Question 17 - A-Level Maths

AQA Paper 3 2018 June — Question 17 12 marks

Exam Board	AQA
Module	Paper 3 (Paper 3)
Year	2018
Session	June
Marks	12
Paper	Download PDF ↗
Mark scheme	Download PDF ↗
Topic	Hypothesis test of binomial distributions
Type	Two-tailed test critical region
Difficulty	Standard +0.3 This is a standard two-tailed and one-tailed binomial hypothesis test at A-level. Part (a) requires setting up hypotheses, calculating binomial probabilities, and comparing to significance level. Part (b) requires finding a critical value for a one-tailed test. Both are routine applications of S1/S2 content with no novel insight required, making it slightly easier than average.
Spec	2.05b Hypothesis test for binomial proportion 2.05c Significance levels: one-tail and two-tail

Suzanne is a member of a sports club. For each sport she competes in, she wins half of the matches.

After buying a new tennis racket Suzanne plays 10 matches and wins 7 of them. Investigate, at the 10% level of significance, whether Suzanne's new racket has made a difference to the probability of her winning a match. [7 marks]
After buying a new squash racket, Suzanne plays 20 matches. Find the minimum number of matches she must win for her to conclude, at the 10% level of significance, that the new racket has improved her performance. [5 marks]

Show mark scheme Show mark scheme source

Question 17:

Answer	Marks	Guidance
17 (a)	States both hypotheses correctly for
two-tailed test	AO2.5	B1

H : p = 0.5

0 H : p ≠ 0.5

1 Under null hypothesis

X∼B(10,0.5)

𝑃𝑃(𝑋𝑋 ≥ 7)= 1−𝑃𝑃(𝑋𝑋 ≤ 6)

= 1−0.8281

= 0.172

0.172> 0.05

Accept H

0 There is not sufficient evidence

that Suzanne’s new racket has

made a difference

Answer	Marks	Guidance
States model used PI	AO3.3	M1

Calculates or

𝑃𝑃(𝑋𝑋 ≤ 6) 𝑃𝑃(𝑋𝑋 ≤ 7)

Answer	Marks	Guidance
0.828(1) or 0.945(3)	AO1.1a	M1
Obtains the correct probability for	AO1.1b	A1

Evaluates Binomial model by comparing

𝑃𝑃(𝑋𝑋 ≥ 7)

Answer	Marks	Guidance
with 0.05	AO3.5a	M1

Infers H accepted CSO

Answer	Marks	Guidance
𝑃𝑃(𝑋𝑋 ≥ 70)	AO2.2b	A1

Concludes correctly in context. (FT only

available if previous M1 mark scored

Answer	Marks	Guidance
and B1 scored)	AO3.2a	E1F

Answer	Marks
17(a)	(ALTERNATIVE using critical region)

States both hypotheses correctly for

Answer	Marks	Guidance
two-tailed test	AO2.5	B1

H : p=0.5

0 H : p≠0.5

1 Under null hypothesis

X∼B(10,0.5)

𝑃𝑃(𝑋𝑋 ≤ 1)= 0.0107 𝑜𝑜𝑜𝑜 0.0108

𝑃𝑃(𝑋𝑋 ≥ 9)= 0.0107 𝑜𝑜𝑜𝑜 0.0108

Critical region is

and

not in critical region

𝑋𝑋 ≤ 1 𝑋𝑋 ≥ 9

𝑋𝑋 = 7

Accept H

0 There is not sufficient evidence

that Suzanne’s new racket has

made a difference

Answer	Marks	Guidance
States model used PI	AO3.3	M1
Considers critical region	AO1.1a	M1
Identifies critical region	AO1.1b	A1

Evaluates Binomial model by comparing

Answer	Marks	Guidance
with critical values	AO3.5a	M1

Infers H accepted CSO

Answer	Marks	Guidance
𝑋𝑋 = 7 0	AO2.2b	A1

Correctly concludes in context. ‘Not

sufficient evidence’ or equivalent

Answer	Marks	Guidance
required.	AO3.2a	E1F
Q	Marking Instructions	AO

Answer	Marks	Guidance
17(b)	States model used PI	AO3.3

Require P(Y > y)<0.1

P ( Y ≥13 )=0.1316>0.1

P ( Y ≥14 )=0.0577>0.1

Minimum number of matches

= 14

Expresses condition in terms of a

cumulative probability statement PI by

Answer	Marks	Guidance
sight of	AO3.1b	M1

Tests one appropriate value for y

Answer	Marks	Guidance
𝑃𝑃(𝑌𝑌 ≥ 𝑐𝑐) < 0.1	AO3.2b	R1

Obtains at least two correct cumulative

Answer	Marks	Guidance
probabilities	AO1.1b	A1

Obtains the correct minimum number of

Answer	Marks	Guidance
matches CSO	AO3.2a	A1
Total	12
Marking Instructions	AO	Marks

