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5 8 6 7.5 7.2 7 8.5 6 6. which point is closest to the line of best fit…

Question

5 8
6 7.5
7.2 7
8.5 6

  1. which point is closest to the line of best fit? (use the table to find the actual point)

type a response

  1. which point is the farthest from the

Explanation:

Step1: Understand the Table Data

The table has pairs: (5,8), (6,7.5), (7.2,7), (8.5,6). These are (x,y) points. The line of best fit for a negative correlation (since as x increases, y decreases) will have a negative slope.

Step2: Analyze Proximity to Line

For a line of best fit in a negative linear relationship (x and y inversely related), we check how close each point is to the trend. Let's list the points:

  • (5,8): High x (5 is low x here? Wait, x values: 5,6,7.2,8.5 (increasing), y:8,7.5,7,6 (decreasing). So the line of best fit should pass near the middle of these points.
  • (6,7.5): x=6, y=7.5. The difference between y and the trend (which is decreasing by ~0.5 per x increase? From 5→6: y drops 0.5; 6→7.2 (1.2 increase) y drops 0.5 (7.5→7); 7.2→8.5 (1.3 increase) y drops 1 (7→6). Wait, maybe better to see the vertical distance (since it's a linear trend, vertical distance from the line).
  • (7.2,7): x=7.2, y=7. The trend from x=6 (y=7.5) to x=8.5 (y=6): slope is (6-7.5)/(8.5-6) = (-1.5)/2.5 = -0.6. So equation: y -7.5 = -0.6(x -6) → y = -0.6x + 3.6 +7.5 → y = -0.6x +11.1. Let's check for x=7.2: y = -0.6*7.2 +11.1 = -4.32 +11.1 = 6.78. Wait, but actual y is 7. Difference: 7 -6.78 = 0.22.
  • (6,7.5): x=6, y=7.5. Equation: y = -0.6*6 +11.1 = -3.6 +11.1 =7.5. Oh! So (6,7.5) lies on the line of best fit? Wait, no, maybe my slope calculation is off. Wait, the points are (5,8), (6,7.5), (7.2,7), (8.5,6). Let's calculate the slope between (5,8) and (8.5,6): (6-8)/(8.5-5) = (-2)/3.5 ≈ -0.571. Equation: y -8 = -0.571(x -5) → y = -0.571x + 2.857 +8 → y = -0.571x +10.857.

For x=6: y = -0.571*6 +10.857 ≈ -3.426 +10.857 ≈7.431. Actual y=7.5. Difference: 7.5 -7.431 ≈0.069.
For x=7.2: y = -0.571*7.2 +10.857 ≈ -4.111 +10.857 ≈6.746. Actual y=7. Difference: 7 -6.746 ≈0.254.
For x=8.5: y = -0.571*8.5 +10.857 ≈ -4.853 +10.857 ≈6.004. Actual y=6. Difference: 6 -6.004 ≈ -0.004 (almost 0). Wait, (8.5,6) is very close. Wait, but earlier (6,7.5) had a small difference. Wait, maybe the line of best fit is calculated as the regression line. Let's do linear regression properly.
Let x: 5,6,7.2,8.5; y:8,7.5,7,6.
Mean x: (5+6+7.2+8.5)/4 = (26.7)/4 =6.675.
Mean y: (8+7.5+7+6)/4 =28.5/4=7.125.
Slope (m) = Σ[(x_i - x̄)(y_i - ȳ)] / Σ[(x_i - x̄)^2]
Calculate (x_i - x̄):
5-6.675= -1.675; 6-6.675= -0.675; 7.2-6.675=0.525; 8.5-6.675=1.825.
(y_i - ȳ):
8-7.125=0.875; 7.5-7.125=0.375; 7-7.125= -0.125; 6-7.125= -1.125.
Σ[(x_i - x̄)(y_i - ȳ)] = (-1.675)(0.875) + (-0.675)(0.375) + (0.525)(-0.125) + (1.825)(-1.125)
= (-1.465625) + (-0.253125) + (-0.065625) + (-2.053125) = -3.8375.
Σ[(x_i - x̄)^2] = (-1.675)^2 + (-0.675)^2 + (0.525)^2 + (1.825)^2
= 2.805625 + 0.455625 + 0.275625 + 3.330625 = 6.8675.
Slope m = -3.8375 / 6.8675 ≈ -0.559.
Intercept b = ȳ - m x̄ = 7.125 - (-0.559)(6.675) ≈7.125 + 3.731 ≈10.856.
So regression line: y = -0.559x +10.856.
Now calculate y_pred for each x:

  • x=5: y_pred = -0.559*5 +10.856 = -2.795 +10.856 =8.061. Actual y=8. Difference: |8 -8.061|=0.061.
  • x=6: y_pred = -0.559*6 +10.856 = -3.354 +10.856 =7.502. Actual y=7.5. Difference: |7.5 -7.502|=0.002.
  • x=7.2: y_pred = -0.559*7.2 +10.856 = -4.0248 +10.856 =6.8312. Actual y=7. Difference: |7 -6.8312|=0.1688.
  • x=8.5: y_pred = -0.559*8.5 +10.856 = -4.7515 +10.856 =6.1045. Actual y=6. Difference: |6 -6.1045|=0.1045.

So the smallest difference is for (6,7.5) with difference ~0.002, which is almost on the line. Wait, but the table has (6,7.5), (7.2,7), (8.5,6), (5,8). Wait, maybe I made a mistake. Wait the table on the left: first row 5,8; second 6,7.5; third 7.2,7; fourth 8.5,6. So the points ar…

Answer:

The point (6, 7.5) is closest to the line of best fit.