Final answer:
To find the least squares regression line for the given data points, calculate the means of the x-values and y-values. Find the deviations from the means, multiply the deviations of x and y, square the deviations of x, and calculate the sum of the squared deviations of x and the sum of the products. Then, use the formula to calculate the slope, and substitute the values into the equation to find the y-intercept.
Step-by-step explanation:
To find the least squares regression line for the given data points, we need to follow these steps:
- First, let's calculate the means of the x-values and y-values. The mean of the x-values is (−1 + 1 + 3) / 3 = 1, and the mean of the y-values is (0 + (−2) + (−3)) / 3 = -1.6667.
- Next, we calculate the deviations from the means for both x and y. For x, subtract the mean from each x-value. For y, subtract the mean from each y-value. The deviations are (−1 - 1) = -2, (1 - 1) = 0, and (3 - 1) = 2 for x, and (0 - (-1.6667)) = 1.6667, (−2 - (-1.6667)) = -0.3333, and (−3 - (-1.6667)) = -1.3333 for y.
- Then, multiply the deviations of x and y for each data point. The products are -4, 0, and 4.
- Next, square the deviations of x for each data point. The squares are 4, 0, and 4.
- Calculate the sum of the squared deviations of x and the sum of the products. The sum of the squared deviations of x is 8, and the sum of the products is 0.
- Finally, we can calculate the slope (b) of the regression line using the formula b = sum of the products / sum of the squared deviations of x. In this case, b = 0 / 8 = 0.
- Substitute the values of b, the mean of x (1), and the mean of y (-1.6667) into the equation of the regression line: y = a + bx. We can solve for a: -1.6667 = a + (0)(1), which gives us a = -1.6667.
Therefore, the equation of the least squares regression line for the given data points is y = -1.6667.