Question 17:
--- 17 (a) ---
17 (a) | States both hypotheses correctly for
two-tailed test | AO2.5 | B1 | X = number of matches won
H : p = 0.5
0
H : p ≠ 0.5
1
Under null hypothesis
X∼B(10,0.5)
𝑃𝑃(𝑋𝑋 ≥ 7)= 1−𝑃𝑃(𝑋𝑋 ≤ 6)
= 1−0.8281
= 0.172
0.172> 0.05
Accept H
0
There is not sufficient evidence
that Suzanne’s new racket has
made a difference
States model used PI | AO3.3 | M1
Calculates or
𝑃𝑃(𝑋𝑋 ≤ 6) 𝑃𝑃(𝑋𝑋 ≤ 7)
0.828(1) or 0.945(3) | AO1.1a | M1
Obtains the correct probability for | AO1.1b | A1
Evaluates Binomial model by comparing
𝑃𝑃(𝑋𝑋 ≥ 7)
with 0.05 | AO3.5a | M1
Infers H accepted CSO
𝑃𝑃(𝑋𝑋 ≥ 70) | AO2.2b | A1
Concludes correctly in context. (FT only
available if previous M1 mark scored
and B1 scored) | AO3.2a | E1F
--- 17(a) ---
17(a) | (ALTERNATIVE using critical region)
States both hypotheses correctly for
two-tailed test | AO2.5 | B1 | X = number of matches won
H : p=0.5
0
H : p≠0.5
1
Under null hypothesis
X∼B(10,0.5)
𝑃𝑃(𝑋𝑋 ≤ 1)= 0.0107 𝑜𝑜𝑜𝑜 0.0108
𝑃𝑃(𝑋𝑋 ≥ 9)= 0.0107 𝑜𝑜𝑜𝑜 0.0108
Critical region is
and
not in critical region
𝑋𝑋 ≤ 1 𝑋𝑋 ≥ 9
𝑋𝑋 = 7
Accept H
0
There is not sufficient evidence
that Suzanne’s new racket has
made a difference
States model used PI | AO3.3 | M1
Considers critical region | AO1.1a | M1
Identifies critical region | AO1.1b | A1
Evaluates Binomial model by comparing
with critical values | AO3.5a | M1
Infers H accepted CSO
𝑋𝑋 = 7 0 | AO2.2b | A1
Correctly concludes in context. ‘Not
sufficient evidence’ or equivalent
required. | AO3.2a | E1F
Q | Marking Instructions | AO | Marks | Typical Solution
--- 17(b) ---
17(b) | States model used PI | AO3.3 | M1 | Y∼B(20,0.5)
Require P(Y > y)<0.1
P ( Y ≥13 )=0.1316>0.1
P ( Y ≥14 )=0.0577>0.1
Minimum number of matches
= 14
Expresses condition in terms of a
cumulative probability statement PI by
sight of | AO3.1b | M1
Tests one appropriate value for y
𝑃𝑃(𝑌𝑌 ≥ 𝑐𝑐) < 0.1 | AO3.2b | R1
Obtains at least two correct cumulative
probabilities | AO1.1b | A1
Obtains the correct minimum number of
matches CSO | AO3.2a | A1
Total | 12
Marking Instructions | AO | Marks | Typical Solution

Show LaTeX source

Suzanne is a member of a sports club.

For each sport she competes in, she wins half of the matches.

\begin{enumerate}[label=(\alph*)]
\item After buying a new tennis racket Suzanne plays 10 matches and wins 7 of them.

Investigate, at the 10% level of significance, whether Suzanne's new racket has made a difference to the probability of her winning a match.
[7 marks]

\item After buying a new squash racket, Suzanne plays 20 matches. Find the minimum number of matches she must win for her to conclude, at the 10% level of significance, that the new racket has improved her performance.
[5 marks]
\end{enumerate}

\hfill \mbox{\textit{AQA Paper 3 2018 Q17 [12]}}

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Q1 1 Q2 1 Q3 1 Q4 3 Q5 3 Q6 13 Q7 5 Q8 9 Q9 7 Q10 10 Q11 1 Q12 1 Q13 3 Q14 6 Q15 7 Q16 12 Q17 12 Q18 8