Interpret correlation coefficient value

7

1. Three airport management trainees, Ryan, Sunil and Tim, were each instructed to select a random sample of 12 suitcases from those waiting to be loaded onto aircraft. Each trainee also had to measure the volume, \(x\), and the weight, \(y\), of each of the 12 suitcases in his sample, and then calculate the value of the product moment correlation coefficient, \(r\), between \(x\) and \(y\).
   • Ryan obtained a value of - 0.843 .
   • Sunil obtained a value of + 0.007 .
   Explain why neither of these two values is likely to be correct.
2. Peggy, a supervisor with many years' experience, measured the volume, \(x\) cubic feet, and the weight, \(y\) pounds, of each suitcase in a random sample of 6 suitcases, and then obtained a value of 0.612 for \(r\).
   • Ryan and Sunil each claimed that Peggy's value was different from their values because she had measured the volumes in cubic feet and the weights in pounds, whereas they had measured the volumes in cubic metres and the weights in kilograms.
   • Tim claimed that Peggy's value was almost exactly half his calculated value because she had used a sample of size 6 whereas he had used one of size 12 .
   Explain why neither of these two claims is valid.
3. Quentin, a manager, recorded the volumes, \(v\), and the weights, \(w\), of a random sample of 8 suitcases as follows.

   \(\boldsymbol { v }\)	28.1	19.7	46.4	23.6	31.1	17.5	35.8	13.8
   \(\boldsymbol { w }\)	14.9	12.1	21.1	18.0	19.8	19.2	16.2	14.7

   1. Calculate the value of \(r\) between \(v\) and \(w\).
   2. Interpret your value in the context of this question.

7. In a school there are 148 students in Years 12 and 13 studying Science, Humanities or Arts subjects. Of these students, 89 wear glasses and the others do not. There are 30 Science students of whom 18 wear glasses. The corresponding figures for the Humanities students are 68 and 44 respectively. A student is chosen at random. Find the probability that this student

1. is studying Arts subjects,
2. does not wear glasses, given that the student is studying Arts subjects. Amongst the Science students, \(80 \%\) are right-handed. Corresponding percentages for Humanities and Arts students are 75\% and 70\% respectively. A student is again chosen at random.
3. Find the probability that this student is right-handed.
4. Given that this student is right-handed, find the probability that the student is studying Science subjects.
   1. (a) Describe the main features and uses of a box plot.
      
   Children from schools \(A\) and \(B\) took part in a fun run for charity. The times, to the nearest minute, taken by the children from school \(A\) are summarised in Figure 1. \begin{figure}[h]
   \captionsetup{labelformat=empty} \caption{Figure 1} \includegraphics[alt={},max width=\textwidth]{3d4f7bfb-b235-418a-9411-a4d0b3188254-015_398_1045_946_461}
   \end{figure}
5. 1. Write down the time by which \(75 \%\) of the children in school \(A\) had completed the run.
   2. State the name given to this value.
6. Explain what you understand by the two crosses ( X ) on Figure 1.

8. The lifetimes of bulbs used in a lamp are normally distributed. A company \(X\) sells bulbs with a mean lifetime of 850 hours and a standard deviation of 50 hours.

1. Find the probability of a bulb, from company \(X\), having a lifetime of less than 830 hours.
2. In a box of 500 bulbs, from company \(X\), find the expected number having a lifetime of less than 830 hours. A rival company \(Y\) sells bulbs with a mean lifetime of 860 hours and \(20 \%\) of these bulbs have a lifetime of less than 818 hours.
3. Find the standard deviation of the lifetimes of bulbs from company \(Y\). Both companies sell the bulbs for the same price.
4. State which company you would recommend. Give reasons for your answer.
   \begin{table}[h]
   \captionsetup{labelformat=empty} \caption{}
   \end{table}

2 In the manufacture of fibre optical cable (FOC), flaws occur randomly. Whether any point on a cable is flawed is independent of whether any other point is flawed. The number of flaws in 100 m of FOC of standard diameter is denoted by \(X\).

