Statistics and Probability — Free IB MYP5 Mathematics (Extended) Questions

1QuestionFairness Bias and RandomnessConcept Practice

4 marks~6 minCriterion A

A quality-control technician suspects a six-sided die used in a board-game factory may be biased. The die is rolled 60 times with the following results:

Outcome	1	2	3	4	5	6
Frequency	8	10	9	7	13	13

a

Calculate the relative frequency of rolling a 4. [1]

b

Calculate the absolute difference between the relative frequency of rolling a 4 and its theoretical probability. [2]

c

The factory rejects any die for which the absolute difference between the relative frequency and theoretical probability of any outcome exceeds 0.08 after 60 trials. Advise the technician whether this die should be rejected, justifying your answer with a numerical comparison. [1]

Solutions

2QuestionUsing Probability Trees for Multi-Stage EventsConcept Practice

3 marks~5 minCriterion A

A school supplies coordinator tracks how students choose writing tools. Based on purchasing records, the probability that a student selects a pen is

0.6

and a pencil is

0.4

. Given a pen is chosen, the probability of selecting a blue refill is

0.7

and a black refill is

0.3

. Given a pencil is chosen, the probability of selecting a blue refill is

0.2

and a black refill is

0.8

.

a

Calculate the probability that a randomly selected student chooses a blue refill. [2]

b

The coordinator orders blue refills for exactly half of all students. Advise the coordinator whether this order quantity is appropriate, justifying your answer using your result from part (a). [1]

Solutions

3QuestionCalculating Mean Median and ModeConcept Practice

2 marks~3 minCriterion A

A meteorologist records the daily maximum temperature (°C) in a coastal city over

n

days, labelling each value

x_1, x_2, x_3, \ldots, x_n

.

State the formula for the arithmetic mean

\bar{x}

of these

n

values, using summation notation. [2]

Solutions

4QuestionBox and Whisker Plots for Spread of DataConcept Practice

4 marks~6 minCriterion A

A school's sports coordinator records the sprint times (in seconds) of 30 students during a fitness assessment. The box-and-whisker plot displays the results, with the following five-number summary:

Minimum

= 8.4

s,

Q_1 = 10.2

s, Median

= 11.8

s,

Q_3 = 13.6

s, Maximum

= 16.0

s

Students who score within 4 seconds of the median are classified as "consistent performers."

a

Calculate the range of the sprint times. [1]

b

Calculate the interquartile range (IQR) of the sprint times. [1]

c

Justify whether the middle 50% of students qualify as consistent performers, using the IQR. [2]

Solutions

5QuestionSet Notation and Venn Diargram NotationsConcept Practice

6 marks~9 minCriterion A

A school records which sports students play. The Venn diagram shows the number of students in a class who play basketball (

B

) and/or volleyball (

V

):

- Only basketball: 12
- Only volleyball: 8
- Both sports: 5
- Neither sport: 3

a

Calculate

n(B \cup V)

. [2]

b

A student is chosen at random from the class. Deduce the probability that this student plays exactly one of the two sports, expressing your answer using appropriate set notation. [2]

c

The school requires that more than 70% of the class participate in at least one sport to qualify for an inter-school tournament. Justify whether this class qualifies, supporting your answer with a calculation. [2]

Solutions

6QuestionComparing Experimental and Expected ResultsAssessment Practice

2 marks~3 minCriterion C

A quality-control analyst tests a six-sided die by rolling it 12 times, recording: 2, 5, 1, 3, 6, 2, 4, 1, 3, 5, 2, 6.

a

Deduce the experimental frequency for each outcome and the expected frequency for each outcome if the die is fair.

Outcome	1	2	3	4	5	6
Experimental frequency	___	___	___	___	___	___
Expected frequency	___	___	___	___	___	___ [1]

b

The analyst concludes the die is fair because no single outcome was recorded more than 3 times. Critique this conclusion using the experimental results. [1]

Solutions

7QuestionApplications in Real-World Risk AssessmentAssessment Practice

4 marks~6 minCriterion D

An insurance company records the following data for two groups of drivers over one year.

