The following data are based on information from the Harvard Business Review (Vol. 72, No. 1). Let x be the number of different research programs and let y be the mean number of patents per program. As in any business, a company can spread itself too thin. For example, too many research programs might lead to a decline in overall research productivity. The following data are for a collection of pharmaceutical companies and their research programs:

Complete parts (a) through (e), given ˆ‘x = 90, ˆ‘y = 8.1, ˆ‘x2 = 1420, ˆ‘y2 = 11.83, ˆ‘xy = 113.8, and r ‰ˆ ˆ’0.973.
(a) Draw a scatter diagram displaying the data.
(b) Verify the given sums ˆ‘x, ˆ‘y, ˆ‘x2, ˆ‘y2, and ˆ‘xy and the value of the sample correlation coefficient r.
(c) Find x, y, a, and b. Then find the equation of the least-squares line ŷ = α + bx.

(e) Find the value of the coefficient of determination r2. What percentage of the variation in y can be explained by the corresponding variation in x and the least-squares line? What percentage is unexplained?
(f) Suppose a pharmaceutical company has 15 different research programs. What does the least-squares equation forecast for y = mean number of patents per program?

10 12 14 0 16 18 20 2 1.8 1.71.5 1.41.0 0.7