CAIE S2 2019 March — Question 3 6 marks

Exam BoardCAIE
ModuleS2 (Statistics 2)
Year2019
SessionMarch
Marks6
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TopicApproximating the Poisson to the Normal distribution
TypeHypothesis test for single Poisson mean
DifficultyStandard +0.3 This is a straightforward application of normal approximation to Poisson for hypothesis testing. Students need to set up H₀: λ=32 vs H₁: λ<32, apply continuity correction, standardize using z = (21.5-32)/√32, and compare to critical value. While it requires understanding of approximation and hypothesis testing framework, it's a standard textbook procedure with no novel insight required, making it slightly easier than average.
Spec5.05c Hypothesis test: normal distribution for population mean
3 At factory \(A\) the mean number of accidents per year is 32 . At factory \(B\) the records of numbers of accidents before 2018 have been lost, but the number of accidents during 2018 was 21. It is known that the number of accidents per year can be well modelled by a Poisson distribution. Use an approximating distribution to test at the \(2 \%\) significance level whether the mean number of accidents at factory \(B\) is less than at factory \(A\).
Question 3:
AnswerMarks Guidance
AnswerMarks Guidance
\(H_0: \lambda = 32\), \(H_1: \lambda < 32\)B1 Accept 'population mean' (\(\mu\))
\(X \sim N(32, 32)\)B1 seen or implied
\(\frac{21.5 - 32}{\sqrt{32}}\)M1 Standardise with their values. Allow with no or wrong cc
\(= -1.856\), cv of \(z = -2.054\) (or \(-2.055\) or \(-2.053\))A1
\('1.856' < 2.054\)M1 Valid comparison or comp \(\Phi\)("1.856") with 0.98 i.e. \(0.9682 < 0.98\)
No evidence that fewer accidents at B than at AA1f No contradictions. Note Use of CV method \(x = 20.38\) M1 A1 comparison \(21.5 > 20.38\) M1 conc A1 
## Question 3:

| Answer | Marks | Guidance |
|--------|-------|----------|
| $H_0: \lambda = 32$, $H_1: \lambda < 32$ | B1 | Accept 'population mean' ($\mu$) |
| $X \sim N(32, 32)$ | B1 | seen or implied |
| $\frac{21.5 - 32}{\sqrt{32}}$ | M1 | Standardise with their values. Allow with no or wrong cc |
| $= -1.856$, cv of $z = -2.054$ (or $-2.055$ or $-2.053$) | A1 | |
| $'1.856' < 2.054$ | M1 | Valid comparison or comp $\Phi$("1.856") with 0.98 i.e. $0.9682 < 0.98$ |
| No evidence that fewer accidents at B than at A | A1f | No contradictions. Note Use of CV method $x = 20.38$ M1 A1 comparison $21.5 > 20.38$ M1 conc A1 |

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3 At factory $A$ the mean number of accidents per year is 32 . At factory $B$ the records of numbers of accidents before 2018 have been lost, but the number of accidents during 2018 was 21. It is known that the number of accidents per year can be well modelled by a Poisson distribution. Use an approximating distribution to test at the $2 \%$ significance level whether the mean number of accidents at factory $B$ is less than at factory $A$.\\

\hfill \mbox{\textit{CAIE S2 2019 Q3 [6]}}
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