Theory question 115
StatisticsThe table shows the marks obtained by some candidates in Physics (y) and Mathematics (x) tests.
Mathematics: 43, 46, 48, 39, 30, 60, 8, 45, 40
Physics: 54, 53, 63, 30, 44, 75, 20, 33, 49
(a) (i) Represent this information on a scatter diagram. (ii) Find x̄ and ȳ, the mean of x and y respectively. (iii) Draw the line of best fit to pass through (x̄, ȳ)
(b) Find the equation of the line in (a)(iii).
(c) Use your equation in (b) to find, correct to one decimal place, the mark in Physics for a candidate who scored 28 in mathematics
Show the model answer
(a)(i) Plot the 9 points (x, y) with Mathematics on the horizontal axis
(ii) x̄ = 359/9 ≈ 39.9; ȳ = 421/9 ≈ 46.8
(iii) Draw a straight line through (39.9, 46.8) with roughly equal numbers of points on each side
(b) Least squares: Σxy = 18379, Σx² = 15979
b = (18379 − 9 × 39.89 × 46.78)/(15979 − 9 × 39.89²) ≈ 1585.6/1658.9 ≈ 0.956
y − 46.8 = 0.956(x − 39.9), i.e. y ≈ 0.956x + 8.65 (a line drawn by eye gives a similar equation)
(c) x = 28: y ≈ 0.956(28) + 8.65 ≈ 35.4
The line of best fit must pass through the mean point (x̄, ȳ); read or calculate its gradient.