Statistics Toolkit (DP IB Maths: AI HL)

Exam Questions

3 hours31 questions
1a
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2 marks

Every week an orangutan sanctuary measures the weight of each of its orangutans.

The weights, to the nearest kg, of ALL their 18 adult males are listed below:

52, 57, 63, 80, 56, 66, 101, 68, 55, 96, 70, 62, 66, 64, 99, 91, 55, 92

Using a convenience sample of size six, calculate the mean weight of the male orangutans from the data set above.

1b
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2 marks

Starting from the third data value, take a systematic sample of size six and
re-calculate the mean weight of the male orangutans from the data set above.

1c
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2 marks

Compare your results from parts (a) and (b) and state, with a reason, which sampling method is more reliable.

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2a
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3 marks

A supermarket wants to gather data from its shoppers on how far they have travelled to shop there. One lunchtime an employee is stationed at the door of the shop for half an hour and instructed to ask every customer how far they have travelled.

(i)      State the sampling method the employee is using.

(ii)     Give one advantage and one disadvantage of using this method.

2b
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2 marks

State and briefly describe an alternative method of non-random sampling that the employee could use to obtain the required data for a sample of 30 customers.

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3
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4 marks

A pharmacy sells face masks in a variety of sizes.  Their sales over a week are recorded in the table below:

  Kids Adults
Size Small Large S M L XL
Frequency bold italic f 29 4 8 24 15 4

(i)

Write down the mode for this data.

(ii)

Explain why, in this case, the mode from part (i) would not be particularly helpful to the shop owner when reordering masks.

(iii)
Given that the shop is open every day of the week, calculate the mean number of masks sold per day.

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4
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3 marks

The lengths ( l spacecm) of a sample of nine otters, measured to the nearest centimetre by a wildlife research team, are:

76        77        91        65        63        83        92        61        88

Calculate the mean and standard deviation of the nine recorded lengths.

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5a
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4 marks

Jeanette works for a conservation charity who rescue orphaned otters.  Over many years she records the weight (g) of each otter when it first arrives.  The data is illustrated in the following box and whisker diagram:

ib5-ai-sl-4-1-ib-maths-medium

Using the box plot above:

(i)

Write down the median weight of the otters.

(ii)

Write down the lower quartile.

(iii)
Find the interquartile range.
5b
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3 marks

Otters are then weighed weekly to track their growth.  Summary data on the weights (g) of otters after one month is shown in the table below:

  Weight bold italic g
Smallest weight 125
Range 48
Median 152
Upper Quartile 164
Interquartile Range 33

On the grid, draw a box plot for the information given above.

ib6a-ai-sl-3-4-ib-maths-veryhard

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6a
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4 marks

The heights, in metres, of a flock of 20 flamingos are recorded and shown below:

0.4       0.9       1.0       1.0       1.2       1.2       1.2       1.2       1.2       1.2

1.3       1.3       1.3       1.4       1.4       1.4       1.4       1.5       1.5       1.6

An outlier is an observation that falls either more than 1.5 cross times (interquartile range) above the upper quartile or less than 1.5cross times  (interquartile range) below the lower quartile. 

(i)        Find the values of straight Q subscript 1,straight Q subscript 2  and straight Q subscript 3.

(ii)       Find the interquartile range.

(iii)      Identify any outliers.

 

6b
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3 marks

Using your answers to part (a), draw a box plot for the data.

ib6-ai-sl-4-1-ib-maths-medium

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7a
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3 marks

120 competitors enter an elimination race for charity.  Runners set off from the same start running as many laps of the course as possible.  Their total distance is tracked and the competitor who runs the furthest over a 6-hour period is the winner.  The distances runners achieved are recorded in the table below:

Distance bold italic d (miles) Frequency bold italic f
25 ≤ d < 30 8
30 ≤ d < 35 10
35 ≤ d < 40 32
40 ≤ d < 45 54
45 ≤ d < 50 10
50 ≤ d < 55 6

On the grid below, draw a cumulative frequency graph for the information in the table.

ib6b-ai-sl-4-1-ib-maths-medium

7b
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3 marks

Use your graph to find an estimate for the median and interquartile range.

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8a
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2 marks

Police check the speed of vehicles travelling along a stretch of highway.  The cumulative frequency curve below summarises the data for the speeds, in kmph, of 80 vehicles:

ib8a-ai-sl-4-1-ib-maths-medium

Use the graph to find an estimate for the median speed.

8b
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3 marks

The speed limit for this section of road is 80 kmph.

Vehicles travelling above the speed limit are issued with a speeding ticket. Those travelling more than 10% over the speed limit are pulled over.  Use the graph to estimate the percentage of vehicles that the police pull over.

