Questions: A data set is given below. (a) Draw a scatter diagram. Comment on the type of relation that appears to exist between x and y. (b) Given that x̄=3.5000, sx=2.3452, ȳ=3.9000, sy=1.8783, and r=-0.9580; determine the least-squares regression line. (c) Graph the least-squares regression line on the scatter diagram drawn in part (a). x 0 2 3 4 6 6 y 5.9 5.6 5.0 3.4 1.6 1.9 (a) Choose the correct graph below. A. B. C. D.

Solution Steps

To draw a scatter diagram, plot the given data points (x, y) on a graph:

• (0, 5.9)
• (2, 5.6)
• (3, 5.0)
• (4, 3.4)
• (6, 1.6)
• (6, 1.9)

By observing the scatter diagram, we can comment on the type of relation between x and y. The points seem to show a negative correlation, meaning as x increases, y tends to decrease.

Given:

• $\bar{x} = 3.5000$
• $s_x = 2.3452$
• $\bar{y} = 3.9000$
• $s_y = 1.8783$
• $r = -0.9580$

The formula for the least-squares regression line is: $$y = mx + b$$ where: $$m = r \left(\frac{s_y}{s_x}\right)$$ $$b = \bar{y} - m\bar{x}$$

Calculate the slope (m): $$m = -0.9580 \left(\frac{1.8783}{2.3452}\right) \approx -0.767$$

Calculate the y-intercept (b): $$b = 3.9000 - (-0.767 \times 3.5000) \approx 6.5845$$

So, the least-squares regression line is: $$y = -0.767x + 6.5845$$

Final Answer