Statistics 3 (S3): Regression and correlationEdexcel International A Level Further Maths: Topic test

1. 1
   A planner ranks six towns, $A$ to $F$, from $1$ (highest) to $6$ (lowest) by average household income and by the number of electric vehicle charging points per $1000$ residents. The income ranks of $A$ to $F$ are $1$, $2$, $3$, $4$, $5$, $6$ and the charging point ranks are $3$, $1$, $2$, $5$, $4$, $6$ respectively.

   (a)

   Find $\sum d^2$, where $d$ is the difference between the two ranks of a town.

   [1 mark]

   • A$0$
   • B$8$
   • C$6$
   • D$4$

   (b)

   Find Spearman's rank correlation coefficient.

   [1 mark]

   • A$0.229$
   • B$-0.371$
   • C$0.784$
   • D$0.771$

   (c)

   The planner has only the ranks of the towns, not the actual incomes or numbers of charging points. Explain why Spearman's rank correlation coefficient can be used here but the product moment correlation coefficient cannot.

   [2 marks]

   Total for question 1: 4 marks

2. 2
   A farmer ranks seven fields from $1$ (lowest) to $7$ (highest) by rainfall during the growing season. The fields with rainfall ranks $1$ to $7$ have wheat yield ranks $2$, $1$, $4$, $3$, $7$, $5$, $6$ respectively.

   (a)

   Suppose two fields had exactly the same yield, so that they would otherwise occupy ranks $3$ and $4$. What rank should each of these two fields be given?

   [1 mark]

   • A$3$
   • B$4$
   • C$3.5$
   • D$7$

   (b)

   Which of the following is a limitation of Spearman's rank correlation coefficient?

   [1 mark]

   • AIt measures only the strength of association between the ranks and says nothing about how much the yield changes for a given change in rainfall
   • BIt can only be used when rainfall and yield are Normally distributed
   • CIt is affected by extreme values in the same way as the product moment correlation coefficient
   • DIt always takes a value between $0$ and $1$

   (c)

   Given that $\sum d^2=10$, calculate Spearman's rank correlation coefficient for the seven fields.

   [2 marks]

   Total for question 2: 4 marks

3. 3
   Eight cities, $A$ to $H$, are ranked from $1$ (highest) to $8$ (lowest) by an air pollution index and by the number of hospital admissions for breathing problems per $10\,000$ residents. The pollution ranks of $A$ to $H$ are $1$, $2$, $3$, $4$, $5$, $6$, $7$, $8$ and the admission ranks are $2$, $1$, $5$, $3$, $4$, $8$, $6$, $7$ respectively.

   (a)

   Calculate Spearman's rank correlation coefficient between the two sets of ranks.

   [3 marks]

   (b)

   Test, at the $5\%$ significance level, whether there is positive correlation between pollution rank and admission rank. The critical value for $n=8$ for a one-tailed test at the $5\%$ level is $0.6429$.

   [4 marks]

   Total for question 3: 7 marks

4. 4
   A sports scientist records the weekly training time, in hours, and the time, in minutes, taken to ride a $40$ km time trial for nine cyclists. The training times are $6$, $11$, $8$, $14$, $9$, $12$, $5$, $10$, $7$ and the corresponding times for the time trial are $63.5$, $58.2$, $61.0$, $55.4$, $60.3$, $58.9$, $64.8$, $59.6$, $60.7$.

   (a)

   Calculate Spearman's rank correlation coefficient for these data. Test, at the $5\%$ significance level, whether there is negative rank correlation between training time and time trial time. The critical value for $n=9$ for a one-tailed test at the $5\%$ level is $0.6000$.

   [6 marks]

   (b)

   The product moment correlation coefficient for the same data is $r=-0.959$. Test, at the $5\%$ significance level, whether there is negative correlation, given that the critical value for $n=9$ for a one-tailed test at the $5\%$ level is $0.5822$. Explain why the hypotheses are written in terms of $\rho$ rather than $r$. State one assumption required for this test that is not required for the test in part (a), and explain why the result does not prove that more training causes faster times.

   [6 marks]

   Total for question 4: 12 marks

5. 5
   A researcher records the age and the resale price of $12$ randomly selected cars of the same model and calculates the product moment correlation coefficient $r=-0.62$. She tests, at the $5\%$ significance level, whether there is any correlation between age and resale price. The critical values for $n=12$ are $\pm0.5760$.