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9a
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2 marks

The following cumulative frequency curve shows the number of hours, h, students took to complete their online driving course.  The data is taken from 80 students, randomly selected from a large sample over a 12 month period.

ib9a-ai-sl-4-1-ib-maths-medium

Find the median number of hours spent completing the online driving course.

9b
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2 marks

Find the number of students whose online course time was within 1 hour of the median.

9c
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2 marks

Calculate the interquartile range.

9d
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3 marks

The same information is represented by the following table.

Hours comma h 0 less than h less or equal than 2 2 less than h less or equal than 4 4 less than h less or equal than 7 7 less than h less or equal than 10
Frequency 9 p q 6

Find the value of p and the value of q.

9e
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3 marks

It is known that 10% of students take longer than d hours to complete the online driving course.

Find the value of d.

9f
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3 marks

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10a
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2 marks

For her IA, Mia decides to investigate the popularity of the social media platform SMEsocial amongst 16 to 18 year olds. She thinks of two possible methods for collecting data to analyse.

Method 1: Use visitor statistics from the SMEsocial website.

Method 2: Survey pupils from her school on their usage of SMEsocial.

State one advantage and one disadvantage of method 2 outlined above.

10b
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1 mark

Mia decides to go ahead with the survey of pupils in her school and designs a questionnaire.  She randomly asks 15 students to complete it.

State the type of sampling method used.

10c
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1 mark

One of the questions that Mia asks on the survey is the following:

“How long do you spend on SMEsocial each week?”

State one criticism of the question used by Mia.

10d
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1 mark

Describe one way in which Mia could increase the reliability of her investigation.

10e
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1 mark

Describe one way that Mia could improve the validity of her data.

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11a
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1 mark

Nils believes that people with longer legs can run quicker.  He decides to test his hypothesis by measuring the outer leg length of 10 boys aged between 10 and 12, and then using a data logger to record the time taken for each participant to run 100 metres.  The results are shown on the scatter diagram below.

q11a-4-1-medium-ib-ai-hl-maths

Nils notices that the scatter graph appears to support his hypothesis. Recalling that the world record for running 100 metres is 9.58 seconds, however, he believes that there may be a problem with the data that he has collected.

Based on the information given in the question and the data, explain what the data suggests about the students.

11b
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2 marks

Explain what could be wrong with the data collected and how it may have occurred.

11c
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1 mark

State what Nils should do to confirm the reliability of the data.

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1a
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3 marks

The following cumulative frequency curve shows the distance travelled, in kilometres, to work by 160 people in Cape Town, South Africa, during 2021.

ib1a-ai-sl-4-1-ib-maths-hard

Rounding your answer to the nearest half kilometre, use the graph to find the

(i)

median distance.

(ii)

lower quartile.

(iii)
upper quartile.
1b
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3 marks

Draw a box-and-whisker diagram to represent this sample.

1c
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2 marks

Using your answers from part (a), calculate the maximum distance that can be travelled by someone and still not be considered an outlier.

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2a
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1 mark

The following cumulative frequency curve shows the amount of water, in litres, that 60 people drink over a month.

ib2a-ai-sl-4-1-ib-maths-hard

State whether the data is discrete or continuous.

2b
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6 marks

Complete the following frequency table.

  Litres of water, straight L 0 less or equal than L less or equal than 30  30 less than L less or equal than 50 50 less than L less or equal than 70 70 less than L less or equal than 90
  Frequency        
  Cumulative Frequency         

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3a
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3 marks

The following cumulative frequency curve shows the number of hours spent gaming per week by 120 high school students.

ib3a-ai-sl-4-1-ib-maths-hard

Find the

(i)

median

(ii)
interquartile range.
3b
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2 marks

Calculate the percentage of students that spent less than 17 hours gaming per week.

3c
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1 mark

The 120 students were chosen randomly by sampling 60 senior students and 60 junior students. The school has the same number of senior and junior students.

Write down the sampling method used.

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4a
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1 mark

The histogram below shows the weights of kiwifruit, each measured to the nearest gram.

q5a-4-1-hard-ib-ai-sl-maths

Write down the modal weight of the kiwifruits.

4b
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2 marks

Find the median weight of the kiwifruits.

4c
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4 marks

Write down two inequalities that represent the weight, w, of a kiwifruit that is considered an outlier.

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5a
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2 marks

A group of people who use a gym participated in a research survey and the ages of the participants were recorded in the following table:

Age comma space in space years space left parenthesis straight a right parenthesis 15 less or equal than a less than 18 18 less or equal than a less than 30 30 less or equal than a less than 50 50 less or equal than a less than 65 65 less or equal than a less than 80
Frequency 4 x 34 28 14

It is known that 34 less than x less than 40.