1. State a further assumption needed for \(X\) to be well modelled by a Poisson distribution. Assume now that \(X\) can be well modelled by the distribution \(\operatorname { Po } ( 0.7 )\).
2. Find the probability that in 300 m of FOC of standard diameter there are exactly 3 flaws. The number of flaws in 100 m of FOC of a larger diameter has the distribution \(\mathrm { Po } ( 1.6 )\).
3. Find the probability that in 200 m of FOC of standard diameter and 100 m of FOC of the larger diameter the total number of flaws is at least 4 . Judith believes that mathematical ability and chess-playing ability are related. She asks 20 randomly chosen chess players, with known British Chess Federation (BCF) ratings \(X\), to take a mathematics aptitude test, with scores \(Y\). The results are summarised as follows. $$n = 20 , \Sigma x = 3600 , \Sigma x ^ { 2 } = 660500 , \Sigma y = 1440 , \Sigma y ^ { 2 } = 105280 , \Sigma x y = 260990$$
   1. Calculate the value of Pearson's product-moment correlation coefficient \(r\).
   2. State an assumption needed to be able to carry out a significance test on the value of \(r\).
   3. Assume now that the assumption in part (b) is valid. Test at the \(5 \%\) significance level whether there is evidence that chess players with higher BCF ratings are better at mathematics.
   4. There are two different grading systems for chess players, the BCF system and the international ELO system. The two sets of ratings are related by $$\text { ELO rating } = 8 \times \text { BCF rating } + 650$$ Magnus says that the experiment should have used ELO ratings instead of BCF ratings. Comment on Magnus's suggestion.
   5. Calculate the value of Pearson's product-moment correlation coefficient \(r\).
   6. State an assumption needed to be able to carry out a significance test on the value of \(r\).
   7. Assume now that the assumption in part (b) is valid. Test at the \(5 \%\) significance level whether there is evidence that chess players with higher BCF ratings are better at mathematics.
   8. There are two different grading systems for chess players, the BCF system and the international ELO system. The two sets of ratings are related by $$\mathrm { ELO } \text { rating } = 8 \times \mathrm { BCF } \text { rating } + 650 .$$ Magnus says that the experiment should have used ELO ratings instead of BCF ratings. Comment on Magnus's suggestion. An environmentalist measures the mean concentration, \(c\) milligrams per litre, of a particular chemical in a group of rivers, and the mean mass, \(m\) pounds, of fish of a certain species found in those rivers. The results are given in the table.

      Question			Answer	Marks	AO	Guidance
      1	(a)		\(\begin{aligned}	0.25 + 0.36 + x + x ^ { 2 } = 1
      	x ^ { 2 } + x - 0.39 = 0
      	x = 0.3 \text { (or } - 1.3 \text { ) }
      	x \text { cannot be negative }
      	\mathrm { E } ( W ) = 2.23
      	\mathrm { E } \left( W ^ { 2 } \right) = \Sigma w ^ { 2 } \mathrm { p } ( w ) \quad [ = 5.83 ]
      	\text { Subtract } [ \mathrm { E } ( W ) ] ^ { 2 } \text { to get } \mathbf { 0 . 8 5 7 1 } \end{aligned}\)	\(\begin{gathered} \text { M1 }
      \text { A1 }
      \text { A1 }
      \text { B1ft }
      \text { B1 }
      \text { M1 }
      \text { A1 }
      { [ 7 ] } \end{gathered}\)	3.1a 1.1b 1.1b 2.3 1.1b 1.1 2.1	Equation using \(\Sigma p = 1\) Correct simplified quadratic Correctly obtain \(x = 0.3\) Explicitly reject other solution 2.23 or exact equivalent only Use \(\Sigma w ^ { 2 } \mathrm { p } ( w )\) Correctly obtain given answer, www	Can be implied Method needed ft on their quadratic Allow for \(\mathrm { E } ( W ) ^ { 2 } = 4.9729\) Need 2.23 or 4.9729 and 5.83 or full numerical \(\Sigma w ^ { 2 } \mathrm { p } ( w )\)
      1	(b)		\(9 \times 0.8571 = 7.7139\)	B1 [1]	1.1b	Allow 7.71 or 7.714
      2	(a)		Flaws must occur at constant average rate (uniform rate)	B1 [1]	1.2	Context (e.g. "flaws") needed Extra answers, e.g. "singly": B0	Not "constant rate" or "average constant rate".
      2	(b)		\(\operatorname { Po(2.1)~or~ } e ^ { - \lambda } \frac { \lambda ^ { 3 } } { 3 ! }\)	M1 A1 [2]	1.1 1.1b	Po(2.1) stated or implied, or formula with \(\lambda = 2.1\) stated Awrt 0.189
      2	(c)		Po(3) \(1 - \mathrm { P } ( \leq 3 )\)	M1 M1 A1 [3]	1.1 1.1 1.1b	\(\operatorname { Po } ( 2 \times 0.7 + 1.6 )\) stated or implied Allow \(1 - \mathrm { P } ( \leq 4 ) = 0.1847\), or from wrong \(\lambda\) Awrt 0.353	Or all combinations \(\leq 3\) \(1 -\) above, not just \(= 3\)

