Bivariate statistical investigations

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Explanation

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Worked solution

1. The data consists of 10 pairs of bivariate measurements: (x, y) where x is the number of zucchini plants and y is the average weekly harvest in kilograms. The data points are: (76, 33), (77, 35), (78, 37), (79, 39), (80, 41), (81, 43), (82, 45), (83, 47), (84, 49), (85, 51).
2. To find the y-intercept of the line of best fit using linear regression, we use the formula for the line of best fit: y = mx + b, where m is the slope and b is the y-intercept. The y-intercept b is calculated as: b = ȳ - m x̄, where ȳ is the mean of y-values and x̄ is the mean of x-values.
3. First, calculate the mean of x-values (x̄). Sum of x-values: 76 + 77 + 78 + 79 + 80 + 81 + 82 + 83 + 84 + 85. This is an arithmetic series with 10 terms, first term 76, last term 85. Sum = (10/2) * (76 + 85) = 5 * 161 = 805. Therefore, x̄ = 805 / 10 = 80.5.
4. Next, calculate the mean of y-values (ȳ). Sum of y-values: 33 + 35 + 37 + 39 + 41 + 43 + 45 + 47 + 49 + 51. This is also an arithmetic series with 10 terms, first term 33, last term 51. Sum = (10/2) * (33 + 51) = 5 * 84 = 420. Therefore, ȳ = 420 / 10 = 42.
5. Now calculate the slope m using the formula: m = Σ[(xi - x̄)(yi - ȳ)] / Σ[(xi - x̄)^2]. We compute the numerator and denominator separately.
6. Compute (xi - x̄) and (yi - ȳ) for each data point. x̄ = 80.5, ȳ = 42. For each point, calculate (xi - 80.5) and (yi - 42).
7. Calculate the product (xi - x̄)(yi - ȳ) for each point and sum them. The values are: (76-80.5)= -4.5, (33-42)= -9 → product = 40.5; (77-80.5)= -3.5, (35-42)= -7 → 24.5; (78-80.5)= -2.5, (37-42)= -5 → 12.5; (79-80.5)= -1.5, (39-42)= -3 → 4.5; (80-80.5)= -0.5, (41-42)= -1 → 0.5; (81-80.5)= 0.5, (43-42)= 1 → 0.5; (82-80.5)= 1.5, (45-42)= 3 → 4.5; (83-80.5)= 2.5, (47-42)= 5 → 12.5; (84-80.5)= 3.5, (49-42)= 7 → 24.5; (85-80.5)= 4.5, (51-42)= 9 → 40.5. Sum of products = 40.5 + 24.5 + 12.5 + 4.5 + 0.5 + 0.5 + 4.5 + 12.5 + 24.5 + 40.5 = 165.
8. Now compute Σ[(xi - x̄)^2]. For each xi: (76-80.5)^2 = (-4.5)^2 = 20.25; (77-80.5)^2 = (-3.5)^2 = 12.25; (78-80.5)^2 = (-2.5)^2 = 6.25; (79-80.5)^2 = (-1.5)^2 = 2.25; (80-80.5)^2 = (-0.5)^2 = 0.25; (81-80.5)^2 = (0.5)^2 = 0.25; (82-80.5)^2 = (1.5)^2 = 2.25; (83-80.5)^2 = (2.5)^2 = 6.25; (84-80.5)^2 = (3.5)^2 = 12.25; (85-80.5)^2 = (4.5)^2 = 20.25. Sum = 20.25 + 12.25 + 6.25 + 2.25 + 0.25 + 0.25 + 2.25 + 6.25 + 12.25 + 20.25 = 82.5.
9. Now compute the slope m = 165 / 82.5 = 2. This is an exact division: 165 ÷ 82.5 = 2.
10. Now compute the y-intercept b = ȳ - m x̄ = 42 - 2 * 80.5 = 42 - 161 = -119.
11. The y-intercept is -119. This is an exact integer, so as a simplified fraction it is -119/1.
12. The y-intercept of the line of best fit is exactly -119. This means that if there were 0 zucchini plants, the model predicts a harvest of -119 kg, which is not physically meaningful but mathematically valid for the regression line.

Reference answer: -119

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