2026-08-29T20:03:39.798528 image/svg+xml Matplotlib v3.10.9, https://matplotlib.org/ © EAGLE BEACON GLOBAL A phone company compares two tariffs. Tariff A's ...

Two scatter graphs are compared by first describing the direction of their trends, and then by how tightly the points cluster around each line of best fit; the equation of each line follows the standard "two points give a gradient and intercept" method.

(a) Both scatter graphs show positive correlation, since cost rises as call minutes rise on both tariffs. However, Tariff A's points lie closer to its line of best fit than Tariff B's points lie to theirs, so Tariff A shows the stronger correlation of the two [2 marks].

(b) Using Tariff A's two given points, \( (100,16) \) and \( (500,32) \), the gradient is \( \dfrac{32-16}{500-100} = \dfrac{16}{400} = 0.04 \). Substituting \( (100,16) \) into \( y=0.04x+c \) gives \( 16=0.04(100)+c=4+c \), so \( c=12 \). Tariff A's line is \( y=0.04x+12 \) [2 marks].

(c) Using Tariff B's two points, \( (100,22) \) and \( (500,30) \), the gradient is \( \dfrac{30-22}{500-100} = \dfrac{8}{400} = 0.02 \). Substituting \( (100,22) \) gives \( 22=0.02(100)+c=2+c \), so \( c=20 \). Tariff B's line is \( y=0.02x+20 \) [2 marks].

(d) Substituting \( x=350 \) into each equation: Tariff A gives \( y=0.04(350)+12=14+12=26 \); Tariff B gives \( y=0.02(350)+20=7+20=27 \). Since \( £26 < £27 \), Tariff A is cheaper for this customer [1 mark].

Tariff B starts with a higher fixed cost (\(c=20\) against \(c=12\)) but a shallower gradient (\(0.02\) against \(0.04\)), so although Tariff A is cheaper at 350 minutes, a heavy caller using far more minutes than 500 would eventually find Tariff B better value, since its cost grows more slowly.