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Date	May 2021	Marks available	1	Reference code	21M.2.SL.TZ2.1
Level	Standard Level	Paper	Paper 2	Time zone	Time zone 2
Command term	State	Question number	1	Adapted from	N/A

Question

A medical centre is testing patients for a certain disease. This disease occurs in $5 %$ of the population.

They test every patient who comes to the centre on a particular day.

It is intended that if a patient has the disease, they test “positive”, and if a patient does not have the disease, they test “negative”.

However, the tests are not perfect, and only $99 %$ of people who have the disease test positive. Also, $2 %$ of people who do not have the disease test positive.

The tree diagram shows some of this information.

Write down the value of

Use the tree diagram to find the probability that a patient selected at random

The staff at the medical centre looked at the care received by all visiting patients on a randomly chosen day. All the patients received at least one of these services: they had medical tests ( $M$ ), were seen by a nurse ( $N$ ), or were seen by a doctor ( $D$ ). It was found that:

$78$ had medical tests,
$45$ were seen by a nurse;
$30$ were seen by a doctor;
$9$ had medical tests and were seen by a doctor and a nurse;
$18$ had medical tests and were seen by a doctor but were not seen by a nurse;
$11$ patients were seen by a nurse and had medical tests but were not seen by a doctor;
$2$ patients were seen by a doctor without being seen by nurse and without having medical tests.

State the sampling method being used.

[1]

$a$ .

[1]

b.i.

$b$ .

[1]

b.ii.

$c$ .

[1]

b.iii.

$d$ .

[1]

b.iv.

will not have the disease and will test positive.

[2]

c.i.

will test negative.

[3]

c.ii.

has the disease given that they tested negative.

[3]

c.iii.

The medical centre finds the actual number of positive results in their sample is different than predicted by the tree diagram. Explain why this might be the case.

[1]

Draw a Venn diagram to illustrate this information, placing all relevant information on the diagram.

[3]

Find the total number of patients who visited the centre during this day.

[2]

Markscheme

convenience sampling (A1)

[1 mark]

$95 %$ A1

[1 mark]

b.i.

$1 %$ A1

[1 mark]

b.ii.

$2 %$ A1

[1 mark]

b.iii.

$98 %$ A1

[1 mark]

b.iv.

$0.95 \times 0.02$ (M1)

$0.019$ A1

[2 marks]

c.i.

$0.05 \times 0.01 + 0.95 \times 0.98$ (M1)(M1)

Note: Award M1 for summing two products and M1 for correct products seen.

$0.932 (0.9315)$ A1

[3 marks]

c.ii.

recognition of conditional probability (M1)

$\frac{0.05 \times 0.01}{0.05 \times 0.01 + 0.95 \times 0.98}$ A1

$0.000537 (0.000536768 \dots)$ A1

Note: Accept $0.000536$ if $0.932$ used.

[3 marks]

c.iii.

EITHER
sample may not be representative of population A1

OR
sample is not randomly selected A1

OR
unrealistic to think expected and observed values will be exactly equal A1

[1 mark]

A1A1A1

Note: Award A1 for rectangle and $3$ labelled circles and $9$ in centre region; A1 for $2, 40, 24$ ; A1 for $18, 1,$ and $11$ .

[3 marks]

$18 + 9 + 1 + 11 + 2 + 40 + 24$ (M1)

$105$ A1

Note: Follow through from the entries on their Venn diagram in part (e). Working required for FT.

[2 marks]

Examiners report

[N/A]

b.i.

[N/A]

b.ii.

[N/A]

b.iii.

[N/A]

b.iv.

[N/A]

c.i.

[N/A]

c.ii.

[N/A]

c.iii.

[N/A]

Syllabus sections

Topic 4—Statistics and probability » SL 4.1—Concepts, reliability and sampling techniques

