4.10 Spearman’s rank correlation questions for IB Maths AI

23 questions on Spearman’s rank correlation, written by Augment in the style of IB exam papers. Six of them are below in full, to answer on paper.

  • IB Mathematics: Applications and Interpretation
  • Part of 4 Statistics and probability
  • SL and HL

Questions by paper and level

SL and HL questions are for students at either level. HL only questions cover what only HL students study.

PaperSL and HLHL onlyAll
Paper 112113
Paper 29110
Every paper21223

Counted on

Questions to try

Written by Augment in the style of IB exams, not past papers. Answer them on paper, then check each one against its markscheme with a free account.

Question 1 · AI-01483Paper 1 · 4 marks

At a winter festival in Harbin, two judges, Mei and Arjun, rank eight ice sculptures, A to H, from best (1) to worst (8).

Sculpture

A

B

C

D

E

F

G

H

Mei's rank

1

2

3

4

5

6

7

8

Arjun's rank

3

1

2

5

4

8

6

7

  1. (a)

    Find Spearman's rank correlation coefficient, rs.

    2 marks

  2. (b)

    Interpret your answer to part (a) in context.

    2 marks

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Question 2 · AI-01703Paper 2 · 17 marks

At a wine festival in Mendoza, two tasters, Valentina and Joaquín, each give a score out of 20 to eight wines, A to H.

Wine

A

B

C

D

E

F

G

H

Valentina's score, v

15

12

17

11

14

18

13

16

Joaquín's score, j

13

14

16

10

14

17

11

12

  1. (a)

    (i) Find Pearson's product-moment correlation coefficient, r.

    (ii) Describe the correlation between the two tasters' scores.

    3 marks

  2. (b)

    Write down the equation of the regression line of j on v.

    2 marks

  3. (c)

    Valentina gives a ninth wine a score of 16.5. Use your regression line to estimate Joaquín's score for this wine.

    2 marks

  4. (d)

    The scores for the eight wines A to H are ranked, with the highest score given rank 1. Valentina's ranks are shown in the table.

    Wine

    A

    B

    C

    D

    E

    F

    G

    H

    Valentina's rank

    4

    7

    2

    8

    5

    1

    6

    3

    Joaquín's rank

    Complete the table by writing down Joaquín's ranks.

    2 marks

  5. (e)

    Find Spearman's rank correlation coefficient, rs.

    2 marks

  6. (f)

    The festival organizer tests, at the 5% significance level, whether there is a positive association between the two tasters' rankings. The critical value for this test is 0.643.

    (i) Write down the alternative hypothesis.

    (ii) State the conclusion of the test. Give a reason for your answer.

    4 marks

  7. (g)

    Valentina later changes her score for wine F from 18 to 19. State, giving a reason, whether

    (i) the value of rs changes;

    (ii) the value of r changes.

    2 marks

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Question 3 · AI-01484Paper 1 · 5 marks

The table shows the scores of eight students, A to H, in a mathematics test and a physics test. The mathematics scores have been ranked, with the highest score given rank 1.

Student

A

B

C

D

E

F

G

H

Mathematics score

78

65

91

52

84

70

59

88

Physics score

72

61

85

58

72

66

49

90

Mathematics rank

4

6

1

8

3

5

7

2

Physics rank

  1. (a)

    Complete the table by writing down the physics ranks.

    2 marks

  2. (b)

    Find Spearman's rank correlation coefficient, rs.

    2 marks

  3. (c)

    Interpret your answer to part (b) in context.

    1 mark

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Question 4 · AI-01704Paper 2 · 13 marks

The table shows the gross domestic product (GDP) per person, g thousand US dollars (USD), and the life expectancy at birth, L years, for ten countries.

g (thousand USD)

1.2

2.8

4.5

7.9

12.4

18.6

25.3

41.7

52.0

85.0

L (years)

62.1

66.8

70.2

73.5

75.1

76.8

78.0

81.2

80.6

82.3

  1. (a)

    Find Pearson's product-moment correlation coefficient, r.

    2 marks

  2. (b)

    Find Spearman's rank correlation coefficient, rs.

    2 marks

  3. (c)

    By considering the data, explain why rs is greater than r.

    2 marks

  4. (d)

    Write down the equation of the regression line of L on g.

    2 marks

  5. (e)

    A researcher uses this regression line to estimate the life expectancy in a country with a GDP per person of 100 thousand USD. Find this estimate.

    2 marks

  6. (f)

    Give two reasons why the estimate in part (e) is unlikely to be reliable.

    2 marks

  7. (g)

    A newspaper reports that "increasing a country's GDP will make its people live longer". Comment on this statement.

    1 mark

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Question 5 · AI-01485Paper 1 · 5 marks

A park ranger in Vermont records the age, a years, and the height, h metres, of eight sugar maple trees.

a (years)

2

4

6

8

10

15

20

30

h (m)

1.1

2.6

4.3

5.6

6.8

8.9

10.3

11.8

  1. (a)

    Write down the value of Spearman's rank correlation coefficient, rs.

    1 mark

  2. (b)

    Find Pearson's product-moment correlation coefficient, r.

    2 marks

  3. (c)

    Explain why rs=1 but r<1 for these data.

    2 marks

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Question 6 · AI-01705Paper 2 · 15 marks

A swimming coach records the age, a years, and the best time for 50 m freestyle, t seconds, of nine swimmers in a masters club.

a (years)

19

22

24

25

27

28

31

33

58

t (s)

24.8

25.1

25.6

25.0

26.2

25.6

26.9

27.4

31.5

  1. (a)

    For the ages of the swimmers, write down

    (i) the lower quartile;

    (ii) the upper quartile.

    2 marks

  2. (b)

    Show that the age of the oldest swimmer is an outlier.

    2 marks

  3. (c)

    Find Pearson's product-moment correlation coefficient, r, for all nine swimmers.

    2 marks

  4. (d)

    Write down the ranks of the nine times, giving the fastest time rank 1.

    2 marks

  5. (e)

    Find Spearman's rank correlation coefficient, rs, for all nine swimmers.

    2 marks

  6. (f)

    The coach removes the oldest swimmer from the data.

    (i) Find the new value of r.

    (ii) Find the new value of rs.

    3 marks

  7. (g)

    Use your answers to parts (c), (e) and (f) to comment on the effect of the outlier on the two correlation coefficients.

    2 marks

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The notes on 4.10 Spearman’s rank correlation

More questions in Statistics and probability

  1. 4.1Collecting data and sampling
  2. 4.2Presenting data
  3. 4.3Measures of central tendency and dispersion
  4. 4.4Correlation and regression
  5. 4.5Concepts of probability
  6. 4.6Combined events
  7. 4.7Discrete random variables
  8. 4.8The binomial distribution
  9. 4.9The normal distribution
  10. 4.10Spearman’s rank correlationThis page
  11. 4.11Hypothesis testing
  12. 4.12Data collection and reliabilityHL only
  13. 4.13Non-linear regressionHL only
  14. 4.14Linear transformations of random variablesHL only
  15. 4.15The central limit theoremHL only
  16. 4.16Confidence intervalsHL only
  17. 4.17The Poisson distributionHL only
  18. 4.18Further hypothesis testingHL only
  19. 4.19Transition matrices and Markov chainsHL only

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