Given the following data points: $(1,2)$, $(2,3)$, $(3,5)$, use the least squares method to find the equation of the regression line in the form $y=a+bx$. Which of the following is the correct regression equation?

• A. $y=1+1.5x$
• B. $y=2+1x$
• C. $y=0.5+1.5x$
• D. $y=0.333+1.5x$

Answer: (D)

Verified step by step guidance

1. First, identify the data points given: (1, 2), (2, 3), and (3, 5). We will use these to calculate the regression line in the form \(y = a + bx\), where \(a\) is the intercept and \(b\) is the slope.
2. Calculate the means of the \(x\) values and the \(y\) values: \(\bar{x} = \frac{1 + 2 + 3}{3}\) and \(\bar{y} = \frac{2 + 3 + 5}{3}\).
3. Compute the slope \(b\) using the formula: \(b = \frac{\sum (x_i - \bar{x})(y_i - \bar{y})}{\sum (x_i - \bar{x})^2}\) where the summations run over all data points.
4. Calculate the intercept \(a\) using the formula: \(a = \bar{y} - b \bar{x}\)
5. Write the regression equation by substituting the values of \(a\) and \(b\) into the equation \(y = a + bx\).