Age group	Under 25 — claims: 120	total drivers: 800
Age group	Over 25 — claims: 80	total drivers: 1200

The relative risk of making a claim is defined as

\text{Relative Risk} = \frac{P(\text{claim} \mid \text{under 25})}{P(\text{claim} \mid \text{over 25})}

a

Calculate

P(\text{claim} \mid \text{under 25})

and

P(\text{claim} \mid \text{over 25})

. [2]

b

Calculate the relative risk of making a claim for drivers under 25 compared to drivers over 25. [1]

c

The insurance company charges drivers under 25 a premium 2.5 times higher than drivers over 25. Justify whether this premium multiplier is supported by the claim data. [1]

Solutions

8QuestionApplications in Real-World Risk AssessmentAssessment Practice

8 marks~12 minCriterion D

A city council is evaluating flood risk in a coastal area. Historical data shows the probability of a flood in any given year is

0.02

. If a flood occurs, the estimated damage cost is 500 000 dollars. The council is considering a flood barrier costing 15 000 dollars per year to maintain.

a

Calculate the expected annual loss due to flooding without the barrier. [2]

b

The council will build the barrier only if the expected annual loss exceeds the annual maintenance cost. Deduce whether the barrier should be built, showing your reasoning clearly. [2]

c

Advise the council whether relying solely on the expected value model is sufficient to guide this decision, considering factors such as climate change, inflation, and non-monetary impacts. [4]

Solutions

9QuestionOrganizing Data for Effective AnalysisAssessment Practice

6 marks~9 minCriterion C

A hospital is analysing post-surgery recovery times (in days) for a sample of 24 patients:

5, 6, 7, 7, 8, 8, 8, 9, 9, 10, 10, 10, 11, 11, 12, 13, 14, 15, 16, 18, 20, 22, 25, 30

The hospital is considering three methods to organise this data: a stem-and-leaf plot, a histogram, and a grouped frequency table.

a

Construct a stem-and-leaf plot for the recovery times. Include a key. [2]

b

A grouped frequency table uses the intervals 5–9, 10–14, 15–19, 20–24, 25–30. Explain one way in which this grouping could lead to a misinterpretation of the data compared with your stem-and-leaf plot. [2]

c

The hospital administrator must decide whether to allocate additional nursing staff to patients whose recovery exceeds 14 days. Advise the administrator which representation — the stem-and-leaf plot or the grouped frequency table — to use when making this staffing decision, and justify your advice with a specific reference to the data. [2]

Solutions

10QuestionData collection and generation (including surveys)Assessment Practice

5 marks~8 minCriterion A

The diagram shows the graph of

f(x) = x^2 - 6x + k

for

0 \le x \le 6

, where

k

is a constant. The graph passes through the point

(1, 3)

.

a

Show that

k = 8

. [1]

b

Find the coordinates of the vertex of the graph. [2]

c

State the values of

m

for which

f(x) = m

has exactly one solution in the domain

0 \le x \le 6

. [2]

Solutions

11QuestionDesigning Surveys and QuestionnairesAssessment Practice

6 marks~9 minCriterion D

A school wants to estimate the average number of hours students spend studying each week. During exam week, the survey team asks every student present in the library. These 80 students report an average of 18 hours per week. The remaining 720 students, surveyed separately, report an average of 6 hours per week.

a

Identify the type of sampling bias present in this survey design and explain how it causes the library-only result to overestimate the true school-wide average. [2]

b

Calculate the true average weekly study hours for all 800 students. Show your working clearly. [2]

c

The school principal states: "An average of 7.2 hours per week is too low to reflect real student effort — we should use the library figure of 18 hours when reporting to parents."

Advise the principal on whether the figure of 18 hours should be used in the school-wide report, justifying your answer with reference to what the calculated average actually represents. [2]

Solutions

12QuestionData collection and generation (including surveys)Assessment Practice

5 marks~8 minCriterion B

The diagram shows the first four figures of a pattern made from small squares arranged in an L-shape.

The table below gives the number of small squares

S

in each figure.

Figure number ( $n$ )	1	2	3	4
Number of squares ( $S$ )	3	5	7	9

a

Write down the number of small squares in Figure 5. [1]

b

Find a rule for the number of small squares

S

in Figure

n

, giving your answer in the form

S = an + b

where

a

and

b

are integers to be determined. [2]

c

A student claims that no figure in this pattern can contain exactly 50 small squares. Justify whether the student is correct. [2]

Solutions

13QuestionUsing Probability Trees for Multi-Stage EventsAssessment Practice

7 marks~11 minCriterion B

A quality-control engineer tests microchips by running them through a sequence of independent diagnostic checks. At each check, a chip either passes (P) or fails (F), each with equal probability.

