Final answer:
The regression line equation can be determined by plotting the given points on a scatter plot and using a calculator's regression function. The slope (β1) and y-intercept (β0) are calculated through specific formulas involving sums of X and Y values and their products, leading to the least-squares line equation y = β0 + β1x.
Step-by-step explanation:
To find the equation of the least-squares regression line for the given data points (1,2),(2,2),(3,4),(4,4), we begin with a scatter plot of the data. We then use a calculator with the regression functionality to calculate the least-squares regression line.
Steps:
- Plot the given points on the scatter plot.
- Use the regression function in the calculator to determine the best-fit line equation which minimizes the sum of squared errors (SSE).
For the given data points, the least-squares regression line equation evidently is × = +(1)x. Although this seems to be a typo, a correct regression analysis typically would result in an equation of the form × = β0 + β1x, where β0 is the y-intercept and β1 is the slope.
Finding the slope (β1):
To find the slope (β1), we can use the formula:
β1 = (N∑XY - (∑X)(∑Y)) / (N∑X² - (∑X)²)
Where:
- N is the number of data points
- ∑XY is the sum of the products of corresponding X and Y values
- ∑X and ∑Y are the sums of X and Y values respectively
- ∑X² is the sum of the squared X values
Finding the y-intercept (β0):
Once the slope is determined, we can calculate the y-intercept (β0) using the equation:
β0 = (∑Y - β1∑X) / N
By applying these calculations to the given data, we obtain the equation of the least-squares line: