Questions: a. Using the pairs of values for all 10 points, find the equation of the regression line. b. After removing the point with coordinates (8,3), use the pairs of values for the remaining 9 points and find the equation of the regression line. c. Compare the results from parts (a) and (b). a. What is the equation of the regression line for all 10 points? ŷ = + × (Round to three decimal places as needed)

Solution Steps

The given graph has 10 points. The coordinates of these points are: (1, 6), (2, 6), (3, 6), (4, 6), (5, 6), (6, 6), (7, 6), (8, 6), (9, 6), (8, 3).

Calculate the mean of the x-coordinates and y-coordinates. \[ \bar{x} = \frac{1+2+3+4+5+6+7+8+9+8}{10} = \frac{53}{10} = 5.3 \] \[ \bar{y} = \frac{6+6+6+6+6+6+6+6+6+3}{10} = \frac{57}{10} = 5.7 \]

Use the formula for the slope \( b \): \[ b = \frac{\sum{(x_i - \bar{x})(y_i - \bar{y})}}{\sum{(x_i - \bar{x})^2}} \] Calculate the numerator: \[ \sum{(x_i - \bar{x})(y_i - \bar{y})} = (1-5.3)(6-5.7) + (2-5.3)(6-5.7) + \ldots + (8-5.3)(3-5.7) \] \[ = (-4.3)(0.3) + (-3.3)(0.3) + (-2.3)(0.3) + (-1.3)(0.3) + (-0.3)(0.3) + (0.7)(0.3) + (1.7)(0.3) + (2.7)(0.3) + (3.7)(0.3) + (2.7)(-2.7) \] \[ = -1.29 - 0.99 - 0.69 - 0.39 - 0.09 + 0.21 + 0.51 + 0.81 + 1.11 - 7.29 \] \[ = -7.1 \] Calculate the denominator: \[ \sum{(x_i - \bar{x})^2} = (1-5.3)^2 + (2-5.3)^2 + \ldots + (8-5.3)^2 \] \[ = 18.49 + 10.89 + 5.29 + 1.69 + 0.09 + 0.49 + 2.89 + 7.29 + 13.69 + 7.29 \] \[ = 68.1 \] \[ b = \frac{-7.1}{68.1} = -0.104 \]

Use the formula for the intercept \( a \): \[ a = \bar{y} - b\bar{x} \] \[ a = 5.7 - (-0.104 \times 5.3) \] \[ a = 5.7 + 0.5512 = 6.2512 \]

Final Answer