Question 8 - A-Level Maths

AQA Further AS Paper 2 Statistics 2020 June — Question 8 9 marks

Exam Board	AQA
Module	Further AS Paper 2 Statistics (Further AS Paper 2 Statistics)
Year	2020
Session	June
Marks	9
Paper	Download PDF ↗
Mark scheme	Download PDF ↗
Topic	Poisson distribution
Type	Poisson hypothesis test
Difficulty	Challenging +1.2 This is a hypothesis test for Poisson distributions requiring students to set up a two-tailed test with combined parameter λ=25, calculate P(X≤16), and find the actual significance level. While it involves multiple steps and understanding of Type I error, the mechanics are straightforward applications of standard Further Maths statistics techniques without requiring novel insight or complex problem-solving.
Spec	5.02j Poisson formula: P(X=x) = e^(-lambda)*lambda^x/x!5.02k Calculate Poisson probabilities 5.05b Unbiased estimates: of population mean and variance

There are two hospitals in a city. Over a period of time, the first hospital recorded an average of 20 births a day. Over the same period of time, the second hospital recorded an average of 5 births a day. Stuart claims that birth rates in the hospitals have changed over time. On a randomly chosen day, he records a total of 16 births from the two hospitals.

Investigate Stuart's claim, using a suitable test at the 5% level of significance. [6 marks]
For a test of the type carried out in part (a), find the probability of making a Type I error, giving your answer to two significant figures. [3 marks]

Show mark scheme Show mark scheme source

Question 8:

Answer	Marks	Guidance
8(a)	States both hypotheses using
correct language.	2.5	B1

0 H : λ ≠ 25

1 X ~ Po(25)

P(X ≤ 16)

= 0.038 > 0.025

Accept H

0 There is no significant evidence to

suggest that the total birth rate in

the two hospitals has changed.

Selects and uses a Poisson model

with λ = 25 to find

P(number of births ≤ 16) or

P(number of births < 16) or the

lower or upper part of the critical

Answer	Marks	Guidance
region (X ≤ 15 or X ≥ 36)	3.3	M1

Obtains 0.038 (AWRT) or both

parts of the critical region (X ≤ 15

Answer	Marks	Guidance
and X ≥ 36)	3.4	A1

Evaluates the Poisson model by

comparing ‘their’ p-value with 0.025

or the sample value with ‘their’

Answer	Marks	Guidance
critical region	3.5a	R1

Infers H not rejected.

0 FT comparison of ‘their’ p-value

Answer	Marks	Guidance
with 0.025 or 0.05	2.2b	E1F

Concludes in context.

(The conclusion must not be

definite.)

FT their incorrect rejection of H if

0 stated or ‘their’ comparison of p-

Answer	Marks	Guidance
value and 0.025/0.05 if not	3.2a	E1
Q	Marking Instructions	AO

Answer	Marks
8(b)	Selects a method to determine the

probability of a Type I error by

considering the probability of the

upper or lower part of the critical

region e.g. by finding P(X ≤ 15) or

P(X ≤ 16)

FT one tailed test used in 8(a).

FT their Po(20) or Po(5) if used in

Answer	Marks	Guidance
8(a).	3.1a	M1

P(X ≤ 16) = 0.03775

P(X ≥ 35) = 0.03384

P(X ≥ 36) = 0.02246

P(Type I error) =

0.02229 + 0.02246

= 0.045

Develops their method by

considering the other part of the

critical region e.g. by finding

P(X ≥ 35) or P(X ≥ 36)

Do not FT one tailed test used in

8(a).

FT their Po(20) or Po(5) if used in

Answer	Marks	Guidance
8(a).	3.1a	M1

Obtains P(Type I error)

= AWRT 0.045

FT one tailed test used in 8(a) .

FT their Po(20) or Po(5) if used in

Answer	Marks	Guidance
8(a).	1.1b	A1
Total	9
Paper total	40

Question 8:
--- 8(a) ---
8(a) | States both hypotheses using
correct language. | 2.5 | B1 | H : λ = 25
0
H : λ ≠ 25
1
X ~ Po(25)
P(X ≤ 16)
= 0.038 > 0.025
Accept H
0
There is no significant evidence to
suggest that the total birth rate in
the two hospitals has changed.
Selects and uses a Poisson model
with λ = 25 to find
P(number of births ≤ 16) or
P(number of births < 16) or the
lower or upper part of the critical
region (X ≤ 15 or X ≥ 36) | 3.3 | M1
Obtains 0.038 (AWRT) or both
parts of the critical region (X ≤ 15
and X ≥ 36) | 3.4 | A1
Evaluates the Poisson model by
comparing ‘their’ p-value with 0.025
or the sample value with ‘their’
critical region | 3.5a | R1
Infers H not rejected.
0
FT comparison of ‘their’ p-value
with 0.025 or 0.05 | 2.2b | E1F
Concludes in context.
(The conclusion must not be
definite.)
FT their incorrect rejection of H if
0
stated or ‘their’ comparison of p-
value and 0.025/0.05 if not | 3.2a | E1
Q | Marking Instructions | AO | Marks | Typical Solution
--- 8(b) ---
8(b) | Selects a method to determine the
probability of a Type I error by
considering the probability of the
upper or lower part of the critical
region e.g. by finding P(X ≤ 15) or
P(X ≤ 16)
FT one tailed test used in 8(a).
FT their Po(20) or Po(5) if used in
8(a). | 3.1a | M1 | P(X ≤ 15) = 0.02229
P(X ≤ 16) = 0.03775
P(X ≥ 35) = 0.03384
P(X ≥ 36) = 0.02246
P(Type I error) =
0.02229 + 0.02246
= 0.045
Develops their method by
considering the other part of the
critical region e.g. by finding
P(X ≥ 35) or P(X ≥ 36)
Do not FT one tailed test used in
8(a).
FT their Po(20) or Po(5) if used in
8(a). | 3.1a | M1
Obtains P(Type I error)
= AWRT 0.045
FT one tailed test used in 8(a) .
FT their Po(20) or Po(5) if used in
8(a). | 1.1b | A1
Total | 9
Paper total | 40

Show LaTeX source

There are two hospitals in a city.

Over a period of time, the first hospital recorded an average of 20 births a day.

Over the same period of time, the second hospital recorded an average of 5 births a day.

Stuart claims that birth rates in the hospitals have changed over time.

On a randomly chosen day, he records a total of 16 births from the two hospitals.

\begin{enumerate}[label=(\alph*)]
\item Investigate Stuart's claim, using a suitable test at the 5% level of significance.
[6 marks]

\item For a test of the type carried out in part (a), find the probability of making a Type I error, giving your answer to two significant figures.
[3 marks]
\end{enumerate}

\hfill \mbox{\textit{AQA Further AS Paper 2 Statistics 2020 Q8 [9]}}

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