AQA Paper 3 2021 June — Question 17 11 marks

Exam BoardAQA
ModulePaper 3 (Paper 3)
Year2021
SessionJune
Marks11
PaperDownload PDF ↗
Mark schemeDownload PDF ↗
TopicHypothesis test of binomial distributions
TypeOne-tailed hypothesis test (upper tail, H₁: p > pβ‚€)
DifficultyStandard +0.3 This is a standard A-level statistics question on binomial distribution and hypothesis testing. Parts (a)-(c) are routine bookwork requiring only recall of binomial assumptions and calculator use. Part (d) is a textbook one-tailed binomial hypothesis test with no complicationsβ€”students follow a standard procedure they've practiced extensively. While worth 11 marks total, it requires no novel insight or problem-solving beyond applying memorized techniques.
Spec2.04b Binomial distribution: as model B(n,p)2.04c Calculate binomial probabilities2.05b Hypothesis test for binomial proportion2.05c Significance levels: one-tail and two-tail
James is playing a mathematical game on his computer. The probability that he wins is 0.6 As part of an online tournament, James plays the game 10 times. Let \(Y\) be the number of games that James wins.
  1. State two assumptions, in context, for \(Y\) to be modelled as \(B(10, 0.6)\) [2 marks]
  2. Find \(P(Y = 4)\) [1 mark]
  3. Find \(P(Y \geq 4)\) [2 marks]
  4. After practising the game, James claims that he has increased his probability of winning the game. In a random sample of 15 subsequent games, he wins 12 of them. Test at a 5% significance level whether James's claim is correct. [6 marks]
Question 17:
AnswerMarks Guidance
17(a)States one correct binomial
assumption in context3.5b E1
game is independent of him
winning another game
The probability of James winning
remains constant at 0.6 from game
to game
States a second correct
binomial assumption in context
eg each time he plays he can
only win or not win or the
number games is fixed or
winning one game is
independent of him winning
another game
Condone omission of 0.6 from
AnswerMarks Guidance
the statement3.5b E1
Subtotal2
QMarking Instructions AO
AnswerMarks Guidance
17(b)Obtains correct probability
AWRT 0.113.1b B1
Subtotal1
QMarking Instructions AO
AnswerMarks
17(c)Calculates either
or
using the𝑃𝑃 B(iπ‘Œπ‘Œno≀m3ia)l =
d0i.0st5r4ib7u6tion 𝑃𝑃 (π‘Œπ‘Œ ≀ 4) =
0o.r1 6623
states
or
subtrac𝑃𝑃ts(π‘Œπ‘Œ thβ‰₯ei4r )st=ate1dβˆ’ v a𝑃𝑃l(uπ‘Œπ‘Œe ≀of3 )
AnswerMarks Guidance
from 13.1b M1
𝑃𝑃(π‘Œπ‘Œ β‰₯ 4) = 1βˆ’π‘ƒπ‘ƒ(π‘Œπ‘Œ ≀ 3)
= 1βˆ’0.05476
= 0.94524
𝑃𝑃O ( bπ‘Œπ‘Œta≀ins3 ) correct probability
AnswerMarks Guidance
AWFW [0.94, 0.95]1.1b A1
Subtotal2
QMarking Instructions AO
AnswerMarks Guidance
17(d)States both hypotheses
correctly for a one-tailed test2.5 B1
: p = 0.6
: p > 0.6
𝐻𝐻0
X B(15,0.6)
𝐻𝐻 1
P(∼X β‰₯ 12) = 1 – P
= 1 – 0.9094
= 0.0905 (𝑋𝑋 ≀ 11)
0.0905 > 0.05 so accept
There is insufficient evide𝐻𝐻n 0 ce to
suggest that the probability of
James winning the game has
increased.
Uses correct binomial model to
obtain either P or
or or
(𝑋𝑋 ≀ 11)
P 𝑃𝑃( I 𝑋𝑋by≀ c1ri2tic ) al re𝑃𝑃g (𝑋𝑋ionβ‰₯ 13)
AnswerMarks Guidance
𝑃𝑃(𝑋𝑋 β‰₯ 14)3.3 M1
𝑋𝑋 β‰₯ 13 π‘œπ‘œπ‘Ÿπ‘Ÿ 𝑋𝑋 β‰₯ 14
Obtains the correct probability
for P(X β‰₯12)or
AnswerMarks Guidance
obtains correct critical region1.1b A1
𝑋𝑋 β‰₯ 13
Evaluates binomial model by
comparing theirP(X β‰₯12)with
0.05 or
Compares 12 with their critical
region and makes their
