A-LevelMathematicsProbability and statistics9709/61 · Oct/Nov 2021 · Q68 Marks

The random variable T denotes the time, in seconds, for 100 m races run by Tania. T is normally distributed with mean µ and variance σ². A random sample of 40 races run by Tania gave the following results. n = 40 Σt = 560 Σt² = 7850

(a)Calculate unbiased estimates of µ and σ².[3]
(b)The random variable S denotes the time, in seconds, for 100 m races run by Suki. S has the independent distribution N(14.2, 0.3). Using your answers to part (a), find the probability that, in a randomly chosen 100 m race, Suki's time will be at least 0.1 s more than Tania's time.[5]

Mark scheme

Marking points from the official mark scheme (Oct/Nov 2021). The mark or mark type is shown in brackets.

[B1] est µ = 560/40 = 14 accept 14
[M1] est σ² = (1/39) * (7850 - 560²/40) or (1/39) * (7850 - 14²)
[A1] 0.25641 or 0.256 (3sf) (Accept 10/39. Without 40/39 i.e. biased: est σ² = 0.25 M0 A0.)
[B1 FT] E(S-T) = 14.2 – '14' [= 0.2] (FT their 14.)
[B1 FT] Var(S-T) = 0.3 + '0.256' [= 0.55641] (FT their 0.256 including FT biased. Var(S-T) = 0.55.)
[M1] (0.1 - '0.2') / √'0.55641' [= -0.134] (Standardising with their values (note biased gives -0.135). FT their E & Var.)
[M1] P(S-T > 0.1) = 1 – Φ(‘−0.134') = Φ(‘0.134') (Finding correct area consistent with their values. Similar scheme for P(T-S) < –0.1. Similar scheme for S – T – 0.1 > 0. And T – S + 0.1 < 0.)
[A1] 0.553 (3sf) (Use of biased gives 0.554 (3sf) can score the A1.)

On Oracle Prep you can write or photograph your own answer and have it marked by AI against this mark scheme, with feedback on what you missed.