a

Write down the number of distinct outcomes after 1 check, after 2 checks, and after 3 checks. Identify the pattern in your sequence. [2]

b

Deduce a formula for the total number of distinct outcomes after

n

checks, each with

m

equally likely results. Justify your formula using the structure of a probability tree. [2]

c

The engineer runs 4 checks on each chip. Construct a probability tree for this 4-stage process and use your formula to verify the total number of distinct paths. A chip is accepted only if it passes all 4 checks. Advise the engineer whether this acceptance criterion is suitable for a production line, justifying your answer using probability. [3]

Solutions

14QuestionUsing Probability Trees for Multi-Stage EventsAssessment Practice

10 marks~15 minCriterion D

A hospital uses a rapid antigen test to screen for a disease with a prevalence of 2%. The test has a sensitivity of 90% — the probability of a positive result given the disease is present — and a specificity of 95% — the probability of a negative result given the disease is absent.

a

Construct a probability tree diagram for two stages: disease status (has disease / no disease) and test result (positive / negative). Clearly label all branches with their probabilities. [3]

b

Using your tree, calculate the probability that a randomly selected individual tests positive. [2]

c

Calculate the positive predictive value (PPV): the probability that an individual actually has the disease given a positive test result. [2]

d

The disease prevalence rises to 10% during an outbreak and falls to 0.5% in a low season. Calculate the PPV in each case, then advise the hospital whether the test is suitable as a standalone screening tool across all three prevalence levels, justifying your recommendation with reference to false positive and false negative consequences and the limitations of this model. [3]

Solutions

15QuestionReal-Life Applications Games Genetics and Decision-MakingAssessment Practice

2 marks~3 minCriterion C

In genetic counselling, both parents are confirmed carriers of a recessive disorder. The probability that any one child inherits the disorder is

0.25

.

Explain why the assumption of independent assortment is necessary to calculate the probability that two children both inherit the disorder, and identify one limitation of this assumption in real genetic contexts. [2]

Solutions

16QuestionCalculating Mean Median and ModeAssessment Practice

6 marks~9 minCriterion B

A city's air-quality monitoring station records the daily Air Quality Index (AQI). The mean AQI for the first 5 days of a month is 12, and the AQI on day 6 is 18.

a

Calculate

M_6

, the mean AQI for the first 6 days. [2]

b

Deduce a general formula for

M_{n+1}

, the mean of the first

(n+1)

days, in terms of

M_n

,

n

, and

a_{n+1}

. Show all working. [2]

c

The AQI values for days 7 through 10 are 20, 15, 11, and 9. Health guidelines state that a monthly mean AQI above 13 triggers a public advisory. Using your formula from part (b), advise the monitoring station whether a public advisory should be issued after day 10. Justify your answer. [2]

Solutions

17QuestionChoosing the Appropriate MeasureAssessment Practice

4 marks~6 minCriterion D

A school librarian records the number of books read last month by 15 students.

Number of books $(x)$	1	2	3	4	5	20
Frequency $(f)$	4	3	3	2	2	1

a

Calculate the mean number of books read. [1]

b

Determine the median number of books read. [1]

c

The librarian uses the mean to report the "typical" reading behaviour of the group. Advise the librarian whether the mean or the median is the more appropriate measure to use, justifying your answer with reference to the data. [2]

Solutions

18QuestionCalculating Mean Median and ModeAssessment Practice

2 marks~3 minCriterion C

A teacher records test scores for Class A (22 students, total score 1584) and Class B (18 students, total score 1332).

a

Explain why the mean score is a more appropriate measure than the total score for comparing the performance of the two classes. [1]

b

Identify one limitation of using the mean score to compare the performance of these two classes. [1]

Solutions

19QuestionRange and standard deviationAssessment Practice

6 marks~9 minCriterion B

The diagram shows Figures 1 to 4 of a pattern made from dots arranged in a triangular grid.

Figure 1 has 1 dot, Figure 2 has 3 dots, Figure 3 has 6 dots, and Figure 4 has 10 dots.

a

Write down the number of dots added when going from Figure 3 to Figure 4. [1]

b

Find a rule for the number of dots

D

in Figure

n

, giving your answer in the form

D = \frac{n(n+1)}{2}.

Verify your rule for Figure 3. [3]

c

A student claims that Figure 16 is the first figure in the pattern to contain more than 100 dots. Justify whether the student is correct. [2]

Solutions

20QuestionCalculating the interquartile rangeAssessment Practice

6 marks~9 minCriterion D

A city council collects annual household income data (in thousands of dollars) for two neighbourhoods.

