Loreto is a manager at the Da Vinci health centre. If the mean rate of patients arriving at the health centre exceeds 1.5 per minute then Loreto will employ extra staff. It is assumed that the number of patients arriving in any given time period follows a Poisson distribution.

let $X$ be the random variable "number of patients arriving in a minute", such that $X \sim \operatorname{Po}(m)$. $\mathrm{H}_{0}: m=1.5$ A1 $\mathrm{H}_{1}: m>1.5$ A1

Note: Allow a value of 270 for $m$. Award at most A0A1 if it is not clear that it is the population mean being referred to e.g $\mathrm{H}_{0}$: The number of patients is equal to 1.5 every minute $\mathrm{H}_{1}$: The number of patients exceeds 1.5 every minute. Referring to the "expected" number of patients or the use of $\mu$ or $\lambda$ is sufficient for A1A1.

under $\mathrm{H}_{0}$ let $Y$ be the number of patients in 3 hours $Y \sim \operatorname{Po}(270)$ (A1) $\mathrm{P}(Y \geq 320)(=1-\mathrm{P}(Y \leq 319))=0.00166(0.00165874)$ (M1)A1 since $0.00166<0.05$ R1 (reject $\mathrm{H}_{0}$) Loreto should employ more staff A1

$\mathrm{H}_{0}$: The probability of a patient waiting less than 20 minutes is $0.95$ A1 $\mathrm{H}_{1}$: The probability of a patient waiting less than 20 minutes is less than 0.95 A1