CIE A Level Maths: Probability & Statistics 1

Topic Questions

1.2 Representation of Data

1a
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4 marks

The train journey times, in minutes, between March and Peterborough, are illustrated in the box and whisker diagram below.q1-easy-2-2-data-presentation-edexcel-a-level-maths-statistics

Using the box and whisker diagram above to find

(i)
the median journey time
(ii)
the lower and upper quartiles
(iii)
the interquartile range.
1b
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3 marks

The above times show those for a weekday.  The table below summarises the times for the same journey on a Saturday.

 

Journey Time

Fastest

16

Lower quartile

18

Median

19

Upper Quartile

20

Slowest

25

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

q2b-1-2-easy-ial-sl-maths-statistics

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2a
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3 marks

In the 2019 Red Bull Paper Wings World Finals, 55 contestants flew their paper aeroplanes in the Airtime Pre-Eliminations round.  The flight times achieved by the contestants’ paper aeroplanes are shown in the table below.

Time, t seconds Frequency bold italic f
0 ≤ t < 4 12
4 ≤ t < 8 25
8 ≤ t <12 16
12 ≤ t <16 2

On the grid below, draw a cumulative frequency graph for the information in the table.q2a-easy-2-2-data-presentation-edexcel-a-level-maths-statistics

2b
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1 mark

Use your graph to find an estimate of the median time.

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3a
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4 marks

The beats per minute (bpm) of 60 randomly selected drum ‘n’ bass songs were recorded and the data is summarised in the table below.

b (bpm) Frequency Class width Frequency density
140 ≤ b <160 10 160-140=20 10÷20=0.5
160 ≤ b <170 20 10  
170 ≤ b <175 20 5  
175 ≤ b <180 10 5  
(i)
Complete the column ‘Frequency density’ – the first one has been done for you.

(ii)

On the grid below, draw a histogram to represent these data.q3a-easy-2-2-data-presentation-edexcel-a-level-maths-statistics
3b
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2 marks

Estimate how many of the 60 songs had less than 150 beats per minute.

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4a
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4 marks

A local ambulance service is looking to cut down the times it takes to respond to 999 calls.  The ambulance service manager recorded the response times, in minutes, on 15 occasions.  These are given below.

4 8 12 9 7
14 6 5 8 7
9 10 7 3 6

                       

(i)
Find the median of the response times.

(ii)
Find the upper and lower quartiles, and the interquartile range.
4b
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3 marks

On the grid, draw a box plot for the information given above.q4b-easy-2-2-data-presentation-edexcel-a-level-maths-statistics

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5
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5 marks

To quality control the elasticity of elastic bands, a company selects random elastic bands from the end of their production line and has a machine stretch them until they snap.  The length, measured in millimetres, of an elastic band at the moment it snaps is recorded.  The incomplete histogram and frequency table below show the results.q5-easy-2-2-data-presentation-edexcel-a-level-maths-statistics

Snap length, l (mm) Frequency Class width Frequency density
100 ≤ l <150 5 150 - 100 = 50 5÷50=0.1
150 ≤ l <175     ÷25=0.4
175 ≤ l <200      
200 ≤ l < 225 15 225 - 200 = 25  
225 ≤ l <275 10    


Use the information to complete both the histogram and frequency table.

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6a
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4 marks

In the 2019 Red Bull Paper Wings World Finals, 40 contestants flew their paper aeroplanes in the Distance Pre-Eliminations round. The distances achieved by the contestants’ paper aeroplanes are shown in the cumulative frequency diagram below.q6-easy-2-2-data-presentation-edexcel-a-level-maths-statistics

Use the cumulative frequency graph to estimate

(i)
the median distance travelled by the contestants’ paper aeroplanes
(ii)
the upper and lower quartiles
(iii)
the interquartile range.
(iii)
the interquartile range.
6b
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2 marks

The top 9 contestants in the Pre-Eliminations round qualified for the Super Finals round.  Estimate the minimum distance a contestants’ paper aeroplane would’ve need to have flown in order for them to have qualified for the Super Finals.

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7a
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4 marks

Kungawo is investigating the lengths of earthworms he finds in his garden.

