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Date	November 2021	Marks available	5	Reference code	21N.2.AHL.TZ0.7
Level	Additional Higher Level	Paper	Paper 2	Time zone	Time zone 0
Command term	State	Question number	7	Adapted from	N/A

Question

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.

Loreto performs a hypothesis test to determine whether she should employ extra staff. She finds that $320$ patients arrived during a randomly selected $3$ -hour clinic.

Loreto is also concerned about the average waiting time for patients to see a nurse. The health centre aims for at least $95 %$ of patients to see a nurse in under $20$ minutes.

Loreto assumes that the waiting times for patients are independent of each other and decides to perform a hypothesis test at a $10 %$ significance level to determine whether the health centre is meeting its target.

Loreto surveys $150$ patients and finds that $11$ of them waited more than $20$ minutes.

Write down null and alternative hypotheses for Loreto’s test.

[2]

a.i.

Using the data from Loreto’s sample, perform the hypothesis test at a $5 %$ significance level to determine if Loreto should employ extra staff.

[5]

a.ii.

Write down null and alternative hypotheses for this test.

[2]

b.i.

Perform the test, clearly stating the conclusion in context.

[5]

b.ii.

Markscheme

let $X$ be the random variable “number of patients arriving in a minute”, such that $X ~ Po (m)$

$H_{0} : m = 1.5$ A1

$H_{1} : m > 1.5$ 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
$H_{0} :$ The number of patients is equal to 1.5 every minute
$H_{1} :$ The number of patients exceeds 1.5 every minute.
Referring to the “expected” number of patients or the use of $μ$ or $λ$ is sufficient for A1A1.

[2 marks]

a.i.

under $H_{0}$ let $Y$ be the number of patients in $3$ hours

$Y ~ Po (270)$ (A1)

$P (Y \geq 320) (= 1 - P (Y \leq 319)) = 0.00166 (0.00165874)$ (M1)A1

since $0.00166 < 0.05$ R1

(reject $H_{0}$ )

Loreto should employ more staff A1

[5 marks]

a.ii.

$H_{0}$ : The probability of a patient waiting less than $20$ minutes is $0.95$ A1

$H_{1}$ : The probability of a patient waiting less than $20$ minutes is less than $0.95$ A1

[2 marks]

b.i.

Under $H_{0}$ let $W$ be the number of patients waiting more than $20$ minutes

$W ~ B (150, 0.05)$ (A1)

$P (W \geq 11) = 0.132 (0.132215 \dots)$ (M1)A1

since $0.132 > 0.1$ R1

(fail to reject $H_{0}$ )

insufficient evidence to suggest they are not meeting their target A1

Note: Do not accept “they are meeting target” for the A1. Accept use of $B (150, 0.95)$ and $P (W \leq 139)$ and any consistent use of a random variable, appropriate $p$ -value and significance level.

[5 marks]

b.ii.

Examiners report

In part (a) there was a general lack of consistency in how candidates wrote down their null and alternative hypotheses. It was surprising how many candidates solved a Poisson $PDF$ rather than $CDF$ to find their $p$ -value. This suggests a lack of understanding of the nature of distributions or more specifically the concepts of hypothesis testing. In part (b), which was challenging, there were issues for many candidates in interpreting the situation. This is understandable since it was difficult, but as previously mentioned interpretation is a general issue in the paper. When writing down the conclusion of the tests, there was often very loose use of the terms accept/reject and candidates seemed unclear of the significance and importance of the correct use of these terms.

a.i.

a.ii.

b.i.

b.ii.

Syllabus sections

Topic 4—Statistics and probability » SL 4.8—Binomial distribution

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Find the probability that Arianne scores at least $5$ points in the competition.
21N.2.AHL.TZ0.3d.iii:
Find the probability that Arianne scores at least $5$ points and less than $8$ points.
21N.2.AHL.TZ0.7a.i:
Write down null and alternative hypotheses for Loreto’s test.
21N.2.AHL.TZ0.7a.ii:
Using the data from Loreto’s sample, perform the hypothesis test at a $5 %$ significance level to determine if Loreto should employ extra staff.
21N.2.AHL.TZ0.7b.i:
Write down null and alternative hypotheses for this test.

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