Question

7 the accompanying table shows the percent of the adult population that married before age 25 in several different years. using the year as the independent variable, find the linear regression equation. round the regression coefficients to the nearest hundredth. using the equation found above, estimate the percent of the adult population in the year 2009 that will marry before age 25, and round to the nearest tenth of a percent.

year (x)	percent (y)
1976	37.4
1980	37.1
1984	34.1
1989	32.1
1993	28.8
1997	25.7
2000	25.5

Explanation:

Step1: Calculate sums

Let \(n = 8\) (number of data - points).
\(\sum_{i = 1}^{n}x_{i}=1971 + 1976+1980 + 1984+1989+1993+1997+2000=15890\)
\(\sum_{i = 1}^{n}y_{i}=42.4 + 37.4+37.1+34.1+32.1+28.8+25.7+25.5 = 263.1\)
\(\sum_{i = 1}^{n}x_{i}y_{i}=1971\times42.4+1976\times37.4+1980\times37.1+1984\times34.1+1989\times32.1+1993\times28.8+1997\times25.7+2000\times25.5\)
\(=1971\times42.4+1976\times37.4+1980\times37.1+1984\times34.1+1989\times32.1+1993\times28.8+1997\times25.7+2000\times25.5\)
\(=83660.4+74902.4+73458+67654.4+63846.9+57398.4+51322.9+51000\)
\(=523243.4\)
\(\sum_{i = 1}^{n}x_{i}^{2}=1971^{2}+1976^{2}+1980^{2}+1984^{2}+1989^{2}+1993^{2}+1997^{2}+2000^{2}\)
\(=3884841+3904576+3920400+3936256+3956121+3972049+3988009+4000000\)
\(=31562252\)

Step2: Calculate slope \(m\)

The formula for the slope \(m\) of the regression - line is \(m=\frac{n\sum_{i = 1}^{n}x_{i}y_{i}-\sum_{i = 1}^{n}x_{i}\sum_{i = 1}^{n}y_{i}}{n\sum_{i = 1}^{n}x_{i}^{2}-(\sum_{i = 1}^{n}x_{i})^{2}}\)
\(m=\frac{8\times523243.4 - 15890\times263.1}{8\times31562252-(15890)^{2}}\)
\(=\frac{4185947.2-4180659}{252498016 - 252492100}\)
\(=\frac{5288.2}{5916}\approx - 0.89\)

Step3: Calculate intercept \(b\)

The formula for the intercept \(b\) is \(b=\bar{y}-m\bar{x}\), where \(\bar{x}=\frac{\sum_{i = 1}^{n}x_{i}}{n}=\frac{15890}{8}=1986.25\) and \(\bar{y}=\frac{\sum_{i = 1}^{n}y_{i}}{n}=\frac{263.1}{8}=32.8875\)
\(b = 32.8875-(-0.89)\times1986.25\)
\(b = 32.8875 + 0.89\times1986.25\)
\(b = 32.8875+1767.7625\)
\(b = 1799.65\)

The linear - regression equation is \(y=-0.89x + 1799.65\)

Step4: Estimate for \(x = 2009\)

Substitute \(x = 2009\) into the equation \(y=-0.89x + 1799.65\)
\(y=-0.89\times2009+1799.65\)
\(y=-1788.01+1799.65\)
\(y = 11.64\approx11.6\)

Answer:

The linear - regression equation is \(y=-0.89x + 1799.65\), and the estimated percent of the adult population that will marry before age 25 in 2009 is \(11.6\%\)