Write down

(i)
the modal class.

(ii)
the mid interval value of the modal class.
5b
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2 marks

Determine the class in which the lower quartile lies.

5c
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2 marks

Calculate the mean age of participants between the ages of 30 and 80.

5d
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1 mark

The participants in this survey were chosen by selecting every person who entered the gym who was not wearing headphones.

Write down the type of sampling method used.

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6a
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4 marks

The table below shows the distribution of deliveries made by a group of food delivery drivers on a working day in Berlin.

Deliveries

6

7

8

9

10

11

Frequency

15

18

22

41

12

5

Find

(i)
the mean number of deliveries made.

(ii)
the standard deviation.

(iii)
the median.

(iv)
the interquartile range.
6b
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2 marks

Determine if a delivery driver who made 4 deliveries would be considered an outlier.

6c
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1 mark

The delivery drivers were selected for the survey by ordering their names alphabetically, then selecting every 20th number.

Identify the sampling technique used in the sampling method.

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7a
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3 marks

The following table shows the number of passes made by 11 players on a rugby team:

Player

1

2

3

4

5

6

7

8

9

10

11

Number of passes

12

18

22

41

9

18

22

28

30

21

18

Write down

(i)
the mean.

(ii)
the median.

(iii)
the mode.
7b
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3 marks

Find the interquartile range.

7c
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3 marks

Determine if any of the players would be considered an outlier, and if so, state which player(s).

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8a
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4 marks

The table below shows the number of TVs school students have at their home:

Number of TVs

0

1

2

3

4

5

Frequency

12

42

56

42

30

15

Find

(i)
the mean.

(ii)
the standard deviation.

(iii)
the median.

(iv)
the interquartile range.
8b
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2 marks

Determine if a student who has seven TVs would be considered an outlier.

8c
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1 mark

The students were selected for the survey by randomly selecting student ID numbers, using a random number generator.

Identify the sampling technique used in the sampling method.

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9a
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4 marks

The number of days off taken by employees in a company during a two-year period was recorded. The data was organised into a box and whisker diagram as shown below:

q9a-4-1-hard-ib-ai-sl-maths

For this data, write down

(i)
the maximum number of days off by an employee during the two years.

(ii)
the median.

(iii)
the interquartile range.
9b
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2 marks

Steven claims that this box and whisker diagram can be used to infer that the percentage of employees who took between 14 and 30 days off is greater than the percentage of employees who took 25 days or more off.

State whether Steven is correct. Justify your answer.

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10a
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1 mark

The table below shows the points scored per game from two basketball players, Karo and Anna, across 9 games:

Anna

22

25

27

22

21

20

31

29

28

Karo

17

12

8

6

19

18

20

19

96

State a statistical measure that would be helpful for a coach who wants to measure the consistency of players scoring performances.

10b
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6 marks

For both players, find

(i)
the mean.

(ii)
the standard deviation.

(iii)
the median.

(iv)
the range.

(v)
the interquartile range.
10c
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1 mark

Determine whether the mean or median is a better representation of Karo’s scoring ability. Justify your answer.

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11a
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2 marks

Ashika runs a maths revision course three months before the final exam. In order to determine the current levels of the students Ashika gives them a practice exam the day before the course starts. On the first day of the course Ashika gives the students another test which has similar questions to the first test.

State whether Ashika is testing the reliability or validity of her method for determining the current levels of the students. State the name of the test she uses.

11b
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3 marks

Ashika wants to determine whether the course improves the students’ levels so she asks the following question to each of the students after the revision course:

“Do you think that this course will help you to achieve at least a level 5 in your maths exam?”

Explain whether this question has content validity. Give two reasons for your answer.

11c
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2 marks

Explain whether asking the question anonymously would increase its validity.

11d
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1 mark

After the final exam, Ashika analyses how well the levels from the test at the start of her revision course predict the levels in the final exam.

State the name of the validity test that Ashika is using.

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1a
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2 marks

A group of Netflix subscribers participated in a research survey and the ages of participants were recorded in the following table.

Age, in years open parentheses a close parentheses 15 less or equal than a less than 25 25 less or equal than a less than 35 35 less or equal than a less than 45 45 less or equal than a less than 55 55 less or equal than a less than 65
Number of participants 11 62 56 x 12

It is known that 56 less than x less than 62..

Write down

(i)

the modal class

(ii)
the mid interval value of the modal class.
1b
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2 marks

Determine the class in which the upper quartile lies.

1c
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2 marks

Using the mid-interval values the mean of the data can be estimated to be 39.95.

Find the value of x.

1d
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1 mark

The participants in this survey were chosen by randomly selecting people entering a supermarket. However, to be more efficient, the surveyor only selected people who were in groups of at least 3.

