Product moment and Spearman's rank correlationEdexcel A-Level Further Maths: Subtopic test

1. 1
   A researcher records the hours of sleep, $x$, and the reaction time, $y$ ms, of each of 8 randomly chosen students. The summary statistics are $S_{xx}=62.5$, $S_{yy}=40$ and $S_{xy}=-45$.

   (a)

   Calculate the product moment correlation coefficient between $x$ and $y$.

   [1 mark]

   • A$-0.72$
   • B$0.9$
   • C$-1.125$
   • D$-0.9$

   (b)

   Each reaction time is recoded as $w=100-2y$. Find the product moment correlation coefficient between $x$ and $w$.

   [1 mark]

   • A$-0.9$
   • B$0.9$
   • C$0.45$
   • D$-1.8$

   (c)

   Interpret the value of $r$ between $x$ and $y$ in context, and state an assumption about the population needed to use $r$ in a hypothesis test.

   [2 marks]

   Total for question 1: 4 marks

2. 2
   Two judges each rank six paintings, $A$ to $F$, from 1 (best) to 6 (worst). Judge 1 ranks $A$ to $F$ as $3, 1, 2, 5, 4, 6$ respectively. Judge 2 ranks them as $2, 1, 4, 5, 3, 6$ respectively.

   (a)

   Find $\sum d^2$, where $d$ is the difference between the two ranks given to each painting.

   [1 mark]

   • A$6$
   • B$4$
   • C$0$
   • D$12$

   (b)

   Calculate Spearman's rank correlation coefficient for these rankings.

   [1 mark]

   • A$0.171$
   • B$0.838$
   • C$0.829$
   • D$0.143$

   (c)

   State, with a reason, whether Spearman's rank correlation coefficient or the product moment correlation coefficient is more appropriate for these data.

   [2 marks]

   Total for question 2: 4 marks

3. 3
   Seven students sit a mathematics test and a physics test. Their scores, in the order of students $1$ to $7$, are: mathematics $56, 61, 61, 70, 48, 75, 66$ and physics $50, 58, 49, 64, 44, 70, 60$.

   (a)

   Rank the mathematics scores and the physics scores from lowest (rank 1) to highest, and hence show that $\sum d^2=3.5$.

   [3 marks]

   (b)

   Calculate Spearman's rank correlation coefficient, using the formula with $\sum d^2=3.5$. Interpret your answer in context and explain why the formula gives only an approximate value for these data.

   [4 marks]

   Total for question 3: 7 marks

4. 4
   A café owner records, over 10 days, the midday temperature $t\,^{\circ}$C and the number of cold drinks sold $s$. The data are coded as $x=t-15$ and $y=\frac{s-200}{10}$, giving $\sum x=30$, $\sum y=40$, $\sum x^2=250$, $\sum y^2=358$ and $\sum xy=270$.

   (a)

   (i) Calculate $S_{xx}$, $S_{yy}$ and $S_{xy}$, and hence the product moment correlation coefficient between $x$ and $y$.
   (ii) Write down the product moment correlation coefficient between $t$ and $s$, giving a reason.

   [6 marks]

   (b)

   (i) State the condition that the data must satisfy for the product moment correlation coefficient to be valid, and how a scatter diagram helps to check it.
   (ii) The café owner defines $w=100-4y$. Write down the product moment correlation coefficient between $x$ and $w$, justifying your answer.
   (iii) Spearman's rank correlation coefficient for the same data is $0.97$. Comment on what this suggests about the relationship between $t$ and $s$.

   [6 marks]

   Total for question 4: 12 marks

End of questions