October 2021 Paper 3 Q1

EdexcelCurrent spec7 marksIncludes samplingBinomial Distribution Data Processing

(a) State one disadvantage of using quota sampling compared with simple random sampling. (1)

In a university 8% of students are members of the university dance club.

A random sample of 36 students is taken from the university.

The random variable \(X\) represents the number of these students who are members of the dance club.

(b) Using a suitable model for \(X\), find

(i) \(\mathrm{P}(X = 4)\)

(ii) \(\mathrm{P}(X \geqslant 7)\) (3)

Only 40% of the university dance club members can dance the tango.

A random sample of 50 students is taken from the university.

(d) Find the probability that fewer than 3 of these students are members of the university dance club and can dance the tango. (2)

Mark scheme (a)

Scheme	Marks	AO
Disadvantage: e.g. Not random; cannot use (reliably) for inferences	B1	1.1b
	(1)

Notes

B1 for a suitable disadvantage:

Allow (B1)	Do NOT allow (B0)
Not random or less random (o.e.)	Not representative
Cannot use (reliably) for inferences	Less accurate
(More likely to be) biased	Any comment based on time or cost
	Any mention of skew
	Any mention of non-response

Mark scheme (b)

Scheme	Marks	AO
[Sight or correct use of] \(X \sim \mathrm{B}(36, 0.08)\)	M1	3.3
(i) \(\mathrm{P}(X = 4) = 0.167387\ldots\) awrt 0.167	A1	1.1b
(ii) \([\mathrm{P}(X \geqslant 7) = 1 - \mathrm{P}(X \leqslant 6) =]\ 0.022233\ldots\) awrt 0.0222	A1	1.1b
	(3)

Notes

M1 for sight of \(\mathrm{B}(36, 0.08)\) Allow in words: binomial with \(n = 36\) and \(p = 0.08\)
may be implied by one correct answer to 2sf or sight of \(\mathrm{P}(X \leqslant 6) = 0.97776\ldots\) i.e. awrt 0.98
Allow for \(36\mathrm{C}4 \times 0.08^4 \times 0.92^{32}\) as this is “correct use”

(i) 1^st A1 for awrt 0.167 NB An answer of just awrt 0.167 scores M1(\(\Rightarrow\))1^st A1

(ii) 2^nd A1 for awrt 0.0222

Mark scheme (c)

Scheme	Marks	AO
P(In dance club and dance tango) \(= 0.4 \times 0.08 =\) 0.032 or \(\underline{\underline{\dfrac{4}{125}}}\) or 3.2%	B1	1.1b
	(1)

Notes

B1 for 0.032 o.e. (Can allow for sight of \(0.4 \times 0.08\))

Mark scheme (d)

Scheme	Marks	AO
[Let \(T\) = those who can dance the Tango. Sight or use of] \(T \sim \mathrm{B}(50, \text{``}0.032\text{''})\)	M1	3.3
\([\mathrm{P}(T \lt 3) = \mathrm{P}(T \leqslant 2) =]\ 0.7850815\ldots\) awrt 0.785	A1	1.1b
	(2)
	(7 marks)

Notes

M1 for sight of \(\mathrm{B}(50, \text{``}0.032\text{''})\) ft their answer to (c) provided it is a probability \(\neq 0.08\)
may be implied by correct answer
or sight of \([\mathrm{P}(T \leqslant 3)] = 0.924348\ldots\) i.e. awrt 0.924 or \(\mathrm{P}(T \leqslant 2)\) as part of \(1 - \mathrm{P}(T \leqslant 2)\) calc.

A1 for awrt 0.785

MR Allow MR of 50 (e.g. 30) provided clearly attempting \(\mathrm{P}(T \leqslant 2)\) and score M1A0