Harbourview (15 households, sorted):
35, 42, 48, 51, 53, 55, 58, 62, 65, 68, 72, 78, 85, 92, 500 000

Westside (15 households, sorted):
30, 33, 36, 39, 42, 45, 48, 51, 54, 57, 60, 63, 66, 69, 72

The mean household income for Harbourview is approximately 33 393 thousand dollars and the standard deviation is approximately 128 924 thousand dollars.

a

Calculate the interquartile range (IQR) for each neighbourhood. [2]

b

Justify which pair of summary statistics — median and IQR, or mean and standard deviation — better represents the typical household income in Harbourview. [2]

c

Advise the council whether the IQR alone is sufficient for comparing income spread across neighbourhoods when allocating social services funding, referring to the Harbourview data. [2]

Solutions

21QuestionComparing Variability Between Data SetsAssessment Practice

4 marks~6 minCriterion C

A school is comparing the consistency of test scores between two Grade 10 classes. The scores (out of 100) are:

Class A: 45, 62, 68, 70, 72, 75, 78, 82, 85, 90

Class B: 55, 56, 58, 60, 61, 62, 63, 65, 67, 95

a

Calculate the range for each class. [1]

b

Calculate the interquartile range (IQR) for each class. [2]

c

The school principal concludes that Class B is more consistent because its range is smaller than Class A's. Critique this conclusion, referring to the IQR values and the effect of the outlier in Class B. [1]

Solutions

22QuestionConstructing and Interpreting Box PlotsAssessment Practice

6 marks~9 minCriterion B

A sports analyst records the recovery times (in minutes) of 23 athletes after a training session. The ordered values are:

4, 8, 12, 15, 18, 21, 24, 27, 30, 33, 36, 39, 42, 45, 48, 51, 54, 57, 60, 63, 66, 69, 72

Quartile positions for datasets of size

n = 7, 11, 15, 19

are given below.

n = 7

:

Q_1

at position 2, median at position 4,

Q_3

at position 6

n = 11

:

Q_1

at position 3, median at position 6,

Q_3

at position 9

n = 15

:

Q_1

at position 4, median at position 8,

Q_3

at position 12

n = 19

:

Q_1

at position 5, median at position 10,

Q_3

at position 15

a

Deduce a general rule, in terms of

n

, for the positions of the median,

Q_1

, and

Q_3

in an ordered dataset of

n

values. [2]

b

Using your rules from part (a), construct the five-number summary for the 23 recovery times, showing all position calculations. [2]

c

The analyst claims that the middle 50% of athletes recover within a 36-minute window. Justify whether the interquartile range supports this claim. [2]

Solutions

23QuestionConstructing and Interpreting Box PlotsAssessment Practice

4 marks~6 minCriterion C

A sports analyst is comparing the consistency of two athletes' training scores. She notes that larger sample sizes may produce more stable quartiles. She examines two datasets of daily training scores.

Dataset A (8 values): 42, 45, 48, 50, 52, 55, 58, 60

Dataset B (20 values): 30, 35, 40, 45, 48, 49, 50, 50, 50, 50, 50, 51, 52, 53, 55, 58, 62, 68, 72, 80

a

Calculate the five-number summary and IQR for each dataset. [2]

b

Construct box plots for both datasets on the same scale from 30 to 80. [1]

c

The analyst concludes that Dataset B's scores are more consistent because its IQR is smaller. Critique her conclusion, referring to both the IQR and the range of each dataset. [1]

Solutions

24QuestionCumulative Frequency Graphs and CurvesAssessment Practice

5 marks~8 minCriterion A

Two cumulative frequency graphs display test scores (out of 100) for 30 students in each of Class A and Class B.

Class A: median

= 62

, lower quartile

= 48

, upper quartile

= 76

. The curve rises steeply between scores of 50 and 80.

