4.10 Spearman's rank correlationIB Maths: Applications and Interpretation HL: Subtopic test

1. 1
   Six students sat a maths test and a physics test. Their marks (maths, physics) were $(45,52)$, $(60,58)$, $(72,80)$, $(38,40)$, $(55,61)$ and $(80,75)$. In both subjects the highest mark is given rank $1$.

   (a)

   What is the physics rank of the student who scored $61$?

   [1 mark]

   • A$2$
   • B$4$
   • C$3$
   • D$5$

   (b)

   Spearman's rank correlation coefficient $r_s$ for the maths and physics marks is closest to which value?

   [1 mark]

   • A$0.943$
   • B$-0.886$
   • C$0.114$
   • D$0.886$

   (c)

   Interpret the value of $r_s$ in context.

   [2 marks]

   Total for question 1: 4 marks

2. 2
   Seven dishes in a cookery competition were each given a score out of $10$ by two judges, $X$ and $Y$. In the order of the dishes, judge $X$ gave $7,5,9,5,8,6,4$ and judge $Y$ gave $8,6,9,7,7,5,3$. The highest score is given rank $1$.

   (a)

   What rank is given to each of the two dishes that judge $X$ scored $5$?

   [1 mark]

   • A$5.5$
   • B$5$
   • C$6$
   • D$4.5$

   (b)

   Use your GDC to find Spearman's coefficient $r_s$ for the two judges' scores.

   [1 mark]

   • A$0.799$
   • B$0.791$
   • C$0.209$
   • D$-0.791$

   (c)

   Write down the ranks given by judge $Y$ to the seven dishes, in the order of the dishes.

   [2 marks]

   Total for question 2: 4 marks

3. 3
   A scientist counts the bacteria in a culture, $y$ thousand, $x$ hours after the start of an experiment. For $x=1,2,3,4,5,6,7,8$ the counts are $2,4,9,17,31,52,85,140$. Use your GDC.

   (a)

   (i) Find Pearson's product-moment correlation coefficient $r$ between $x$ and $y$.
   (ii) Find Spearman's rank correlation coefficient $r_s$, justifying your value.

   [3 marks]

   (b)

   The scientist says that, because $r<1$, the counts are not perfectly related to time. Comment on this, and state which coefficient is more appropriate.

   [4 marks]

   Total for question 3: 7 marks

4. 4
   A tutor records the hours, $x$, that eight students revised for a test and their scores, $y$ (%). The hours were $2,4,5,6,8,9,10,30$ and the scores, in the same order, were $40,52,50,58,64,70,75,45$. The student who revised for $30$ hours was unwell on the day. Use your GDC.

   (a)

   (i) Find Pearson's product-moment correlation coefficient $r$ for the eight students.
   (ii) Write down the ranks of the hours and of the scores (rank $1$ for the lowest value) and find Spearman's $r_s$.
   (iii) Explain why $r_s$ is a more reliable measure than $r$ here.

   [6 marks]

   (b)

   The tutor now leaves out the student who revised for $30$ hours.
   (i) Find $r$ and $r_s$ for the remaining seven students.
   (ii) Compare how much each coefficient changed, and explain what this shows.
   (iii) State a conclusion about revision hours and scores for the seven students.

   [6 marks]

   Total for question 4: 12 marks

End of questions