# Question 2:

## Part 2(a):
| Answer | Mark | Guidance |
|--------|------|----------|
| $P(R \geq 23) = 0.8517...$ awrt **0.852** | B1 | awrt 0.852 |
| | **(1)** | |

## Part 2(b):
| Answer | Mark | Guidance |
|--------|------|----------|
| $R \sim Po(28)$, $A \sim Po(16)$; $Y = R + A \rightarrow Y \sim Po(44)$ | M1 | For combining distributions and sight or use of $Po(28+16[=44])$. Condone $28+16=42$ followed by awrt 0.061 |
| $P(Y = 42) = 0.05866...$ awrt **0.0587** | A1 | awrt 0.0587 |
| | **(2)** | |

## Part 2(c):
| Answer | Mark | Guidance |
|--------|------|----------|
| $P(\text{less than 80 passengers checked in}) = 0.183...$ | B1 | awrt 0.18 may be implied by awrt 27.5 for the mean |
| $X \sim B(150, \text{"0.183..."})$, mean $= 150 \times \text{"0.183..."}$ $[= 27.48...]$ | M1 | Setting up a new model $B(150, \text{"0.183"})$ and using $np$ to calculate the mean |
| $T \sim Po(\text{"27.4..."})$ and $1 - P(T \leq 24)$ | M1 | Using the model $Po(\text{their } np)$ and using or writing $1 - P(T \leq 24)$ |
| $= 1 - 0.2922...$ awrt **0.708** | A1 | awrt 0.708 |
| | **(4)** | |

## Part 2(d):
| Answer | Mark | Guidance |
|--------|------|----------|
| $H_0: \lambda = 84$; $H_1: \lambda < 84$ (allow 28 for both) | B1 | Both hypotheses correct using $\lambda$ or $\mu$. Allow 28 instead of 84 |
| $J \sim Po(84)$ | M1 | Writing or using $Po(84)$ |
| $P(J \leq 67) = 0.03[246...]$ **or** CR $J \leq 68$ | A1 | awrt 0.03 or $J \leq 68$ |
| $0.03... < 0.05$ or $67 \leq 68$ or 67 is in the critical region or 67 is significant or Reject $H_0$. There is evidence at the 5% level of significance that the system is working **slower** than normal. | A1cao | dep on previous M mark awarded and a probability found. Drawing a correct inference in context – need the word *slower* or support for Alex's complaint |
| | **(4)** | |

---