Question

Find the equation for the least squares regression line of the data described below. Faced with budget cuts, a state commission is looking into closing some of its public libraries. To help minimize the negative impact of the closures, the commission gathered data on local school and library facilities.

The number of schools, x, and the number of libraries, y, were recorded for each town.

schools libraries
2 2
3 6
6 7
11 9
11 7

Round your answers to the nearest thousandth.

y= __x+___

Answer 1

To determine the least squares regression line for the given data (number of schools versus number of libraries), one must calculate the slope and y-intercept using the sum of x and y values, the sum of the products of x and y values, and the sum of the squares of x values. This results in an equation of the form ŷ = a + bx.

Step-by-step explanation:

To find the equation for the least squares regression line from the provided data, we need to apply statistical methods that summarize the relationship between the independent variable (number of schools, x) and the dependent variable (number of libraries, y). Typically, this method involves calculating the slope and y-intercept of the regression line which has the form Ÿ = a + bx.

First, let's denote the sum of all x-values, the sum of all y-values, the sum of the product of x and y-values, and the sum of the squares of x-values with Σx, Σy, Σxy, and Σx² respectively. Using these summations, we calculate the slope (b) using the formula b = (Σxy - (Σx×Σy)/n) / (Σx² - (Σx)²/n) where n is the number of data points. Subsequently, compute the y-intercept (a) using the formula a = (Σy/n) - b(Σx/n). After calculating a and b, plug these values back into the regression equation to obtain the line of best fit. Remember that this line is only an estimate and will not perfectly predict future values, but rather minimizes the sum of the squares of the residuals. Rounding of the calculated coefficients is done to the nearest thousandths as requested.