Class B: median

= 58

, lower quartile

= 37

, upper quartile

= 79

. The curve rises more gradually across the full range.

a

State the interquartile range (IQR) for each class. [2]

b

Justify which class performed better overall and which class was more consistent, referring to the median, IQR, and the shape of the cumulative frequency curves. [3]

Solutions

25QuestionCumulative Frequency Graphs and CurvesAssessment Practice

2 marks~3 minCriterion D

A factory produces steel rods with a target length of 200 mm. A cumulative frequency graph is drawn from a sample of 500 rods. The graph passes through the points (198, 50) and (202, 430).

a

Deduce the number of rods outside the acceptable tolerance range of 198–202 mm. [1]

b

Advise the factory manager whether the production process requires corrective action, justifying your answer using the result from part (a). [1]

Solutions

26QuestionCalculating Combined ProbabilitiesAssessment Practice

2 marks~3 minCriterion D

A quality control inspector tests two independent components in a smartphone: the battery and the screen. The probability that a battery is defective is

0.03

; the probability that a screen is defective is

0.02

. The factory's acceptance threshold states that no more than

5\%

of phones should have at least one defective component.

a

Calculate the probability that a randomly selected phone has at least one defective component. [1]

b

Justify whether this production line meets the factory's acceptance threshold, and identify one reason why the independence assumption may not hold in a real assembly line. [1]

Solutions

27QuestionSolving Problems using tree diagrams and Venn DiagramsAssessment Practice

6 marks~9 minCriterion C

A pharmaceutical company conducts a clinical trial for a new drug. In the trial, 70% of patients receive the drug and 30% receive a placebo. Of those receiving the drug, 95% show improvement. Of those receiving the placebo, 8% show improvement.

a

Construct a tree diagram to represent this situation, labelling all branches with their probabilities. [2]

b

Determine the probability that a randomly selected patient shows improvement. [2]

c

A patient selected at random has shown improvement. The pharmaceutical company claims this result confirms the drug is highly effective. Using your calculated probability, advise the company on whether this claim is justified. [2]

Solutions

28QuestionMutually exclusive eventsAssessment Practice

6 marks~9 minCriterion B

The Venn diagram shows two mutually exclusive events

X

and

Y

inside a sample space

S

.

Three figures each give a pair of mutually exclusive events with their probabilities.

Figure 1	$P(X) = 0.2$	$P(Y) = 0.3$
Figure 2	$P(X) = 0.4$	$P(Y) = 0.15$
Figure 3	$P(X) = 0.25$	$P(Y) = 0.35$

a

State the value of

P(X \cap Y)

for any pair of mutually exclusive events, and explain what this means in context. [1]

b

Find

P(X \cup Y)

for each figure. Record your results below, then describe the pattern you observe between

P(X \cup Y)

,

P(X)

, and

P(Y)

.

Figure 1	$P(X) = 0.2$	$P(Y) = 0.3$	$P(X \cup Y) =$ ?
Figure 2	$P(X) = 0.4$	$P(Y) = 0.15$	$P(X \cup Y) =$ ?
Figure 3	$P(X) = 0.25$	$P(Y) = 0.35$	$P(X \cup Y) =$ ? [2]

c

A fourth pair of mutually exclusive events has

P(X \cup Y) = 0.72

and

P(X) = 3 \times P(Y)

. Find

P(Y)

, and hence justify whether there is sufficient probability remaining in the sample space for a third mutually exclusive event

Z

with

P(Z) = 0.3

. [3]

Solutions

29QuestionData visualizations and infographicsAssessment Practice

5 marks~8 minCriterion A

The diagram shows the graph of

f(x) = x^2 - 6x + k

for

0 \le x \le 6

, where

k

is a constant. The graph passes through the point

(1, 3)

.

a

Show that

k = 8

. [1]

b

Find the coordinates of the vertex of the graph. [2]

c

State the values of

m

for which

f(x) = m

has exactly one solution in the domain

0 \le x \le 6

, and justify your answer. [2]

Solutions

30QuestionData visualizations and infographicsAssessment Practice

7 marks~11 minCriterion D

The table below shows monthly sales (in thousands of dollars) for three products over six months.

Month	Jan	Feb	Mar	Apr	May	Jun
Product A	10	12	15	8	20	18
Product B	8	10	12	14	16	15
Product C	5	7	9	11	13	12

a

Calculate the total sales for Product C over the six months, and hence find the overall total sales across all products and all months. [2]

b

A rival company claims that Product C accounts for more than 27% of total sales over the six months. Deduce whether this claim is correct, showing all working. [2]

c

The company sets a target: in a seventh month, Product A must achieve sales high enough so that Product A's share of the new seven-month overall total exceeds 40%. Products B and C each sell 15 thousand dollars in month 7. Find the minimum integer value of Product A's month-7 sales (in thousands of dollars) that meets this target, and justify whether this target is a realistic expectation for Product A based on its six-month sales record. [3]

Solutions

31QuestionData visualizations and infographicsAssessment Practice

6 marks~9 minCriterion C

A group of 60 students were asked to choose their favourite fruit. The results are shown below.