He records his results in a stem-and-leaf diagram as shown below.

       n space equals space 23                                   

4

2     represents a length of 42 mm

 

 

3

2   6

4

5   8   8   9

5

1   2   2   3   3   3   4   5   5   7   8   9                           

6

0   2   4

7

1   3

(i)    Find the median, lower quartile and upper quartile.

(ii)   Find the interquartile range. 

7b
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1 mark

State one advantage of using a stem-and-leaf diagram to represent data.

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8a
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2 marks

A meteorologist records the maximum wind speed (gust) per day using a device located in their garden.  The speeds, measured in knots, for a fifteen-day period during June are shown in the box-and-whisker diagram (box plot) below.

1-2-s-q--q8a-easy-cie-a-level-statistics

(i)
Give a reason why a box plot is a useful way to represent these data.
(ii)
 The box plot omits two outliers; 24 knots and 68 knots.

Explain why it would not be useful to include the outliers in the box plot. 

8b
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2 marks

The box plot below shows the speeds recorded by the meteorologist for a fifteen‑day period in July.

1-2-sq---q8b---easy-cie-a-level-statistics

(i)
Explain how the box plot for July shows that 10% of the maximum wind speeds recorded were over 59 knots.
(ii)

Explain how you cannot tell, from the July box plot, the lowest wind speed of the fifteen days.

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1a
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1 mark

The amounts of time engineers spent dealing with individual faults in a power plant were recorded to the nearest minute.  Data on 30 different faults is summarised in the table below.

Time t (minutes) Frequency bold italic f
90 - 129 6
130 - 169 8
170 - 199 12
200 - 249 4

Give a reason to support the use of a histogram to represent these data.

1b
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4 marks

On the grid below, draw a histogram to represent the data.q1b-hard-2-2-data-presentation-edexcel-a-level-maths-statistics

1c
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3 marks

Estimate the proportion of individual faults on which engineers spent longer than three hours.

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2a
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3 marks

A teacher took 19 students on an international trip. The incomplete box plot below shows part of the summary of the weights, in kg, of the luggage brought by each student.  Each student’s luggage weighed a different value.q2-hard-2-2-data-presentation-edexcel-a-level-maths-statistics

The median weight is 4 space kg more than the lower quartile.  The range of weights is three times the interquartile range of weights.

Use the information above to complete the box plot.

2b
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2 marks

Calculate the proportion of luggage weights which were less than 20 space kg.

2c
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2 marks

Students had to pay an additional fee if the weight of their luggage exceeded 23 kg.

Find the number of students who had to pay the additional fee.

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3a
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2 marks

Remy was studying how long it takes each of his 80 rats to find the exit to a maze.  Every two and a half minutes he records the number of rats which have found the exit which he then represents as a cumulative frequency curve.q3-hard-2-2-data-presentation-edexcel-a-level-maths-statistics

Based on the graph, write down an inequality for the time, t, taken by the fastest rat.  

3b
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3 marks

Remy’s assistant also recorded the actual times taken by the fastest and slowest rats.  She has used this information to begin constructing a box plot to represent the data.

Use the cumulative frequency curve to complete the box plot for the times.q3b-hard-2-2-data-presentation-edexcel-a-level-maths-statistics

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4a
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1 mark

An annual cheese-rolling contest involves participants chasing a 4 kg round of cheese down a steep 200 yard long hillside.  A group of 60 friends participated in the contest and the table below summarises the distances travelled by each before first falling over.

Distance d (yards) Frequency f
0 ≤ d < 40 23
40 ≤ d < 80 11
80 ≤ d < 120 9
120 ≤ d < 160 7
160 ≤ d < 200 6

How many of the 60 friends made it to the bottom without falling over?

4b
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3 marks

Draw a cumulative frequency graph for the information in the table.

q4b-hard-2-2-data-presentation-edexcel-a-level-maths-statistics

4c
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3 marks

The steepest part of the hill is between 100 and 140 yards away from the start. Using your graph, estimate how many people fell during this section of the hill.

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5a
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5 marks

The histogram below shows the masses, in grams, of 80 apples.q5-hard-2-2-data-presentation-edexcel-a-level-maths-statistics

Find estimates for the mean and standard deviation.

