In business, do nice guys finish first or last? Refer to the Nature (March 20, 2008) study of the use of punishment in cooperation games, Given Exercises 1 and 2. Recall that simple linear regression was used to model a player's average payoff (y) as a straight-line function of the number of times punishment was used (x) by the player.
(a) If the researchers want to predict average payoff for a single player who used punishment 10 times, how should they proceed?
(b) If the researchers want to estimate the mean of the average payoffs for all players who used punishment 10 times, how should they proceed?
Exercise 1
In business, do nice guys finish first or last? In baseball, there is an old saying that ‘‘nice guys finish last.'' Is this true in the business world? Researchers at Harvard University attempted to answer this question and reported their results in Nature (March 20, 2008). In the study, Boston-area college students repeatedly played a version of the game ‘‘prisoner's dilemma,'' where competitors choose cooperation, defection, or costly punishment. (Cooperation meant paying 1 unit for the opponent to receive 2 units; defection meant gaining 1 unit at a cost of 1 unit for the opponent; and punishment meant paying 1 unit for the opponent to lose 4 units.) At the conclusion of the games, the researchers recorded the average payoff and the number of times cooperation, defection, and punishment were used for each player. The scattergrams plot average payoff (y) against level of cooperation use, defection use, and punishment use, respectively.
(a) Consider cooperation use (x) as a predictor of average payoff (y). Based on the scattergram, is there evidence of a linear trend?
(b) Consider defection use (x) as a predictor of average payoff (y). Based on the scattergram, is there evidence of a linear trend?
(c) Consider punishment use (x) as a predictor of average payoff (y). Based on the scattergram, is there evidence of a linear trend?
(d) Refer to part c. Is the slope of the line relating punishment use (x) to average payoff (y) positive or negative?
(e) The researchers concluded that ‘‘winners don't punish.'' Do you agree? Explain.
Exercise 2
Physical activity of obese young adults. In a study published in the International Journal of Obesity (January 2007), pediatric researchers measured overall physical activity for two groups of young adults: 13 obese adults and 15 normal weight adults. The researchers recorded the total number of registered movements (counts) of each young adult over a period of time. Baseline physical activity was then computed as the number of counts per minute (cpm). Four years later, the physical activity measurements were taken again, called physical activity at follow-up.
(a) For the 13 obese young adults, the researchers reported a correlation of r = .50 between baseline and follow-up physical activity, with an associated p-value of .07. Give a practical interpretation of this correlation coefficient and p-value.
(b) Refer to part a. Construct a possible scatterplot for the 13 data points that would yield a value of r = .50.
(c) Refer to part a. Compute and interpret the coefficient of determination, r2, relating baseline and follow-up physical activity for obese young adults in a simple linear regression.
(d) For the 15 normal weight young adults, the researchers reported a correlation of r = -.12 between baseline and follow-up physical activity, with an associated p-value of .66. Give a practical interpretation of this correlation coefficient and p-value
(e) Refer to part d. Construct a possible scatterplot for the 15 data points that would yield a value of r = -.12.
(f) Refer to part d. Compute and interpret the coefficient of determination, r2, relating baseline and follow-up physical activity for normal weight young adults in a simple linear regression.