Chapter 9.2, Problem 16P

For Problems 7–18, please do the following.

1. (a) Draw a scatter diagram displaying the data.
2. (b) Verify the given sums ∑x, ∑y, ∑x², ∑y², and ∑xy and the value of the sample correlation coefficient r.
3. (c) Find $\bar{x}$, $\bar{y}$, a, and b. Then find the equation of the least-squares line $\hat{y}=a+bx$.
4. (d) Graph the least-squares line on your scatter diagram. Be sure to use the point $\left(\bar{x},\bar{y}\right)$ as one of the points on the line.
5. (e) Interpretation 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?

16. Research: Patents 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 ∑x = 90, ∑y = 8.1, ∑x² = 1420, ∑y² = 11.83, ∑xy = 113.8, and r ≈ –0.973.

1. (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?