   (a)

   Which are the correct hypotheses?

   [1 mark]

   • A$\mathrm{H}_0:\rho=0$ and $\mathrm{H}_1:\rho\neq0$
   • B$\mathrm{H}_0:r=0$ and $\mathrm{H}_1:r\neq0$
   • C$\mathrm{H}_0:\rho=0$ and $\mathrm{H}_1:\rho<0$
   • D$\mathrm{H}_0:\rho\neq0$ and $\mathrm{H}_1:\rho=0$

   (b)

   Which is the correct conclusion?

   [1 mark]

   • ADo not reject $\mathrm{H}_0$, because $r$ is negative
   • BReject $\mathrm{H}_0$; $r$ is beyond $-0.5760$, so there is evidence of correlation between the age and resale price of cars of this model
   • CReject $\mathrm{H}_0$, because $-0.62$ is less than $0.05$
   • DDo not reject $\mathrm{H}_0$, because $-0.62$ lies between $-1$ and $1$

   (c)

   State the assumption about the population that is needed for this test to be valid, and describe how a scatter diagram of the data could be used to check it.

   [2 marks]

   Total for question 5: 4 marks

6. 6
   A museum ranks ten exhibits by the number of visitors and by the mean visitor rating. It calculates Spearman's rank correlation coefficient $r_s=0.52$ and tests, at the $5\%$ significance level, for positive correlation between the two sets of ranks. For $n=10$ the critical value for a one-tailed test at the $5\%$ level is $0.5636$.

   (a)

   Which is the correct conclusion?

   [1 mark]

   • AReject $\mathrm{H}_0$, because $0.52$ is positive
   • BReject $\mathrm{H}_0$, because $0.52$ is greater than $0.05$
   • CDo not reject $\mathrm{H}_0$; there is insufficient evidence of positive correlation between the ranks
   • DAccept $\mathrm{H}_0$, which proves that there is no correlation

   (b)

   A second museum ranks $12$ exhibits and also finds $r_s=0.52$. For $n=12$ the one-tailed critical value at the $5\%$ level is $0.5035$. Which statement is correct?

   [1 mark]

   • ADo not reject $\mathrm{H}_0$, because the value of $r_s$ is the same as for the first museum
   • BDo not reject $\mathrm{H}_0$, because the critical value is larger for a larger sample
   • CReject $\mathrm{H}_0$, because the significance level is higher for a larger sample
   • DReject $\mathrm{H}_0$; the same value of $r_s$ is now significant because the critical value is smaller for a larger sample

   (c)

   Write down the hypotheses for the first museum's test, in context.

   [2 marks]

   Total for question 6: 4 marks

7. 7
   A forester measures the trunk diameter $x$ cm and the height $y$ m of $15$ randomly chosen trees of one species. The data are summarised by $S_{xx}=126.4$, $S_{yy}=85.6$ and $S_{xy}=47.8$.

   (a)

   Calculate the product moment correlation coefficient $r$.

   [3 marks]

   (b)

   Test, at the $5\%$ significance level, whether there is positive correlation between trunk diameter and height, given that for $n=15$ the one-tailed critical values are $0.4409$ at the $5\%$ level and $0.5923$ at the $1\%$ level. State whether your conclusion changes at the $1\%$ level.

   [4 marks]

   Total for question 7: 7 marks

8. 8
   A microbiologist measures the area, in mm$^2$, of a bacterial colony every $2$ hours. At $2$, $4$, $6$, $8$, $10$, $12$, $14$, $16$, $18$ and $20$ hours the areas are $0.8$, $1.1$, $1.7$, $2.6$, $4.0$, $6.1$, $9.0$, $13.2$, $19.5$ and $28.1$ respectively.

   (a)

   Find Spearman's rank correlation coefficient for these data and test, at the $5\%$ significance level, for positive rank correlation, given that for $n=10$ the one-tailed critical value is $0.5636$. Explain why this value of the coefficient does not mean that the points lie on a straight line.

   [6 marks]

   (b)

   The product moment correlation coefficient for the same data is $r=0.912$. Test, at the $5\%$ significance level, for positive correlation, given that for $n=10$ the one-tailed critical value is $0.5494$. Explain why $r$ is less than $1$ even though $r_s=1$, state which coefficient better describes the strength of this association and why, and state the value of $r_s$ if all the areas were doubled.

   [6 marks]

   Total for question 8: 12 marks

End of questions