Question 10.5

For Exercises 10.1 and 10.2, see page 488; for 10.3 and 10.4, see page 490; for 10.5, see pages 493–494; for 10.6 to 10.8, see pages 498–499; for 10.9 and 10.10, see page 500; and for 10.11 and 10.12, see page 502.

Question 10.5

10.5 Research and development spending.

The National Science Foundation collects data on the research and development spending by universities and colleges in the United States.³ Here are the data for the years 2008–2011:

Year	2008	2009	2010	2011
Spending (billions of dollars)	51.9	54.9	58.4	62.0

Page 494

Make a scatterplot that shows the increase in research and development spending over time. Does the pattern suggest that the spending is increasing linearly over time?
Find the equation of the least-squares regression line for predicting spending from year. Add this line to your scatterplot.
For each of the four years, find the residual. Use these residuals to calculate the standard error $s$ . (Do these calculations with a calculator.)
Write the regression model for this setting. What are your estimates of the unknown parameters in this model?
Use your least-squares equation to predict research and development spending for the year 2013. The actual spending for that year was $63.4 billion. Add this point to your plot, and comment on why your equation performed so poorly.

(Comment: These are time series data. Simple regression is often a good fit to time series data over a limited span of time. See Chapter 13 for methods designed specifically for use with time series.)

10.5

(a) The spending is increasing linearly over time. (b) $\hat{y} = - 6735.3 + 3.38 x$ .
(c) 0.17, −0.21, −0.09, 0.13. $s = 0.22136$ .
(d) $SPENDING = β_{0} + β_{1} \times YEAR + ε$ . The estimate for $β_{0}$ is −6735.5, the estimate for $β_{1}$ is 3.38, and the estimate for $ε$ is 0. (e) 68.63.