Write down the type of sampling method used.

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2a
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2 marks

25 people are invited to the premier of the movie “ICE”, and they are asked to give the movie a score out of 1-10. The table below shows the distribution of the scores.

  Score   1   2   3   4   5   6   7   8   9   10
  Frequency   1   2   1   4   5   5   1 a   3  b

It is known that a greater than b.

Find the value of a spaceand  the value of b.

2b
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4 marks

Draw a bar chart of the data on the grid below.

ib2b-ai-sl-4-1-ib-maths-veryhard

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3a
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1 mark

A shoe store wants to know which shoe size is the most popular and so they record the shoe sizes of 30 customers.

  Score   7   7.5   8   8.5   9   9.5     10   11   12   13
  Frequency   5   1   4 p   3   3   2 1 q 1

State whether the above data is continuous or discrete.

3b
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2 marks

It is known that p equals 4 q.

Find the value of

(i)

p

(ii)
q.
3c
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3 marks

Write down

(i)

the mean

(ii)

the median

(iii)

the mode.

3d
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1 mark

State which statistical measure is most useful for the shoe store. Justify your answer.

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4a
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3 marks

A data set has a mean of 22 and a standard deviation of 6.

Each element of the data set has 4 subtracted from it.

Find the value of

(i)

the new mean

(ii)
the new standard deviation.
4b
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3 marks

After each element has 4 subtracted from it, each element is divided by  2 over 3 .

Find the value of

(i)

the new mean

(ii)
the new variance.

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5a
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1 mark

The following box and whisker diagram shows the number of social media posts made by a group of content creators over a week.

ib5a-ai-sl-4-1-ib-maths-veryhard

State whether the data is discrete or continuous.

5b
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3 marks

It is given that 7 a equals 4 b and a plus b equals 22.

Calculate the value of

(i)

a

(ii)
b
5c
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4 marks

A content creator made k posts, where k less than a. Given that k spaceis an outlier find the maximum value of k.

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6a
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2 marks

The following table displays the percentage change of an investor’s portfolio over 12 months.

Month Percentage change
1 negative 5.51
2 6.86
3 4.00
4 1.67
5 2.18
6 0.17
7 1.43
8 negative 2.31
9 3.31
10  negative 3.35
11  1.81
12 negative 3.24

Calculate

(i)

the mean

(ii)
the standard deviation.
6b
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1 mark

State which statistical measure, calculated in part (a), gives an indication of the volatility of the investor’s portfolio.

6c
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3 marks

The investor’s portfolio value at the beginning of the 12 months was $6000.

Calculate the value of the portfolio at the end of the 12 months. Give your answer to 2 decimal places.

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7a
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4 marks

At a swimming competition the mean time of the first four swimmers is 28.2 seconds. The time for the fifth and sixth swimmers are then recorded and the mean time of the first six swimmers is 29.8 seconds. The difference between the fifth and sixth swimmer’s time is 0.4 seconds.

Find the time achieved by

(i)

the fifth swimmer

(ii)
the sixth swimmer.
7b
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1 mark

The first swimmers time is 25.7 seconds.

Calculate the range in times of the swimmers.

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8a
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3 marks

The table below shows the average temperature,T space degree C , in a city over a normal year

(not a leap year).

Temperature negative 5 less or equal than T less than 5 5 less or equal than T less than 15 15 less or equal than T less than 25 25 less or equal than T less than 35
Frequency 124 p 109 q

It is given that p equals 11 q.

Calculate the values of

(i)

p

(ii)
q
8b
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3 marks

Using your GDC, estimate the value of

(i)

the mean

(ii)
the standard deviation.
8c
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2 marks

It Is given that the mean temperature of the city over the year is 14.2 space degree straight C.

Calculate the percentage error between your estimate of the mean temperature, found in part (b) (i), and the actual mean temperature.

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9a
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1 mark

Ann is the product manager of a bicycle company and she is testing a new type of tyre. Ann wants to test whether using the new tyres allows cyclists to reach higher speeds. She asks 10 volunteers to cycle 30 miles and to record their times. An extract of the data is shown below:

Volunteer

Time (hours)

A

1.05

B

1.75

C

1.50

D

1.25

It is known that the recording devices had two settings for displaying time. Ann checks and finds that volunteer C took 1 hour and 30 minutes whereas volunteer D took 1 hour and 25 minutes.

Explain why Ann will need to verify the time of volunteer A.

9b
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3 marks

Ann runs the experiment again but this time she records the times herself. She asks the volunteers to repeat the process three times.

State three factors which Ann should control to try to ensure that her measurements are reliable.

9c
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2 marks

It is later discovered that the volunteers completed all three journeys during the same day. Explain how this would affect the reliability of the measurements.

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