A recent study in Great Britain examines the relationship between the number of friends an individual has on Facebook and grey matter density in the areas of the brain associated with social perception and associative memory. The data are available in the dataset FacebookFriends and the relevant variables are GMdensity (normalized z-scores of grey matter density in the relevant regions) and FBfriends (the number of friends on Facebook). The study included 40 students at City University London. A scatterplot of the data is shown in Figure 9.9 and computer output for both correlation and regression is shown below.

Figure 9.9

(a) Use the scatterplot to determine whether any of the study participants had grey matter density scores more than two standard deviations from the mean.

(b) Use the scatterplot to determine whether we should have any significant concerns about the conditions being met for using a linear model with these data.
(c) What is the correlation between these two variables? What is the p-value from a test of the correlation? What is the conclusion of the test, in context?
(d) What is the least squares line to predict the number of Facebook friends based on the normalized grey matter density score? What number of Facebook friends is predicted for a person whose normalized score is 0? Whose normalized score is +1? Whose normalized score is ˆ’1?
(e) What is the p-value for a test of the slope? Compare it to the p-value for the test of correlation.

(f) What is R² for this linear model? Interpret it in context.

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Pearson correlation of GMdensity and FBfriends = 0.436 P-Value = 0.005 The regression equation is FBfriends = 367 + 82.4 GMdensity SE Coef 26.35 Coef Predictor P 0.000 Constant 366.64 13.92 82.45 GMdensity 27.58 2.99 0.005 R-Sq = 19.0% S= 165.716 R-Sqladj) = 16.9% 800 700 600 500 400 300 200 100 0. -2 -1 1 GMdenslty FBfriends