- Organize data with \(t = \text{Year} - 1980\) and use the least squares method to find the slope and intercept.
- Calculate the regression line as \(y = 1.4471x - 2780.7623\).
- Estimate voter turnout for the year 2020 by substituting \(x = 2020\) into the equation.
- The estimated number of voters for 2020 is \(\boxed{142.38}\) million.
Explanation
- Data Analysis and Setup
To find the line of best fit for the data, we represent the year as \(x\) and the number of voters (in millions) as \(y\). The data points \((x_i, y_i)\) are:
(1980, 86.5), (1984, 92.7), (1988, 91.6), (1992, 104.4), (1996, 96.3), (2000, 122.3).
To simplify, let \(t = x - 1980\). Then the points become \((t_i, y_i)\):
(0, 86.5), (4, 92.7), (8, 91.6), (12, 104.4), (16, 96.3), (20, 122.3).
The number of data points is \(n = 6\). We calculate the following sums:
\(\sum t_i = 0 + 4 + 8 + 12 + 16 + 20 = 60\)
\(\sum y_i = 86.5 + 92.7 + 91.6 + 104.4 + 96.3 + 122.3 = 593.8\)
\(\sum t_i^2 = 0^2 + 4^2 + 8^2 + 12^2 + 16^2 + 20^2 = 880\)
\(\sum t_i y_i = 0(86.5) + 4(92.7) + 8(91.6) + 12(104.4) + 16(96.3) + 20(122.3) = 6343.2\)
- Determining the Line of Best Fit
Using the least squares method, the slope \(m\) and the t-intercept \(b\) for the line \(y = mt + b\) are:
\[m = \frac{n \sum t_i y_i - \sum t_i \sum y_i}{n \sum t_i^2 - (\sum t_i)^2} = \frac{6(6343.2) - 60(593.8)}{6(880) - (60)^2} = \frac{38059.2 - 35628}{5280 - 3600} = \frac{2431.2}{1680} \approx 1.4471\]
\[b = \frac{\sum y_i - m \sum t_i}{n} = \frac{593.8 - 1.4471(60)}{6} = \frac{593.8 - 86.826}{6} \approx 84.4957\]
The line equation in terms of \(t\) is \(y = 1.4471t + 84.4957\).
To convert this to the original year variable \(x\), we substitute \(t = x - 1980\):
\[y = 1.4471(x - 1980) + 84.4957 = 1.4471x - 2865.258 + 84.4957 = 1.4471x - 2780.7623\]
Rounding to four decimal places, the line of best fit is approximately \(y = 1.4471x - 2780.7623\).
- Estimating Voters for 2020
To estimate the number of voters in the year \(x = 2020\), we substitute \(x = 2020\) into the linear model:
\[y = 1.4471(2020) - 2780.7623 \approx 2923.142 - 2780.7623 = 142.3797\]
Rounding to the nearest hundredth, the estimated number of voters in the 2020 election is \(142.38\) million.
Examples
Linear regression is used frequently by pollsters and political analysts to predict future trends. By examining historical voter participation data, they can build models to estimate turnout for upcoming elections. This helps organizations allocate resources, such as staffing and ballot materials, effectively. Beyond politics, this same method is applied in business for sales forecasting, in climate science for trend analysis, and in sports to track athlete performance improvement over time.