Show 80 related questions

22M.2.AHL.TZ1.3a:
Name the type of sampling that best describes the method used by the Principal.
22M.2.SL.TZ1.3b:
Determine if the Netherlands’ score is an outlier for this data. Justify your answer.
19M.2.AHL.TZ1.H_3b.ii:
the standard deviation.
18N.2.AHL.TZ0.H_10b.ii:
an estimate for the standard deviation of the number of emails received per working day.
EXM.2.SL.TZ0.5d:
State an assumption that the company is making, in order to use a t-test.
EXM.2.SL.TZ0.5c.i:
The new drug.
EXM.2.SL.TZ0.5b:
Calculate the number of volunteers in the sample under the age of 30.
17M.1.SL.TZ1.S_9c:
The line $y = kx - 5$ is a tangent to the curve of $f$ . Find the values of $k$ .
18N.2.AHL.TZ0.H_10d:
Suppose that the probability of Archie receiving more than 10 emails in total on any one day is 0.99. Find the value of λ.
EXM.2.SL.TZ0.5e:
State the hypotheses for this t-test.
22M.1.SL.TZ2.7a:
Show that the test score of $25$ would not be considered an outlier.
18M.1.AHL.TZ2.H_3b.i:
Find μ, the expected value of X.
18N.2.AHL.TZ0.H_10a.ii:
Using this distribution model, find the standard deviation of $X$ .
18N.2.AHL.TZ0.H_10b.i:
an estimate for the mean number of emails received per working day.
19M.2.SL.TZ1.S_9b:
Find u.
18M.1.SL.TZ1.S_2a:
Find the value of the interquartile range.
17M.1.SL.TZ1.S_6a.i:
Write down the gradient of the curve of $f$ at P.
17M.2.SL.TZ2.S_8b.i:
Write down the coordinates of A.
EXM.2.SL.TZ0.5f:
Find the p-value for this t-test.
EXM.2.SL.TZ0.5c.ii:
The current drug.
EXM.1.SL.TZ0.5c:
Give an example of a set of data with 7 numbers in it that does have an outlier, justify this fact by stating the Interquartile Range.
19M.1.SL.TZ2.S_9d:
The following diagram shows the graph of $f$ for 0 ≤ $x$ ≤ 3. Line $M$ is a tangent to the graph of $f$ at point P.

Given that $M$ is parallel to $L$ , find the $x$ -coordinate of P.
18M.1.AHL.TZ2.H_3b.ii:
Find P(X > μ).
SPM.1.SL.TZ0.3b:
Find the value of $k$ .
EXM.2.SL.TZ0.5g:
State the conclusion of this test, in context, giving a reason.
17N.1.SL.TZ0.S_5a:
Find $(g \circ f)(x)$ .
19M.2.SL.TZ1.S_9d.iii:
Hence or otherwise, find the obtuse angle formed by the tangent line to $f$ at $x = 8$ and the tangent line to $f$ at $x = 2$ .
17M.2.SL.TZ2.S_8b.ii:
Write down the rate of change of $f$ at A.
18N.2.AHL.TZ0.H_10a.i:
Using this distribution model, find ${\text{P}}\left( {X < 60} \right)$ .
18M.1.SL.TZ1.S_2b:
One student sent k text messages, where k > 11 . Given that k is an outlier, find the least value of k.
18N.2.AHL.TZ0.H_10e:
Now suppose that Archie received exactly 20 emails in total in a consecutive two day period. Show that the probability that he received exactly 10 of them on the first day is independent of λ.
19M.2.SL.TZ1.S_9d.i:
Find $\left( {f \circ f} \right)\left( x \right)$ .
17M.2.SL.TZ2.S_8c.i:
Find the coordinates of B.
18N.2.AHL.TZ0.H_10c:
Give one piece of evidence that suggests Willow’s Poisson distribution model is not a good fit.
19M.2.SL.TZ1.S_9c:
Find the acute angle between $y = x$ and $L$ .
18N.1.SL.TZ0.S_10b.ii:
Hence, find the equation of L in terms of $a$ .
18N.1.SL.TZ0.S_10a:
Find the coordinates of P.
16N.2.SL.TZ0.S_10a:
(i) Find the value of $c$ .

(ii) Show that $b = \frac{\pi }{6}$ .

(iii) Find the value of $a$ .
EXM.2.SL.TZ0.5a:
State the name for this type of sampling technique.
SPM.1.SL.TZ0.3a:
State whether the data is discrete or continuous.
17M.1.SL.TZ1.S_6b:
Determine the concavity of the graph of $f$ when $4 < x < 5$ and justify your answer.
17M.1.SL.TZ1.S_9b:
Find the value of $a$ .
16N.2.AHL.TZ0.H_11d:
(i) Hence show that $X$ has two modes ${m_1}$ and ${m_2}$ .

(ii) State the values of ${m_1}$ and ${m_2}$ .
18N.1.SL.TZ0.S_10b.i:
Find $f'\left( x \right)$ .
17M.1.SL.TZ1.S_6a.ii:
Find the equation of the normal to the curve of $f$ at P.
19M.1.SL.TZ2.S_9b:
Line $L$ passes through the origin and has a gradient of ${\text{tan}}\,\theta$ . Find the equation of $L$ .
17M.1.SL.TZ1.S_9a:
Find the value of $p$ .
19M.1.SL.TZ2.S_9a:
Find the value of ${\text{tan}}\,\theta$ .
18N.1.SL.TZ0.S_10c:
The graph of $f$ has a local minimum at the point Q. The line L passes through Q.

