Questions: Government funding: The following table presents the budget (in millions of dollars) for selected organizations that received U.S. government funding for arts and culture in both 2006 and last year. Organization 2006 Last Year --------- Organization 1 460 450 Organization 2 247 236 Organization 3 142 159 Organization 4 124 168 Organization 5 95 156 Organization 6 18 41 Organization 7 2 3 Part: 0 / 3 Part 1 of 3 Compute the least-squares regression line for predicting last year's budget from the 2006 budget. Round the slope and y-intercept to four decimal places as needed. The equation for the least-squares regression line is ŷ = .

Solution Steps

To compute the least-squares regression line, we need to find the slope and y-intercept that minimize the sum of squared differences between the observed values and the values predicted by the line. This involves calculating the means of the 2006 and last year's budgets, the covariance between these two sets of data, and the variance of the 2006 budgets. The slope is the covariance divided by the variance, and the y-intercept is the mean of last year's budgets minus the slope times the mean of the 2006 budgets.

To find the least-squares regression line, we first calculate the means of the 2006 and last year's budgets. The mean of the 2006 budgets is \(\bar{x} = 155.4286\) and the mean of last year's budgets is \(\bar{y} = 173.2857\).

Next, we calculate the covariance between the 2006 and last year's budgets and the variance of the 2006 budgets. The covariance is given by: \[ \text{Cov}(X, Y) = \sum (x_i - \bar{x})(y_i - \bar{y}) = 135731.1429 \] The variance of the 2006 budgets is: \[ \text{Var}(X) = \sum (x_i - \bar{x})^2 = 148395.7143 \]

The slope \(m\) of the regression line is calculated as: \[ m = \frac{\text{Cov}(X, Y)}{\text{Var}(X)} = \frac{135731.1429}{148395.7143} = 0.9147 \] The y-intercept \(b\) is calculated as: \[ b = \bar{y} - m \cdot \bar{x} = 173.2857 - 0.9147 \cdot 155.4286 = 31.1219 \]

Final Answer