Fruit	Apple	Banana	Orange	Grape
Students	21	15	12	12

a

Show that the sector angle for Apple in a pie chart is

126°

. [1]

b

A student claims that the combined sector angle for Banana and Orange equals the sector angle for Apple. Determine whether this claim is correct, showing all working. [2]

c

Two students are chosen at random from the group, one after the other, without replacement. The probability that both students chose Apple is

\dfrac{7}{59}

. Justify whether this result suggests that Apple is a genuinely popular choice among the group, or whether it could reasonably be explained by chance. [3]

Solutions

32QuestionData visualizations and infographicsAssessment Practice

6 marks~9 minCriterion B

The diagram shows Figures 1 to 4 of a pattern made from small equilateral triangles.

Figure 1	1 shaded triangle	0 unshaded triangles.
Figure 2	2 shaded triangles	1 unshaded triangle.
Figure 3	3 shaded triangles	3 unshaded triangles.
Figure 4	4 shaded triangles	6 unshaded triangles.

The total number of small triangles in Figure

n

is denoted

T(n)

.

a

Write down the values of

T(1)

,

T(2)

,

T(3)

, and

T(4)

. [1]

b

Find a rule for

T(n)

in the form

T(n) = an^2 + bn

where

a

and

b

are constants to be determined. [3]

c

The number of unshaded triangles in Figure

n

is given by

U(n) = \frac{n^2 - n}{2}.

Using your rule from part (b), deduce a simplified expression for the number of shaded triangles

S(n)

in Figure

n

, and justify why

S(n)

must always equal

n

. [2]

Solutions

33QuestionCorrelation using technology (value and interpretation)Assessment Practice

6 marks~9 minCriterion D

The table below shows the number of hours studied per week (

x

) and the test score out of 50 (

y

) for five students.

x

: 2, 4, 6, 8, 10

y

: 20, 28, 35, 42, 48

A student claims that a linear model fits this data well and uses the regression line to make predictions.

a

Using your calculator, find the Pearson correlation coefficient

r

. Give your answer to 3 significant figures. [1]

b

The regression line of

y

on

x

has the form

y = ax + b

. Using your calculator, find the values of

a

and

b

to 3 significant figures. Hence find the predicted score for a student who studies for 7 hours per week. [3]

c

A sixth student studies for 15 hours per week and scores 49 out of 50. The teacher uses the regression line to predict this score. Advise the teacher whether the regression line should be used to predict scores for students who study 15 hours per week, justifying your answer with two distinct mathematical reasons. [2]

Solutions

34QuestionCorrelation using technology (value and interpretation)Assessment Practice

10 marks~15 minCriterion B

The diagram shows four figures in a pattern made from small squares arranged in an L-shape.

Figure 1 has 3 squares, Figure 2 has 7 squares, Figure 3 has 13 squares, Figure 4 has 21 squares.

a

Find the number of squares added between consecutive figures, and hence deduce the number of squares in Figure 5. [3]

b

Show that the total number of squares

S

in Figure

n

is given by

S = n^2 + n + 1

by verifying the formula for

n = 1, 2, 3, 4

and explaining how each term relates to the structure of the L-shape. [4]

c

Justify why no figure in this pattern can contain exactly 100 squares. [3]

Solutions

35QuestionCorrelation using technology (value and interpretation)Assessment Practice

11 marks~17 minCriterion A

A student records the number of unit squares in each figure of a growing pattern. The diagram shows Figures 1, 2, and 3.

The number of unit squares

S

in Figure

n

follows a rule of the form

S = an^2 + bn + c

where

a

,

b

, and

c

are integers.

