Further Statistics 2: CorrelationEdexcel A-Level Further Maths: Topic test

1. 1
   The product moment correlation coefficient between $x$ and $y$ for a set of paired data is $r=0.72$. New variables are defined by $u=5x-3$ and $v=10-2y$.

   (a)

   What is the product moment correlation coefficient between $u$ and $v$?

   [1 mark]

   • A$0.72$
   • B$-0.72$
   • C$-1.44$
   • D$0.28$

   (b)

   A test for zero correlation using $r$ is to be carried out on a random sample of pairs $(x,y)$. Which condition is needed for the critical values of $r$ to be valid?

   [1 mark]

   • ABoth $x$ and $y$ take whole-number values only
   • B$x$ and $y$ have the same mean
   • CThe pairs $(x,y)$ come from a bivariate Normal distribution
   • DThe data are ranked with no ties

   (c)

   Spearman's rank correlation coefficient between $x$ and $y$ is $0.65$. Write down its value between $u$ and $v$, giving a reason.

   [2 marks]

   Total for question 1: 4 marks

2. 2
   A wine critic ranks six wines, $P$ to $U$, from $1$ (best) to $6$ (worst) for taste, and a shop ranks the same wines from $1$ (cheapest) to $6$ (most expensive). The critic's ranks for $P$ to $U$ are $1,2,3,4,5,6$ and the price ranks are $2,1,4,3,6,5$.

   (a)

   What is the value of $\sum d^2$, where $d$ is the difference between the two ranks of a wine?

   [1 mark]

   • A$0$
   • B$12$
   • C$36$
   • D$6$

   (b)

   What is Spearman's rank correlation coefficient for these data, to $3$ significant figures?

   [1 mark]

   • A$0.829$
   • B$0.171$
   • C$0.833$
   • D$0.971$

   (c)

   Give two reasons why Spearman's rank correlation coefficient is used here in preference to the product moment correlation coefficient.

   [2 marks]

   Total for question 2: 4 marks

3. 3
   A cyclist records, on $10$ randomly chosen rides, the mean gradient $g$ of the route and the mean speed $s$ of the ride. The data are summarised by $S_{gg}=84.5$, $S_{ss}=210.6$ and $S_{gs}=-112.4$. Assume the data come from a bivariate Normal distribution.

   (a)

   Calculate the product moment correlation coefficient between $g$ and $s$.

   [3 marks]

   (b)

   Test, at the $5\%$ level of significance, whether there is evidence of negative correlation between the gradient and the speed. The critical value for $n=10$ at the $5\%$ level for a one-tailed test is $0.5494$. State your hypotheses and your conclusion in context.

   [4 marks]

   Total for question 3: 7 marks

4. 4
   Eight club sprinters each run $100$ m and $400$ m. Their times in seconds, in the order of runners $1$ to $8$, are: $100$ m: $11.2,\ 11.5,\ 11.9,\ 11.5,\ 12.3,\ 12.0,\ 11.8,\ 12.6$; $400$ m: $52.1,\ 54.0,\ 53.4,\ 56.2,\ 55.8,\ 57.5,\ 53.4,\ 58.9$. Rank $1$ is given to the shortest time.

   (a)

   Calculate Spearman's rank correlation coefficient for these data, dealing with the tied values in the usual way.

   [6 marks]

   (b)

   Test, at the $5\%$ significance level, whether there is any association between the $100$ m and $400$ m times. The critical value for a two-tailed test with $n=8$ at the $5\%$ level is $0.7381$. State your hypotheses and your conclusion in context, and comment on how the tied values affect your calculation.

   [6 marks]

   Total for question 4: 12 marks

5. 5
   A marine biologist records the water temperature and the number of jellyfish at each of $20$ randomly chosen sites. She assumes that the pairs come from a bivariate Normal distribution. The product moment correlation coefficient is $r=0.41$. She tests for positive correlation at the $5\%$ level, for which the critical value is $0.3783$.

