Calculating Linear Regression Equation from Data and Predicting Grades

Let's solve the problem step by step using the provided data: | Homework Grade (x) | Test Grade (y) | |--------------------|:--------------:| | 80 | 84 | | 80 | 70 | | 70 | 65 | | 56 | 56 | | 86 | 73 | | 81 | 69 | | 75 | 80 | | 58 | 44 | **Step 1: Compute means \(\bar{x}\) and \(\bar{y}\):** \[ \bar{x} = \frac{80+80+70+56+86+81+75+58}{8} = \frac{586}{8} = 73.25 \] \[ \bar{y} = \frac{84+70+65+56+73+69+80+44}{8} = \frac{541}{8} = 67.625 \] --- **Step 2: Compute the slope (m) and y-intercept (b):** Recall the formulas: \[ m = \frac{\sum (x_i - \bar{x})(y_i - \bar{y})}{\sum (x_i - \bar{x})^2} \] \[ b = \bar{y} - m \bar{x} \] --- **Step 3: Set up the data for calculations:** Let’s compute the terms for the numerator (for m) and the denominator: | x | y | x - 73.25 | y - 67.625 | (x-73.25)*(y-67.625) | (x-73.25)^2 | |-----|-----|-----------|------------|----------------------|-------------| | 80 | 84 | 6.75 | 16.375 | 110.531 | 45.563 | | 80 | 70 | 6.75 | 2.375 | 16.031 | 45.563 | | 70 | 65 | -3.25 | -2.625 | 8.531 | 10.563 | | 56 | 56 | -17.25 | -11.625 | 200.531 | 297.563 | | 86 | 73 | 12.75 | 5.375 | 68.531 | 162.563 | | 81 | 69 | 7.75 | 1.375 | 10.656 | 60.063 | | 75 | 80 | 1.75 | 12.375 | 21.156 | 3.063 | | 58 | 44 | -15.25 | -23.625 | 360.844 | 232.563 | Sum for numerator: \[ \sum (x-\bar{x})(y-\bar{y}) = 110.531 + 16.031 + 8.531 + 200.531 + 68.531 + 10.656 + 21.156 + 360.844 = 796.811 \] Sum for denominator: \[ \sum (x-\bar{x})^2 = 45.563 + 45.563 + 10.563 + 297.563 + 162.563 + 60.063 + 3.063 + 232.563 = 857.504 \] --- **Step 4: Find the slope (m)** \[ m = \frac{796.811}{857.504} \approx 0.93 \] --- **Step 5: Find the y-intercept (b):** \[ b = 67.625 - (0.93 \times 73.25) = 67.625 - 68.123 = -0.498 \approx -0.5 \] --- **Step 6: Write the regression equation (rounded to the nearest tenth):** \[ \boxed{y = 0.9x - 0.5} \] --- **Step 7: Find y for \(x = 62\):** \[ y = 0.9 \times 62 - 0.5 = 55.8 - 0.5 = 55.3 \] Rounding to the nearest integer: **55** --- ## **Final Answers** **Regression Equation:** \[ \boxed{y = 0.9x - 0.5} \] **Projected Test Grade for a homework grade of 62:** \[ \boxed{55} \]