Figure ( $n$ )	1	2	3
Squares ( $S$ )	6	14	26

A second pattern has the rule

T = 3n^2 + 2n + 1

.

a

Find the values of

a

,

b

, and

c

. [3]

b

Show that

S = 62

when

n = 5

, and find the figure number that first contains more than 200 unit squares. [4]

c

A designer claims that both patterns will eventually produce figures with identical numbers of unit squares, and plans to use that figure number as a design checkpoint. Justify whether the designer's claim is valid, and interpret what your answer means for the design plan. [4]

Solutions

36QuestionCorrelation using technology (value and interpretation)Assessment Practice

10 marks~15 minCriterion C

The table below shows the number of hours spent studying per week (

x

) and the end-of-term test score (

y

, out of 100) for eight students.

x

(hours): 2, 3, 5, 6, 7, 8, 10, 11

y

(score): 45, 50, 55, 60, 65, 70, 75, 80

a

Find the equation of the least-squares regression line of

y

on

x

, giving your answer in the form

y = mx + c

with

m

and

c

correct to 3 significant figures. [3]

b

A ninth student studied for 4 hours. Use your regression equation from part (a) to predict their test score. Hence determine whether this prediction is an interpolation or an extrapolation, and explain why this affects its reliability. [3]

c

A student claims: "Because the correlation coefficient

r

is very close to 1, the regression line proves that studying more hours causes higher test scores." Critique this claim, identifying one mathematical reason and one contextual reason why it is not fully justified. [4]

Solutions

37QuestionBox plots and quartilesAssessment Practice

10 marks~15 minCriterion B

The table below shows five datasets of increasing sample size. All datasets share the same minimum (20), maximum (80), and median (50), but have different interquartile ranges.

Dataset A	$n = 8$	$Q_1 = 35$	$Q_3 = 65$
Dataset B	$n = 12$	$Q_1 = 38$	$Q_3 = 62$
Dataset C	$n = 16$	$Q_1 = 40$	$Q_3 = 60$
Dataset D	$n = 20$	$Q_1 = 42$	$Q_3 = 58$
Dataset E	$n = 24$	$Q_1 = 44$	$Q_3 = 56$

a

Calculate the interquartile range for each dataset and describe how the IQR changes as

n

increases, including the size of each successive decrease. [3]

b

A sixth dataset has

n = 28

and follows the same pattern. Deduce the IQR for this dataset and write down a general rule for the IQR in terms of

n

, valid for

n \geq 12

. [3]

c

Justify why the linear rule found in part (b) must eventually break down, referring to the positions of

Q_1

and

Q_3

relative to the fixed median and the constraints imposed by the minimum and maximum. [4]

Solutions

38QuestionBox plots and quartilesAssessment Practice

6 marks~9 minCriterion C

The following 11 test scores are listed in ascending order.

18,\ 22,\ 25,\ 28,\ 31,\ 33,\ 36,\ 39,\ 42,\ 45,\ 50

a

State the median and interquartile range (IQR) of this data set. [2]

b

A twelfth score,

s

, is added to the data set. The new median is 34. Find the value of

s

. [2]

c

A student claims that adding

s

to the data set does not change the IQR. Justify whether this claim is correct. [2]

Solutions

39QuestionBox plots and quartilesAssessment Practice

6 marks~9 minCriterion D

A school surveys a random sample of 7 students and records the number of minutes each student spends on homework in one week. The results are:

120, \ 150, \ 180, \ 200, \ 220, \ 250, \ 300

The school wishes to use a box plot of this sample to draw conclusions about all 200 students in the year group.

a

State the median and interquartile range (IQR) of this data set. [2]

b

A new student joins the sample. The mean of all 8 values is 196.25 minutes. Find the number of minutes this student spent on homework. Give your answer as an exact value. [2]

c

The school claims the box plot of the original 7 students reliably represents the homework habits of all 200 students. Assess whether this claim is reasonable, referring to both the sample size and one other statistical consideration. [2]

Solutions

40QuestionBox plots and quartilesAssessment Practice

9 marks~14 minCriterion A

The ordered test scores (out of 100) of 15 students are shown below.

3, \ 45, \ 52, \ 58, \ 61, \ 65, \ 68, \ 70, \ 72, \ 75, \ 78, \ 80, \ 82, \ 85, \ 90

a

Find the median, lower quartile

Q_1

, and upper quartile

Q_3

of the data. [3]

b

A value is defined as an outlier if it lies below

Q_1 - 1.5 \times \text{IQR}

or above

Q_3 + 1.5 \times \text{IQR}

, where

\text{IQR} = Q_3 - Q_1

.

Show that 3 is the only outlier in this dataset. [3]

c

A second class of 15 students sits the same test. Their scores have the same median and the same interquartile range as the first class, but a larger standard deviation.

Justify whether the second class's results should be considered more or less consistent than the first class's results, referring to both measures in your answer. [3]

Solutions

Statistics and Probability — Free MYP5 Mathematics (Extended) Practice Questions