   (a)

   Which pair of hypotheses should she use?

   [1 mark]

   • A$\mathrm{H}_0:\rho=0$, $\mathrm{H}_1:\rho>0$
   • B$\mathrm{H}_0:\rho=0.41$, $\mathrm{H}_1:\rho>0.41$
   • C$\mathrm{H}_0:r=0$, $\mathrm{H}_1:r>0$
   • D$\mathrm{H}_0:\rho>0$, $\mathrm{H}_1:\rho=0$

   (b)

   What is the correct conclusion at the $5\%$ level?

   [1 mark]

   • ADo not reject $\mathrm{H}_0$: there is no correlation
   • BReject $\mathrm{H}_0$: there is evidence of positive correlation
   • CReject $\mathrm{H}_0$: there is proof that temperature causes the jellyfish numbers
   • DDo not reject $\mathrm{H}_0$: the value $0.41$ is too small to be a correlation

   (c)

   She repeats the test at the $1\%$ level, for which the critical value is $0.5155$. State her new conclusion and explain why it differs from the conclusion at the $5\%$ level.

   [2 marks]

   Total for question 5: 4 marks

6. 6
   Twelve flats are ranked from $1$ (largest) to $12$ (smallest) by monthly rent and, separately, by distance from the city centre. Spearman's rank correlation coefficient between the two sets of ranks is $r_s=-0.62$.

   (a)

   Which statement best describes the result?

   [1 mark]

   • AFlats with higher rents tend to be further from the city centre
   • BThere is no relationship between rent and distance
   • CFlats with higher rents tend to be nearer the city centre
   • DThe distance from the centre causes the rent to fall

   (b)

   For which relationship is Spearman's rank correlation coefficient more suitable than the product moment correlation coefficient?

   [1 mark]

   • AOne that is exactly linear with Normal errors
   • BOne in which $y$ rises and then falls
   • COne in which every $x$ value is identical
   • DOne that is always decreasing but not a straight line

   (c)

   Rents fall quickly with distance close to the centre and then level off further out. Explain why $r_s$ may be closer to $-1$ than the product moment correlation coefficient for the same flats.

   [2 marks]

   Total for question 6: 4 marks

7. 7
   A zoologist measures the shoulder height $x$ cm and the mass $y$ kg of $8$ randomly chosen adult dogs of one breed, assuming a bivariate Normal distribution. The data are coded as $u=x-40$ and $v=\frac{y-5}{2}$, giving $\sum u=34$, $\sum v=28$, $\sum u^2=204$, $\sum v^2=140$ and $\sum uv=167$.

   (a)

   Calculate the product moment correlation coefficient between $u$ and $v$.

   [3 marks]

   (b)

   State the product moment correlation coefficient between $x$ and $y$, giving a reason. Hence test, at the $5\%$ level, $\mathrm{H}_0:\rho=0$ against $\mathrm{H}_1:\rho\neq0$, given that the two-tailed critical value for $n=8$ is $0.7067$.

   [4 marks]

   Total for question 7: 7 marks

8. 8
   An ecologist ranks $12$ mountain sites by altitude and by mean wind speed, each from lowest ($1$) to highest ($12$). The differences $d$ between the two ranks for the sites are $1,-2,0,3,-1,2,-3,1,0,-1,2,-2$. Wind speed rises with altitude quickly at first and then more slowly.

   (a)

   Calculate Spearman's rank correlation coefficient. Test, at the $5\%$ level, whether there is a positive association between altitude and mean wind speed, given that the critical value for a one-tailed test with $n=12$ is $0.5035$.

   [6 marks]

   (b)

   A colleague proposes using the product moment correlation coefficient on the measured values instead. Evaluate this proposal, referring to the shape of the relationship, the units used for altitude and the conditions for the test.

   [6 marks]

   Total for question 8: 12 marks

End of questions