented subjects with a series of gambles with different odds and different amounts of winnings and losses. From a subject’s choices, they constructed a measure of “behavioral loss aversion.” Higher scores show greater sensitivity to losses. Observing brain activity while subjects made their decisions pointed to specific brain regions. Here are data for 16 subjects on behavioral loss aversion and “neural loss aversion,” a measure of activity in one region of the brain:1

a. Make a scatterplot that shows how behavior responds to brain activity.

b. Describe the overall pattern of the data. There is one clear outlier. What is the behavioral score associated with this outlier?

c. Find the correlation r between neural and behavioral loss aversion both with and without the outlier. Does the outlier have a strong influence on the value of r? By looking at your plot, explain why adding the outlier to the other data points causes r to increase.

d. Find the least-squares line for predicting y from x, leaving out the outlier, and add the line to your plot.

e. The outlier lies very close to your regression line. Looking at the plot, you now expect that adding the outlier will increase the correlation but will have little effect on the least-squares line. Explain why.

f. Find the correlation and the equation of the least-squares line with and without the outlier. Your results verify the expectations from (e).

Neural -50 -39.1 -25.9 -26.7 -28.6 -19.8 -17.6 5.5
Behave 0.08 0.81 0.01 0.12 0.68 0.11 0.36 0.34
Neural 2.6 20.7 12.1 15.5 28.8 41.7 55.3 155.2
Behave 0.53 0.68 0.99 1.04 0.66 0.86 1.29 1.94

2. Always plot your data! The table on the next page presents four sets of data prepared by the statistician Frank Anscombe to illustrate the dangers of calculating without first plotting the data.2

a. Without making scatterplots, find the correlation and the least-squares regression line for all four data sets. What do you notice? Use the regression line to predict y for x = 10.

b. Make a scatterplot for each of the data sets and add the regression line to each plot.

1 From a graph in Sabrina M. Tom et al., “The neural basis of loss aversion in decision-making under risk,” Science, 315 (2007), pp. 515–518.

2 Frank J. Anscombe, “Graphs in statistical analysis,” American Statistician, 27 (1973), pp. 17–21.

Homework #5
Business Statistics with Computer Applications II

c. In which of the four cases would you be willing to use the regression line to describe the dependence of y on x? Explain your answer in each case.

Data Set A
x 10 8 13 9 11 14 6 4 12 7 5
y 8.04 6.95 7.58 8.81 8.33 9.96 7.24 4.26 10.84 4.82 5.68

Data Set B
x 10 8 13 9 11 14 6 4 12 7 5
y 9.14 8.14 8.74 8.77 9.26 8.1 6.13 3.1 9.13 7.26 4.74

Data Set C
x 10 8 13 9 11 14 6 4 12 7 5
y 7.46 6.77 12.74 7.11 7.81 8.84 6.08 5.39 8.15 6.42 5.73

Data Set D
x 8 8 8 8 8 8 8 8 8 8 19
y 6.58 5.76 7.71 8.84 8.47 7.04 5.25 5.56 7.91 6.89 12.5