5b
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3 marks

Given that an outlier is classified as any data value falling more than two and a half standard deviations away from the mean, show that there are no outliers in the data above.

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6
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4 marks

Wendy is investigating the running times of 199 movies and finds the following information: 

When the movies are placed in ascending running time order, the 50th movie is an hour long and the 180th movie is two and a half hours long...   
 

   The interquartile range is 63 minutes. 

   The median splits the interquartile range in the ratio 3:4. 

   The middle 80% of the data covers a period of 123 minutes. 

On the grid below draw a box plot for the information above.

1-2-sq---q6a---hard--cie-a-level-statistics

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7a
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2 marks

Aggie runs a local bowls club and wants to attract younger members.  First Aggie needs to find out some information about the ages of current members of the club.  Selecting 35 male members and 35 female members at random, Aggie illustrates the ages of the 70 members on an ordered back-to-back stem-and-leaf diagram as shown below. 

                                                                        n space equals space 70       (35 Female, 35 Male)

           

(represents a female aged 50)     0

5

5     (represents a male aged 55)

 

 

 

Female

 

Male

7

1

 

8

2

4   5

6   2

3

2   7   7   9

5   3   3   1

4

4   4   5   6   6   7   8   8

9   8   8   7   6

5

0   2   2   3   4   6   6   7   8   8   9   9   9

9   9   7   6   4   3   2   1   1   1   0   0

6

2   5   5   6   6

8   7   5   4   2   2

7

4   6

6   6   4

8

2

1

9

 

 

(i)
Justify the use of a stem-and-leaf diagram for the data in this question.
(ii)
State why, despite what the stem-and-leaf diagram shows, there may be some male teenage members of the bowls club. 
7b
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4 marks

Compare the central tendency and variation of the male and female members’ ages.

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1a
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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:q1-medium-2-2-data-presentation-edexcel-a-level-maths-statistics

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.
1b
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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 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.q1b-medium-2-2-data-presentation-edexcel-a-level-maths-statistics

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2a
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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 d (miles) Frequency 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.q2a-medium-2-2-data-presentation-edexcel-a-level-maths-statistics

2b
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3 marks

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

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3a
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5 marks

The total amount of time cleaners spent dealing with unplanned incidents in a supermarket was recorded each day.  Data collected over 49 days is summarised in the table below.

Time t (minutes) Frequency bold italic f
0 ≤ t < 90 9
90 ≤ t < 120 24
120 ≤ t < 200 12
200 ≤ t < 250 4
(i)
Give a reason why a histogram is suitable way of illustrating these data.
(ii)
On the grid below, draw a histogram to represent this data.q3a-medium-2-2-data-presentation-edexcel-a-level-maths-statistics

3b
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2 marks

Estimate how often cleaners spent longer than 3 hours dealing with incidents.

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4a
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3 marks

A taxi firm, JustDrive, records data on the amount of time, to the nearest minute, that customers had to wait before their taxis arrived.  A random sample of 20 times is given below:

6 7 16 30 24
27 20 7 5 8
20 24 27 12 34
32 31 6 19 14

        

Find the median and interquartile range of the waiting times.

4b
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3 marks

On the grid, draw a box plot for the information given above.q4b-medium-2-2-data-presentation-edexcel-a-level-maths-statistics

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5
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4 marks

Filmworld cinemas collected data on the ages of visitors to their cinemas during a 24-hour period.  The incomplete histogram and frequency table show some of the information they collected:q5-medium-2-2-data-presentation-edexcel-a-level-maths-statistics

Age a (years) Frequency bold italic f
0 ≤ a < 5 15
5 ≤ a < 10  
10 ≤ a < 20  
20 ≤ a < 30 12
30 ≤ a < 50 18
50 ≤ a < 60 7


Use the information to complete the histogram and fill out the missing data in the frequency table.

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6a
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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:q6-medium-2-2-data-presentation-edexcel-a-level-maths-statistics

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

6b
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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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7a
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3 marks

An advertisement for a charity is shown on TV at the same time every weekday for four weeks.  To assess the impact of the advert, the charity’s manager decides to record the number of donations the charity receives each day in the hour after the advert is broadcast.  The results are listed below: 

                                    21       27       24       31       17

                                    22       25       26       27       9

                                    32       29       25       24       40

                                    23       22       19       12       14 

Represent these data in a sorted stem-and-leaf diagram.

