Line of Best Fit Worksheet PDF | Download Free PDF | Scatter Plot ... - Free Printable

1. a) Scatterplot: Plot the points (2,77), (5,92), (1,70), (0,63), (4,90), (2,75), (3,84) on graph paper with study hours on the x-axis and grade on the y-axis.

b) To find the line of best fit, calculate the slope and y-intercept using the least squares method.

First, calculate the means:
- Mean of x (study hours): (2+5+1+0+4+2+3)/7 = 17/7 ≈ 2.43
- Mean of y (grade): (77+92+70+63+90+75+84)/7 = 551/7 ≈ 78.71

Calculate the slope (m):
- m = Σ[(xi - x̄)(yi - ȳ)] / Σ[(xi - x̄)²]

Calculate the numerator and denominator:
- Σ[(xi - x̄)(yi - ȳ)] = (2-2.43)(77-78.71) + (5-2.43)(92-78.71) + (1-2.43)(70-78.71) + (0-2.43)(63-78.71) + (4-2.43)(90-78.71) + (2-2.43)(75-78.71) + (3-2.43)(84-78.71)
- = (-0.43)(-1.71) + (2.57)(13.29) + (-1.43)(-8.71) + (-2.43)(-15.71) + (1.57)(11.29) + (-0.43)(-3.71) + (0.57)(5.29)
- = 0.7353 + 34.1653 + 12.4553 + 38.1753 + 17.7253 + 1.5953 + 3.0153 ≈ 107.87

- Σ[(xi - x̄)²] = (-0.43)² + (2.57)² + (-1.43)² + (-2.43)² + (1.57)² + (-0.43)² + (0.57)²
- = 0.1849 + 6.6049 + 2.0449 + 5.9049 + 2.4649 + 0.1849 + 0.3249 ≈ 17.71

- m ≈ 107.87 / 17.71 ≈ 6.09

Calculate the y-intercept (b):
- b = ȳ - m*x̄ = 78.71 - 6.09*2.43 ≈ 78.71 - 14.80 ≈ 63.91

Equation: y = 6.09x + 63.91

c) Predict the grade for 6 hours of study:
- y = 6.09*6 + 63.91 = 36.54 + 63.91 = 100.45

Grade expected: 100.45

d) No, this line cannot go on forever. The model assumes a linear relationship, but in reality, there are limits to how much studying can improve a grade. After a certain point, additional study time may not lead to higher grades due to diminishing returns or other factors like fatigue.

2. a) Scatterplot: Plot the points (67,8.5), (70,9.5), (73.5,11), (75,12), (78,13), (66,8) on graph paper with height on the x-axis and shoe size on the y-axis.

b) To find the line of best fit, calculate the slope and y-intercept using the least squares method.

First, calculate the means:
- Mean of x (height): (67+70+73.5+75+78+66)/6 = 429.5/6 ≈ 71.58
- Mean of y (shoe size): (8.5+9.5+11+12+13+8)/6 = 62/6 ≈ 10.33

Calculate the slope (m):
- m = Σ[(xi - x̄)(yi - ȳ)] / Σ[(xi - x̄)²]

Calculate the numerator and denominator:
- Σ[(xi - x̄)(yi - ȳ)] = (67-71.58)(8.5-10.33) + (70-71.58)(9.5-10.33) + (73.5-71.58)(11-10.33) + (75-71.58)(12-10.33) + (78-71.58)(13-10.33) + (66-71.58)(8-10.33)
- = (-4.58)(-1.83) + (-1.58)(-0.83) + (1.92)(0.67) + (3.42)(1.67) + (6.42)(2.67) + (-5.58)(-2.33)
- = 8.3814 + 1.3114 + 1.2864 + 5.7114 + 17.1314 + 12.9914 ≈ 46.81

- Σ[(xi - x̄)²] = (-4.58)² + (-1.58)² + (1.92)² + (3.42)² + (6.42)² + (-5.58)²
- = 20.9764 + 2.4964 + 3.6864 + 11.6964 + 41.2164 + 31.1364 ≈ 111.11

- m ≈ 46.81 / 111.11 ≈ 0.421

Calculate the y-intercept (b):
- b = ȳ - m*x̄ = 10.33 - 0.421*71.58 ≈ 10.33 - 30.18 ≈ -19.85

Equation: y = 0.421x - 19.85

c) Predict the height for a shoe size of 10.5:
- 10.5 = 0.421x - 19.85
- 0.421x = 10.5 + 19.85 = 30.35
- x ≈ 30.35 / 0.421 ≈ 72.09

Height expected: 72.09 inches

d) The slope of the line represents the rate of change of shoe size with respect to height. In this context, it means that for every additional inch in height, the shoe size increases by approximately 0.421 sizes.

3. a) To find the line of best fit, calculate the slope and y-intercept using the least squares method.

First, calculate the means:
- Mean of x (meal cost): (4.75+6.84+12.52+20.42+8.97)/5 = 53.5/5 = 10.7
- Mean of y (tip): (0.50+0.90+1.50+3.00+1.00)/5 = 6.90/5 = 1.38

Calculate the slope (m):
- m = Σ[(xi - x̄)(yi - ȳ)] / Σ[(xi - x̄)²]

Calculate the numerator and denominator:
- Σ[(xi - x̄)(yi - ȳ)] = (4.75-10.7)(0.50-1.38) + (6.84-10.7)(0.90-1.38) + (12.52-10.7)(1.50-1.38) + (20.42-10.7)(3.00-1.38) + (8.97-10.7)(1.00-1.38)
- = (-5.95)(-0.88) + (-3.86)(-0.48) + (1.82)(0.12) + (9.72)(1.62) + (-1.73)(-0.38)
- = 5.236 + 1.8528 + 0.2184 + 15.7464 + 0.6574 ≈ 23.71

- Σ[(xi - x̄)²] = (-5.95)² + (-3.86)² + (1.82)² + (9.72)² + (-1.73)²
- = 35.4025 + 14.8996 + 3.3124 + 94.4784 + 2.9929 ≈ 151.09

- m ≈ 23.71 / 151.09 ≈ 0.157

Calculate the y-intercept (b):
- b = ȳ - m*x̄ = 1.38 - 0.157*10.7 ≈ 1.38 - 1.6799 ≈ -0.2999

Equation: y = 0.157x - 0.3

c) Predict the tip for a meal costing $10.50:
- y = 0.157*10.50 - 0.3 = 1.6485 - 0.3 = 1.3485

Tip expected: $1.35