AS June 2025 Q4

EdexcelAS paperCurrent spec7 marksChi-Squared Tests

4. A 5-question test is taken by 80 students.

Each question was answered either correctly or incorrectly.

Number of questions answered correctly	0	1	2	3	4	5
Observed frequency	15	23	31	9	2	0

(a) Show that the proportion of questions answered correctly by these 80 students is 0.3 (2)

Anisa believes that the number of questions answered correctly can be modelled using a binomial distribution.
In order to test her belief, Anisa first calculates some of the expected frequencies.

Number of questions answered correctly	0	1	2	3	4	5
Expected frequency	13.45	\(r\)	\(s\)	10.58	2.27	0.19

(b) Find the value of \(r\) and the value of \(s\) (3)

Mark scheme (a)

Scheme	Marks	AO
\(\text{mean} = \dfrac{[15 \times 0 +]\, 23 \times 1 + 31 \times 2 + 9 \times 3 + 4 \times 2\, [+ 5 \times 0]}{80}\ [= 1.5]\)	M1	1.1b
\(p = \dfrac{1.5}{5} = 0.3\)*	A1*	1.1b
	(2)

Notes

M1: Attempt to find the mean with at least 3 correct products

A1*: cso fully correct solution leading to given answer 0.3
May be done in a single calculation for M1A1
\(\dfrac{[15 \times 0 +]\, 23 \times 1 + 31 \times 2 + 9 \times 3 + 4 \times 2\, [+ 5 \times 0]}{400} = 0.3\)

Mark scheme (b)

Scheme	Marks	AO
[\(X \sim \mathrm{B}(5, 0.3)\)] \(\mathrm{P}(X = 1) = 0.36015\) or \(\mathrm{P}(X = 2) = 0.3087\)	M1	1.1b
\(r = 80 \times 0.36015 = 28.8\ldots\) awrt 28.8	A1	1.1b
\(s = 80 \times 0.3087 = 24.69\ldots\) awrt 24.7	A1	1.1b
	(3)

Notes

M1: Either correct probability awrt 0.36 or awrt 0.31
(may be implied by a correct value of \(r\) or a correct value of \(s\))

A1: awrt 28.8

A1: awrt 24.7

Mark scheme (c)

Scheme	Marks	AO
Need to combine last 3 columns since \(E_i\) are \(\lt 5\)	B1	2.4
Therefore 4 – 2 degrees of freedom since proportion for Binomial estimated from \(O_i\)	B1	2.4
	(2)
	(7 marks)

Notes

B1: For explaining need to pool last 3 columns/cells and \(E_i \lt 5\)
allow 4 cells remaining after pooling for need to pool last 3 columns/cells
must be referencing expected frequencies (not observed frequencies)

B1: For proportion/parameter/\(p\)/mean/0.3 calculated/estimated (from data/\(O_i\))
and 2 constraints or 4 – 2 or 4 – 1 – 1 which must come from correct reasoning