Find the value of $a$ .
16N.2.AHL.TZ0.H_11b:
Find the values of the constants $a$ and $b$ .
16N.2.AHL.TZ0.H_11a:
Show that ${\text{P}}(X = 3) = 0.001$ and ${\text{P}}(X = 4) = 0.0027$ .
18N.2.SL.TZ0.S_10c:
When $t$ = 0, the volume of water in the container is 2.3 m³. It is known that the container is never completely full of water during the 4 hour period.

Find the minimum volume of empty space in the container during the 4 hour period.
19M.1.SL.TZ2.S_9c:
Find the derivative of $f$ .
EXN.1.SL.TZ0.1b:
The Principal selects the students for the sample by asking those who took part in a previous survey if they would like to take part in another. She takes the first of those who reply positively, up to the maximum needed for the sample.

State which two of the sampling methods listed below best describe the method used.

Stratified Quota Convenience Systematic Simple random
16N.2.SL.TZ0.S_10b:
(i) Write down the value of $k$ .

(ii) Find $g(x)$ .
17M.2.SL.TZ2.S_8d:
Let $R$ be the region enclosed by the graph of $f$ , the $x$ -axis, the line $x = b$ and the line $x = a$ . The region $R$ is rotated 360° about the $x$ -axis. Find the volume of the solid formed.
EXM.1.SL.TZ0.5b:
Hence, show that a data set with only 5 numbers in it cannot have any outliers.
17M.2.SL.TZ2.S_8a:
Find the value of $p$ .
SPM.1.SL.TZ0.3c:
It was not possible to ask every person in the school, so the Headmaster arranged the student names in alphabetical order and then asked every 10th person on the list.

Identify the sampling technique used in the survey.
18N.2.SL.TZ0.S_10a:
Find the volume of the container.
19M.2.SL.TZ1.S_3a:
Find $f'\left( x \right)$ .
21M.2.SL.TZ1.1d:
Determine whether Jason is correct. Support your reasoning.
21M.1.AHL.TZ1.11a:
Identify the type of sampling used by the restaurant manager.
16N.2.AHL.TZ0.H_11e:
Determine the minimum value of $x$ such that the probability Kati receives at least one free gift is greater than 0.5.
18N.2.SL.TZ0.S_10b.ii:
During the interval $p$ < $t$ < $q$ , he volume of water in the container increases by $k$ m³. Find the value of $k$ .
EXM.1.SL.TZ0.5a:
Recalling definitions, such as the Lower Quartile is the $\frac{{n + 1}}{4}th$ piece of data with the data placed in order, find an expression for the Interquartile Range.
17N.1.SL.TZ0.S_5b:
Given that $\mathop {\lim }\limits_{x \to + \infty } (g \circ f)(x) = - 3$ , find the value of $b$ .
19M.2.SL.TZ1.S_9a:
Find the gradient of $L$ .
19M.2.SL.TZ1.S_3b:
The graph of $f$ has a horizontal tangent line at $x = 0$ and at $x = a$ . Find $a$ .
19M.2.SL.TZ1.S_9d.ii:
Hence, write down ${f^{ - 1}}\left( x \right)$ .
21M.2.SL.TZ1.1a:
State which of the two sampling methods, systematic or quota, Jason has used.
18M.1.AHL.TZ2.H_3a:
Find the value of p.
18N.2.SL.TZ0.S_10b.i:
Find the value of $p$ and of $q$ .
16N.2.SL.TZ0.S_10c:
(i) Find $w$ .

(ii) Hence or otherwise, find the maximum positive rate of change of $g$ .
21M.3.AHL.TZ1.2b:
Show that $11$ employees are selected for the sample from the national department.
16N.2.AHL.TZ0.H_11c:
Deduce that $\frac{{{\text{P}}(X = n)}}{{{\text{P}}(X = n - 1)}} = \frac{{0.9(n - 1)}}{{n - 3}}$ for $n > 3$ .
21M.3.AHL.TZ2.1b:
Juliet classifies response $K$ as an outlier and removes it from the data. Suggest one possible justification for her decision to remove it.
EXN.1.SL.TZ0.1a:
Calculate the number of grade $12$ students who should be in the sample.
17M.2.SL.TZ2.S_8c.ii:
Find the the rate of change of $f$ at B.
18M.1.SL.TZ1.S_7:
Consider f(x), g(x) and h(x), for x∈ $\mathbb{R}$ where h(x) = $\left( {f \circ g} \right)$ (x).

Given that g(3) = 7 , g′ (3) = 4 and f ′ (7) = −5 , find the gradient of the normal to the curve of h at x = 3.

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Topic 4—Statistics and probability

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