AnswerMarks Guidance
inference3.5a M1
Infer is not rejected
CSO
AnswerMarks Guidance
Allow𝐻𝐻 r 0 eference to2.2b A1
Concludes correctl y𝐻𝐻 i 1 n context
that there is insufficient
evidence to suggest that the
probability of winning the game
AnswerMarks Guidance
has increased.3.2a R1
Subtotal6
Question Total11
QMarking Instructions AO 
Question 17:
--- 17(a) ---
17(a) | States one correct binomial
assumption in context | 3.5b | E1 | The event of James winning one
game is independent of him
winning another game
The probability of James winning
remains constant at 0.6 from game
to game
States a second correct
binomial assumption in context
eg each time he plays he can
only win or not win or the
number games is fixed or
winning one game is
independent of him winning
another game
Condone omission of 0.6 from
the statement | 3.5b | E1
Subtotal | 2
Q | Marking Instructions | AO | Marks | Typical Solution
--- 17(b) ---
17(b) | Obtains correct probability
AWRT 0.11 | 3.1b | B1 | 0.111
Subtotal | 1
Q | Marking Instructions | AO | Marks | Typical Solution
--- 17(c) ---
17(c) | Calculates either
or
using the𝑃𝑃 B(iπ‘Œπ‘Œno≀m3ia)l =
d0i.0st5r4ib7u6tion 𝑃𝑃 (π‘Œπ‘Œ ≀ 4) =
0o.r1 6623
states
or
subtrac𝑃𝑃ts(π‘Œπ‘Œ thβ‰₯ei4r )st=ate1dβˆ’ v a𝑃𝑃l(uπ‘Œπ‘Œe ≀of3 )
from 1 | 3.1b | M1 | 𝑃𝑃(π‘Œπ‘Œ ≀ 3) = 0.05476
𝑃𝑃(π‘Œπ‘Œ β‰₯ 4) = 1βˆ’π‘ƒπ‘ƒ(π‘Œπ‘Œ ≀ 3)
= 1βˆ’0.05476
= 0.94524
𝑃𝑃O ( bπ‘Œπ‘Œta≀ins3 ) correct probability
AWFW [0.94, 0.95] | 1.1b | A1
Subtotal | 2
Q | Marking Instructions | AO | Marks | Typical Solution
--- 17(d) ---
17(d) | States both hypotheses
correctly for a one-tailed test | 2.5 | B1 | X = number of games won
: p = 0.6
: p > 0.6
𝐻𝐻0
X B(15,0.6)
𝐻𝐻 1
P(∼X β‰₯ 12) = 1 – P
= 1 – 0.9094
= 0.0905 (𝑋𝑋 ≀ 11)
0.0905 > 0.05 so accept
There is insufficient evide𝐻𝐻n 0 ce to
suggest that the probability of
James winning the game has
increased.
Uses correct binomial model to
obtain either P or
or or
(𝑋𝑋 ≀ 11)
P 𝑃𝑃( I 𝑋𝑋by≀ c1ri2tic ) al re𝑃𝑃g (𝑋𝑋ionβ‰₯ 13)
𝑃𝑃(𝑋𝑋 β‰₯ 14) | 3.3 | M1
𝑋𝑋 β‰₯ 13 π‘œπ‘œπ‘Ÿπ‘Ÿ 𝑋𝑋 β‰₯ 14
Obtains the correct probability
for P(X β‰₯12)or
obtains correct critical region | 1.1b | A1
𝑋𝑋 β‰₯ 13
Evaluates binomial model by
comparing theirP(X β‰₯12)with
0.05 or
Compares 12 with their critical
region and makes their
inference | 3.5a | M1
Infer is not rejected
CSO
Allow𝐻𝐻 r 0 eference to | 2.2b | A1
Concludes correctl y𝐻𝐻 i 1 n context
that there is insufficient
evidence to suggest that the
probability of winning the game
has increased. | 3.2a | R1
Subtotal | 6
Question Total | 11
Q | Marking Instructions | AO | Marks | Typical Solution
James is playing a mathematical game on his computer.

The probability that he wins is 0.6

As part of an online tournament, James plays the game 10 times.

Let $Y$ be the number of games that James wins.

\begin{enumerate}[label=(\alph*)]
\item State two assumptions, in context, for $Y$ to be modelled as $B(10, 0.6)$
[2 marks]

\item Find $P(Y = 4)$
[1 mark]

\item Find $P(Y \geq 4)$
[2 marks]

\item After practising the game, James claims that he has increased his probability of winning the game.

In a random sample of 15 subsequent games, he wins 12 of them.

Test at a 5% significance level whether James's claim is correct.
[6 marks]
\end{enumerate}

\hfill \mbox{\textit{AQA Paper 3 2021 Q17 [11]}}
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