Product moment correlation coefficientEdexcel International A Level Maths: Subtopic test

1. 1
   A dealer records the age, $x$ years, and the value, $y$ thousand pounds, of 10 used cars of the same model. The summary statistics are $S_{xx}=40$, $S_{yy}=90$ and $S_{xy}=-48$.

   (a)

   Find the value of the product moment correlation coefficient, $r$.

   [1 mark]

   • A$-1.2$
   • B$0.8$
   • C$-0.8$
   • D$-0.533$

   (b)

   Which statement best interprets this value?

   [1 mark]

   • AThere is strong negative linear correlation: older cars tend to have lower values
   • B80% of the cars lose value as they get older
   • CThere is weak negative linear correlation between age and value
   • DThere is strong positive linear correlation between age and value

   (c)

   The dealer converts the ages into months and the values into pounds. State the new value of $r$ and justify your answer.

   [2 marks]

   Total for question 1: 4 marks

2. 2
   A seaside town records, for each of 12 months, the ice cream sales, $x$ hundred cones, and the number of sunburn cases, $y$, treated at the local clinic. The product moment correlation coefficient for these data is $r=0.91$.

   (a)

   Which statement correctly describes the correlation?

   [1 mark]

   • A91% of ice cream sales lead to sunburn cases
   • BThere is weak positive linear correlation between ice cream sales and sunburn cases
   • CThere is strong negative linear correlation between ice cream sales and sunburn cases
   • DThere is strong positive linear correlation between ice cream sales and sunburn cases

   (b)

   Which is the most appropriate conclusion?

   [1 mark]

   • AEating ice cream causes sunburn
   • BMonths with higher ice cream sales tend to have more sunburn cases, but this does not show that one causes the other
   • CSunburn causes people to buy ice cream
   • DA strong correlation shows that no other variable is involved

   (c)

   Suggest a reason why the two variables are strongly correlated, even though neither is likely to cause the other.

   [2 marks]

   Total for question 2: 4 marks

3. 3
   A teacher records the number of practice tests, $x$, completed by five students and their scores, $y$, out of 5 in a final quiz. The values of $x$ are 1, 2, 3, 4, 5 and the scores are 2, 4, 5, 4, 5. For these data $\sum x=15$, $\sum y=20$, $\sum x^2=55$, $\sum y^2=86$ and $\sum xy=66$.

   (a)

   Find $S_{xx}$, $S_{yy}$ and $S_{xy}$.

   [3 marks]

   (b)

   (i) Calculate the product moment correlation coefficient.
   (ii) Interpret your value in context.
   (iii) The teacher concludes that completing more practice tests causes the scores to rise. Comment on this conclusion.

   [4 marks]

   Total for question 3: 7 marks

4. 4
   A researcher records the age, $x$ years, and the resale price, $y$ pounds, of eight laptops. The values of $x$ are 1, 2, 3, 4, 5, 6, 7, 8 and the prices are 900, 780, 700, 560, 520, 410, 330, 250. For these data $\sum x=36$, $\sum y=4450$, $\sum x^2=204$, $\sum y^2=2831900$ and $\sum xy=16170$.

   (a)

   (i) Find $S_{xx}$, $S_{yy}$ and $S_{xy}$.
   (ii) Calculate the product moment correlation coefficient.
   (iii) Interpret your value in context.

   [6 marks]

   (b)

   (i) The researcher converts the ages into months and the prices into euros using a fixed exchange rate. State the new value of $r$ and give a reason.
   (ii) A ninth laptop, a rare model aged 12 years, is added to the data with a price of 1200 pounds. Without calculating, state and explain the effect on $r$.
   (iii) For a different set of laptops, prices follow the curve $y=\frac{1000}{x}$ exactly. Explain why $r$ would not be $-1$.

   [6 marks]

   Total for question 4: 12 marks

End of questions