7b
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2 marks

The manager decides that the advert is not cost effective unless the median number of donations per day in the hour after broadcast is at least 25. Determine whether the manager should continue to run the TV advert.

7c
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1 mark

Give one advantage of using a stem-and-leaf diagram as opposed to grouping the data into a frequency table.

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1a
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4 marks

The cumulative frequency graph below shows the IQs of 160 employees of a company.q1-veryhard-2-2-data-presentation-edexcel-a-level-maths-statistics

Using the graph, estimate the number of employees whose IQ is within 5 of the median IQ.

1b
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2 marks

The employees who are within the top 10% of IQs are offered training for a management role. Estimate the lowest IQ of an employee who is offered the training.

1c
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2 marks

The employees who are within the bottom 5% of IQs must attend a meeting with the director of the company. Estimate the highest IQ of an employee who must attend a meeting with the director.

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2a
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1 mark

There are 180 dogs in a rescue shelter. The histogram below shows the highest sound level reached by each individual dog’s bark, measured in decibels (dB).q2-veryhard-2-2-data-presentation-edexcel-a-level-maths-statistics

Write down the underlying feature associated with each of the bars in a histogram.

2b
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5 marks

Estimate how many dogs had a bark which ranged between 99 dB and 107 dB.

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3a
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3 marks

Mr Shapesphere, a history teacher, records the time, to the nearest minute, it takes him to mark each student’s essay. The times were summarised in a grouped frequency table and an extract is shown below:

Time t (minutes) Frequency bold italic f
0 - 10 7
11 – 30 16
31 – 35 4


A histogram was drawn to represent these data. The 11 – 30 group was represented by a bar of width 6 cm and height 4.5 cm.

Find the width and height of the 0 – 10 group.

3b
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3 marks

The total area under the histogram was 60.75 cm².

Find the number of essays which Mr Shapesphere recorded as taking longer than 35 minutes, to the nearest minute.

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4a
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3 marks

The grouped frequency table below contains information about the lengths of time of Susie’s calls with her customers. The table was used to draw the cumulative frequency curve also shown below.

Time (t minutes) 4 < t ≤ 8 8 < t ≤ 12 12 < t ≤ 16 16 < t ≤ 20
Frequency 16 a b c

q4-veryhard-2-2-data-presentation-edexcel-a-level-maths-statistics

Use the graph to find the values of a comma b space and space c.

4b
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3 marks

Use the graph to calculate the interquartile range of times for Susie’s calls.

4c
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2 marks

Use the graph to estimate the percentage of customers whose calls lasted longer than 10 minutes.

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5a
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5 marks

Crystal is given an incomplete box plot showing the lengths of 99 unicorn horns. She also knows that the median length is the midpoint of the minimum and maximum lengths and that the range is 2.5 times as big as the interquartile range.

Complete the diagrams below to show that there are two possible distributions given the information above.q5a-1-veryhard-2-2-data-presentation-edexcel-a-level-maths-statistics

5b
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3 marks

The box plot below shows the masses of the 99 unicorn horns.q5b-veryhard-2-2-data-presentation-edexcel-a-level-maths-statistics

Crystal discovers that two masses were recorded incorrectly; 11 kg should have been 8 kg and 9 kg should have been 10 kg. 

Explain why at most one value will need to be changed to fix the box plot.

5c
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2 marks

Explain why it is possible that the box plot will remain unchanged when it is fixed.

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6a
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4 marks

Football’s Premier League was launched in 1992 and the champions at the end of each season (up to and including the 2020-21 season) had scored the following number of goals:

 

                        67       80       80       73       76       68      80

                        97       79       79       74       73       72      72

                        83       80       68      103      78       93      86

                        102     73       68       85       106      95      85       83

 

(i) Justify the use of a stem-and-leaf diagram for these data.
(ii)   Draw a stretched, ordered stem-and-leaf diagram for these data.

6b
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4 marks
(i)
Comment on the shape of the distribution of the data.
(ii)

Which measure of central tendency would be the most appropriate to use with these data? Justify your choice.

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