June 2023 Paper 2 Q12

OCR ACurrent spec4 marksIncludes hypothesis testingBinomial Distribution

12 A student has an ordinary six-sided dice. The student suspects that it is biased against six, so that when it is thrown, it is less likely to show a six than if it were fair.

In order to test this suspicion, the student plans to carry out a hypothesis test at the 5% significance level.

The student throws the dice 100 times and notes the number of times, \(X\), that it shows a six.

(a) Determine the largest value of \(X\) that would provide evidence at the 5% significance level that the dice is biased against six. [3]

Later another student carries out a similar test, at the 5% significance level. This student also throws the dice 100 times.

(b) It is given that the dice is fair.
Find the probability that the conclusion of the test is that there is significant evidence that the dice is biased against six. [1]

Mark scheme (a)

Scheme	Marks	AO
	M1	3.3
\(\mathrm{P}(X \leqslant 11) = 0.0777\) (3 sf) \(\mathrm{P}(X \leqslant 10) = 0.0427\) (3 sf)	M1	3.4
Largest value of \(X\) is 10	A1	3.4
	[3]

Notes

M1: For indication of B(100, 1/6) used and an attempt at any \(\mathrm{P}(X \leqslant a)\) in the right region, i.e. finding \(\mathrm{P}(X \leqslant a)\) for any \(a \leqslant 16\) BC

M1: For finding both \(\mathrm{P}(X \leqslant 10)\) and \(\mathrm{P}(X \leqslant 11)\) BC (allow numerical slip if clear indication that the right distribution is used)
Allow \(\lt\) in both cases if values are correct.

A1: Dep on both M marks so must have found both \(\mathrm{P}(X \leqslant 11)\) and \(\mathrm{P}(X \leqslant 10)\) correctly (condone values correct to 2sf throughout but do not accept answers that are not correct to 2sf, e.g. 0.077 or 0.042 unless there is a clear indication of truncation such as ‘…’)