Standard +0.3 This is a straightforward two-tailed hypothesis test using normal approximation to Poisson. Students must recognize the setup, calculate the test statistic using standard formulas, and compare to critical values. While it requires multiple steps (stating hypotheses, finding expected value, calculating z-statistic, conclusion), each step follows a standard procedure with no novel insight required. Slightly above average difficulty due to the approximation requirement and two-tailed nature, but remains a textbook-style question.
4 At a certain company, computer faults occur randomly and at a constant mean rate. In the past this mean rate has been 2.1 per week. Following an update, the management wish to determine whether the mean rate has changed. During 20 randomly chosen weeks it is found that 54 computer faults occur. Use a suitable approximation to test at the \(5 \%\) significance level whether the mean rate has changed.
Or pop weekly mean \(= 2.1\) etc.; allow 'population mean' not just 'mean'; ft their '42' (Accept alt method \(N(2.1, 2.1/20)\))
B1\(\checkmark\)
\(\frac{53.5-42}{\sqrt{42}}\)
M1
\(= 1.77(4)\) (or \(0.038\) for area comparison)
A1
allow with wrong or no cc.; Accept alt method using \(N(2.1, 2.1/20)\) with or without cc
comp \(1.96\)
M1
Valid comp \(z\) or \(1-\) ('1.774') with \(0.025\) seen
No evidence that mean has changed
A1\(\checkmark\) [6]
allow comp 1.645 if \(H_1: \lambda\) (or \(\mu) > 42\); No contradictions. No ft for \(H_1: \lambda\) (or \(\mu) > 42\); Note – accept other valid methods (e.g. cv method)
## Question 4:
| Answer/Working | Mark | Guidance |
|---|---|---|
| $H_0: \lambda$ (or $\mu) = 42$; $H_1: \lambda$ (or $\mu) \neq 42$; $Po(42) \sim N(42, 42)$ stated or implied | B1 | Or pop weekly mean $= 2.1$ etc.; allow 'population mean' not just 'mean'; ft their '42' (Accept alt method $N(2.1, 2.1/20)$) |
| | B1$\checkmark$ | |
| $\frac{53.5-42}{\sqrt{42}}$ | M1 | |
| $= 1.77(4)$ (or $0.038$ for area comparison) | A1 | allow with wrong or no cc.; Accept alt method using $N(2.1, 2.1/20)$ with or without cc |
| comp $1.96$ | M1 | Valid comp $z$ or $1-$ ('1.774') with $0.025$ seen |
| No evidence that mean has changed | A1$\checkmark$ [6] | allow comp 1.645 if $H_1: \lambda$ (or $\mu) > 42$; No contradictions. No ft for $H_1: \lambda$ (or $\mu) > 42$; Note – accept other valid methods (e.g. cv method) |
4 At a certain company, computer faults occur randomly and at a constant mean rate. In the past this mean rate has been 2.1 per week. Following an update, the management wish to determine whether the mean rate has changed. During 20 randomly chosen weeks it is found that 54 computer faults occur. Use a suitable approximation to test at the $5 \%$ significance level whether the mean rate has changed.
\hfill \mbox{\textit{CAIE S2 2016 Q4 [6]}}