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