A-LevelMathematicsProbability and statistics9709/61 · May/June 2022 · Q76 Marks

In the past, the mean time for Jenny's morning run was 28.2 minutes. She does some extra training and she wishes to test whether her mean time has been reduced. After the training Jenny takes a random sample of 40 morning runs. She decides that if the sample mean run time is less than 27 minutes she will conclude that the training has been effective. You may assume that, after the training, Jenny's run time has a standard deviation of 4.0 minutes.

(a)State suitable null and alternative hypotheses for Jenny's test.[1]
(b)Find the probability that Jenny will make a Type I error.[3]
(c)Jenny found that the sample mean run time was 27.2 minutes. Explain briefly whether it is possible for her to make a Type I error or a Type II error or both.[2]

Mark scheme

Marking points from the official mark scheme (May/June 2022). The mark or mark type is shown in brackets.

[B1] H₀: pop mean run time = 28.2 mins H₁: pop mean run time < 28.2 mins Allow 'μ'. Not 'mean journey time'
[M1] (27 - 28.2) / (4/√40) [= -1.897] For standardising Must have √40
[M1] Φ(<-1.897) = 1 - Φ('1.897') For correct area consistent with these values
[A1] 0.0289 (3 sf)
[M1] H₀ is not rejected so...
[A1] Type II error can be made and Type I error cannot be made Both needed (accept 'only a Type II error could be made')

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.