Question

Choose the best answer from among choices below. A police department wants to find out if a correlation exists between x = the number of muggings per night during warmer nights in a particular park not far from downtown and y = degrees Fahrenheit. This is to help decide how police duties should be allocated on these nights. Some data points for some nights from the spring and summer of the past year are listed below. x:{5,7,8,4,11,12,8,7} y: {79,82,83,81,86,89,91,84,} Compute the coefficient of correlation r. Compute the coefficient of determination r^2. What percentage of the variation in y can be explained by the corresponding variation in x using the least-squares line? What percentage of the variation is unexplained? Select one: a. r = 0.7132; r^2 = 0.5087; 51% explained; 49% unexplained b. r = 0.7217; r^2 = 0.5208; 52% explained; 48% unexplained c. r = 0.7877; r^2 = 0.6204; 62% explained; 38% unexplained d. r = 0.6911; r^2 = 0.4776; 48% explained; 52% unexplained Check

Answer 1

∑x = 5 + 7 + 8 + 4 + 11 + 12 + 8 + 7 = 62
(∑x)^2 = 62^2 = 3,844
x bar = 62 / 8 = 7.75
∑x^2 = 25 + 49 + 64 + 16 + 121 + 144 + 64 + 49 = 522

∑y = 79 + 82 + 83 + 81+ 86 + 89 + 91 + 84 = 675
(∑y)^2 = 455,625
y bar = 675 / 8 = 84.375
∑y^2 = 6,241 + 6,724 + 6,889 + 6,561 + 7,396 + 7,921 + 8,281 + 7,056 = 57,069

∑xy = 395 + 574 + 664 + 324 + 946 + 1,068 + 728 + 588 = 5,287
 
r = (∑xy - n(x bar)(y bar)) / (sqrt(∑x^2 - n(x bar)^2) sqrt(∑y^2 - n(y bar)^2)) = (5,287 - 8(7.75)(84.375)) / (sqrt(522 - 8(7.75)^2) sqrt(57,069 - 8(84.375)^2)) = (5,287 - 5,231.25) / (sqrt(522 - 480.5) sqrt(57,069 - 56,953.125)) = 55.75 / (sqrt(41.5) sqrt(115.875)) = 55.75 / 69.3456 = 0.8039