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 1 460 441 Organization 2 247 230 Organization 3 142 153 Organization 4 124 173 Organization 5 95 159 Organization 6 18 40 Organization 7 2 4 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.

Solution Steps

To compute the least-squares regression line, we need to find the line that best fits the data points. This involves calculating the slope (m) and y-intercept (b) of the line using the formulas derived from the method of least squares. The slope is calculated as the covariance of the two variables divided by the variance of the independent variable (2006 budget), and the y-intercept is calculated using the means of both variables. Once we have these values, we can construct the equation of the line in the form \( \hat{y} = mx + b \).

First, we calculate the mean of the budgets for 2006 and last year. The mean for 2006 is given by:

\[ \text{mean}_{2006} = \frac{460 + 247 + 142 + 124 + 95 + 18 + 2}{7} = 155.4286 \]

Similarly, the mean for last year is:

\[ \text{mean}_{\text{last year}} = \frac{441 + 230 + 153 + 173 + 159 + 40 + 4}{7} = 171.4286 \]

Next, we calculate the covariance between the 2006 budget and last year's budget:

\[ \text{covariance} = \sum (x_i - \text{mean}_{2006})(y_i - \text{mean}_{\text{last year}}) = 132166.7143 \]

The variance of the 2006 budget is:

\[ \text{variance}_{2006} = \sum (x_i - \text{mean}_{2006})^2 = 148395.7143 \]

Using the covariance and variance, we calculate the slope \( m \) of the regression line:

\[ m = \frac{\text{covariance}}{\text{variance}_{2006}} = 0.8906 \]

The y-intercept \( b \) is calculated as:

\[ b = \text{mean}_{\text{last year}} - m \times \text{mean}_{2006} = 32.9981 \]

Final Answer