June 2023 Paper 2 Q9

OCR ACurrent spec6 marksIncludes samplingBinomial Distribution Data Processing

9 A school contains 500 students in years 7 to 11 and 250 students in years 12 and 13. A random sample of 20 students is selected to represent the school at a parents’ evening. The number of students in the sample who are from years 12 and 13 is denoted by \(X\).

(a) State a suitable binomial model for \(X\). [1]

Use your model to answer the following.

(b)

(i) Write down an expression for \(\mathrm{P}(X = x)\). [1]

(ii) State, in set notation, the values of \(x\) for which your expression is valid. [1]

(d) State one disadvantage of using a random sample in this context. [1]

Mark scheme (a)

Scheme	Marks	AO
\(\mathrm{B}\left(20, \frac{1}{3}\right)\)	B1	3.3
	[1]

Notes

B1: Allow ‘\(n = 20, p = \frac{1}{3}\)’ or just \(\left(20, \frac{1}{3}\right)\). Accept \(p = 0.\dot{3}\) but not 0.333

Mark scheme (b)

Scheme	Marks	AO
(i) \({}^{20}\mathrm{C}_x\left(\frac{2}{3}\right)^{(20 - x)}\left(\frac{1}{3}\right)^x\)	B1FT	3.4
	[1]
(ii) \(\{x : x \in \mathbb{Z},\ 0 \leqslant x \leqslant 20\}\)	B1FT	1.2
	[1]

Notes

(b)(i)

B1FT: FT their values of \(n\) and \(p\) and allow \(\dbinom{20}{x}\) as alternative to \({}^{20}\mathrm{C}_x\)
(If decimals used both must be correct to 3sf)

(b)(ii)

B1FT: FT their ‘20’ only
Any acceptable alternative in set notation
Indication that \(x\) is an integer must be present (but condone ‘integer’ or ‘whole number’ or \(x \in \mathbb{N}\))

Mark scheme (c)

Scheme	Marks	AO
\(\mathrm{P}(X \leqslant 9) - \mathrm{P}(X \leqslant 4)\) attempted	M1	3.4
\((= 0.90810 - 0.15151)\) \(= 0.757\) (3 sf)	A1	1.1
	[2]

Notes

M1: Condone \(\mathrm{P}(X \leqslant 9) - \mathrm{P}(X \leqslant 5)\), or \(0.90810 - 0.29721\) or 0.611
May be implied by correct working or answer.
Allow sum of \(\mathrm{P}(X = x)\) for \(x\) from 5 to 9.

A1: BC cao

Mark scheme (d)

Scheme	Marks	AO
May include too many from one year and too few from another year	B1	3.5b
	[1]

Notes

B1: For an explanation (which must be in context) showing clear recognition that a random sample (of this size) would not be representative across the year groups.
Acceptable similar answers in context include:

‘should include roughly equal numbers from each year’
‘not representative of each year group’
‘may not include students from every year group’

Do not accept ‘may end up with more students from Y7-11 than from Y12-13’ (because a representative sample would have this)
Must be in context – so do not accept generic answers without context such as just: ‘It might be biased’ or ‘It’s not representative’ or ‘it should be a stratified sample’

Do not accept answers solely referring to the suitability of the individual students within the sample to represent the school