Bivariate statistical investigations

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Formula

Explanation

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Worked solution

1. The data consists of 10 pairs of bivariate measurements: (x, y) where x is the length of wire in cm and y is the number of beads. The values are: (92, 61), (94, 63), (96, 65), (98, 67), (100, 69), (102, 71), (104, 73), (106, 75), (108, 77), (110, 79).
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 b is the y-intercept. The formula for b is: b = (Σy)(Σx²) - (Σx)(Σxy) / [n(Σx²) - (Σx)²].
3. We compute the required sums: Σx, Σy, Σxy, Σx², and n = 10.
4. Σx = 92 + 94 + 96 + 98 + 100 + 102 + 104 + 106 + 108 + 110. This is an arithmetic series with first term 92, last term 110, and 10 terms. Sum = (10/2) × (92 + 110) = 5 × 202 = 1010.
5. Σy = 61 + 63 + 65 + 67 + 69 + 71 + 73 + 75 + 77 + 79. This is an arithmetic series with first term 61, last term 79, and 10 terms. Sum = (10/2) × (61 + 79) = 5 × 140 = 700.
6. Σxy = (92×61) + (94×63) + (96×65) + (98×67) + (100×69) + (102×71) + (104×73) + (106×75) + (108×77) + (110×79). We compute each product: 92×61 = 5612, 94×63 = 5922, 96×65 = 6240, 98×67 = 6566, 100×69 = 6900, 102×71 = 7242, 104×73 = 7592, 106×75 = 7950, 108×77 = 8316, 110×79 = 8690. Sum = 5612 + 5922 = 11534; +6240 = 17774; +6566 = 24340; +6900 = 31240; +7242 = 38482; +7592 = 46074; +7950 = 54024; +8316 = 62340; +8690 = 71030. So Σxy = 71030.
7. Σx² = 92² + 94² + 96² + 98² + 100² + 102² + 104² + 106² + 108² + 110². We compute each square: 92² = 8464, 94² = 8836, 96² = 9216, 98² = 9604, 100² = 10000, 102² = 10404, 104² = 10816, 106² = 11236, 108² = 11664, 110² = 12100. Sum = 8464 + 8836 = 17300; +9216 = 26516; +9604 = 36120; +10000 = 46120; +10404 = 56524; +10816 = 67340; +11236 = 78576; +11664 = 90240; +12100 = 102340. So Σx² = 102340.
8. Now substitute into the formula for b: b = [Σy × Σx² - Σx × Σxy] / [n × Σx² - (Σx)²]. Numerator = (700 × 102340) - (1010 × 71030). Compute 700 × 102340 = 71,638,000. Compute 1010 × 71030 = 1010 × 71000 = 71,710,000; 1010 × 30 = 30,300; total = 71,740,300. So numerator = 71,638,000 - 71,740,300 = -102,300.
9. Denominator = 10 × 102340 - (1010)² = 1,023,400 - 1,020,100 = 3,300.
10. So b = -102,300 / 3,300. Simplify the fraction: divide numerator and denominator by 100: -1023 / 33. Now divide 1023 by 33: 33 × 31 = 1023. So -1023 / 33 = -31. Therefore, b = -31.
11. The y-intercept of the line of best fit is -31. This means that if no wire were used (x = 0), the model predicts -31 beads, which is not physically meaningful but mathematically valid for the regression line.
12. The result is verified by noting that the data shows a perfect linear pattern: for every 2 cm increase in wire length, the number of beads increases by 2. The slope is 1. Using point (100, 69), the equation is y = x - 31. So when x = 0, y = -31. This confirms the y-intercept is exactly -31.

Reference answer: -31

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