a) Year is on the horizontal axis and population is on the vertical axis. b) Let us first determine the necessary sums: \(\displaystyle\sum\ {x}_{{{i}}}={3990}\)

\(\displaystyle\sum\ {{x}_{{{i}}}^{{{2}}}}={79960670}\)

\(\displaystyle\sum\ {y}_{{{i}}}={5592}\)

\(\displaystyle\sum\ {x}_{{{i}}}\ {y}_{{{i}}}={11183199}\) Next, we can determine \(\displaystyle{S}_{{x x}}\) and \(\displaystyle{S}_{{{x}{y}}}\)

\(\displaystyle{S}_{{x x}}=\ \sum\ {{x}_{{{i}}}^{{{2}}}}\ -\ {\frac{{{\left(\sum\ {x}_{{{i}}}\right)}^{{{2}}}}}{{{n}}}}={79960670}\ -\ {\frac{{{3990}^{{{2}}}}}{{{20}}}}={665}\)

\(\displaystyle{S}_{{{x}{y}}}=\ \sum\ {x}_{{{i}}}{y}_{{{i}}}\ -\ {\frac{{{\left(\sum\ {x}_{{{i}}}\right)}{\left(\sum\ {y}_{{{i}}}\right)}}}{{{n}}}}={11183199}\ -\ {\frac{{{3990}\ \cdot\ {5592}}}{{{20}}}}={1995}\) The estimate b of the slope \(\displaystyle\beta\) is the ratio of \(\displaystyle{S}_{{{x}{y}}}\) and \(\displaystyle{S}_{{x x}}\): \(\displaystyle{b}=\ {\frac{{{S}_{{{x}{y}}}}}{{{S}_{{x x}}}}}=\ {\frac{{{1995}}}{{{665}}}}={3}\) The mean is the sum of all values divided by the number of values: \(\displaystyle\overline{{{x}}}=\ {\frac{{\sum\ {x}_{{{i}}}}}{{{n}}}}=\ {\frac{{{6990}}}{{{20}}}}={1999.5}\)

\(\displaystyle\overline{{{y}}}=\ {\frac{{\sum\ {y}_{{{i}}}}}{{{n}}}}=\ {\frac{{{5592}}}{{{20}}}}={279.6}\) The estimate a of the intercept \(\displaystyle\alpha\) is the average of u decreased by the product of the estimate of the slope and the average of x. \(\displaystyle{a}=\ \overline{{{y}}}\ -\ {b}\overline{{{x}}}={279.6}\ -\ {3}\ \cdot\ {1999.5}=\ -{5718.9}\) General least-squares equation: \(\displaystyle\hat{{{y}}}=\ \alpha\ +\ \beta\ {x}\). Replace \(\displaystyle\alpha\) by \(\displaystyle{a}=\ -{5505.3432}\) and \(\displaystyle\beta\) by \(\displaystyle{b}={2.8930}\) in the general least-squares equation: \(\displaystyle{y}={a}\ +\ {b}{x}=\ -{5718.9}\ +\ {3}{x}\)

c) Let us evaluate the regression line of part (b) at \(\displaystyle{x}={2010}\) and \(\displaystyle{x}={2011}\). \(\displaystyle{y}=\ -{5718.9}\ +\ {3}{\left({2010}\right)}={311.1}\)

\(\displaystyle{y}=\ -{5718.9}\ +\ {3}{\left({2011}\right)}={314.1}\) Thus the predicted U.S. population in 2010 is 311.1 million and predicted U.S. population in 2011 is 314.1 million.