      Question			Answer	Marks	AO	Guidance
      3	(a)		0.4(00)	B2 [2]	1.1 1.1b	SC: if B0, give SC B1 for two of \(S _ { x x } = 12500 , S _ { y y } = 1600 , S _ { x y } = 1790\) and \(S _ { x y } / \sqrt { } \left( S _ { x x } S _ { y y } \right)\)	Also allow SC B1 for equivalent methods using Covariance \	SDs
      3	(b)		Data needs to have a bivariate normal distribution	B1 [1]	1.2	Needs "bivariate normal" or clear equivalent. Not just "both normally distributed"	Allow "scatter diagram forms ellipse"
      3	(c)		\(\mathrm { H } _ { 0 }\) : higher maths scores are not associated with higher BCF grading; \(\mathrm { H } _ { 1 }\) : positively associated CV 0.3783 \(0.400 > 0.3783\) so reject \(\mathrm { H } _ { 0 }\) Significant evidence that higher maths scores are associated with higher BCF grading	B1 B1 M1ft A1ft [4]	2.5 1.1b 2.2b 3.5a	Needs context and clearly onetailed \(O R \rho\) used and defined Not "evidence that ..." Allow 0.378 Reject/do not reject \(\mathrm { H } _ { 0 }\) Contextualised, not too definite Needn't say "positive" if \(\mathrm { H } _ { 1 } \mathrm { OK }\) SC 2-tail: B0; 0.4438, or 0.3783 B1; then M1A0	\(\mathrm { H } _ { 0 } : \rho = 0 , \mathrm { H } _ { 1 } : \rho > 0\) where \(\rho\) is population pmcc (not \(r\) ) FT on their \(r\), but not CV Not "scores are associated ...". FT on their \(r\) only
      3	(d)		It makes no difference as this is a linear transformation	B1 [1]	2.2a	Need both "unchanged" oe and reason, need "linear" or exact equivalent	"oe" includes "their 0.4"
      4	(a)		Neither	B1 [1]	2.5	OE	Not "neither is independent of the other"
      4	(b)		\(c = 2.848 - 0.1567 m\)	B1 B1 B1 [3]	1.1 1.1 1.1	Correct \(a\), awrt 2.85 Correct \(b\), awrt 0.157 Letters correct from correct method (If both wrongly rounded, e.g. \(c = 2.84 - 0.156 m\), give B2)	\(\mathrm { SC } : m\) on \(c\) : \(m = 15.65 - 4.832 c\) : B2 \(y = 15.65 - 4.832 x\) : B1 \(c = 15.65 - 4.832 m : \mathrm { B } 1\) If B0B0, give B1 for correct letters from valid working

      Question		Answer	Marks	AO	Guidance
      4	(c)	\(a\) unchanged, \(b\) multiplied by 2.2 (allow " \(a\) unchanged, \(b\) increases", etc)	B1 [1]	2.2a	oe, e.g. \(c = 2.848 - 0.345 m\); \(m = 7.114 - 2.196 c\)	SC: \(m\) on \(c\) in (b): Both divided by 2.2 B1
      4	(d)	Draw approximate line of best fit Draw at least one vertical from line to point Say that "Best fit" line minimises the sum of squares of these distances	M1 M1 A1 [3]	1.1 2.4 2.4	Needs M2 and "minimises" and "sums of squares" oe	SC: Horizontal(s): full marks (indept of (b))

On a particular day the height above sea level, \(x\) metres, and the mid-day temperature, \(y\)°C, were recorded in 8 north European towns. These data are summarised below \(S_{xx} = 3\,535\,237.5 \quad \sum y = 181 \quad \sum y^2 = 4305 \quad S_{yy} = -23\,726.25\)

1. Find \(S_{yy}\). [2]
2. Calculate, to 3 significant figures, the product moment correlation coefficient for these data. [2]
3. Give an interpretation of your coefficient. [1]

A student thought that the calculations would be simpler if the height above sea level, \(h\), was measured in kilometres and used the variable \(h = \frac{x}{1000}\) instead of \(x\).

1. Write down the value of \(S_{hh}\) [1]
2. Write down the value of the correlation coefficient between \(h\) and \(y\). [1]