Effect of data changes

6 The table shows the total distance travelled, in thousands of miles, and the amount of commission earned, in thousands of pounds, by each of seven sales agents in 2005.

1. (a) Calculate Spearman's rank correlation coefficient, \(r _ { s }\), for these data.
   (b) Comment briefly on your value of \(r _ { s }\) with reference to this context.
   (c) After these data were collected, agent \(A\) found that he had made a mistake. He had actually travelled 19000 miles in 2005. State, with a reason, but without further calculation, whether the value of Spearman's rank correlation coefficient will increase, decrease or stay the same. The agents were asked to indicate their level of job satisfaction during 2005. A score of 0 represented no job satisfaction, and a score of 10 represented high job satisfaction. Their scores, \(y\), together with the data for distance travelled, \(x\), are illustrated in the scatter diagram below.
   
   [diagram]
2. For this scatter diagram, what can you say about the value of
   (a) Spearman's rank correlation coefficient,
   (b) the product moment correlation coefficient?

4

1. The table gives the heights and masses of 5 people.

   Person	\(A\)	\(B\)	\(C\)	\(D\)	\(E\)
   Height (m)	1.72	1.63	1.77	1.68	1.74
   Mass (kg)	75	62	64	60	70

   Calculate Spearman's rank correlation coefficient.
2. In an art competition the value of Spearman's rank correlation coefficient, \(r _ { s }\), calculated from two judges' rankings was 0.75 . A late entry for the competition was received and both judges ranked this entry lower than all the others. By considering the formula for \(r _ { s }\), explain whether the new value of \(r _ { s }\) will be less than 0.75 , equal to 0.75 , or greater than 0.75 .

Each contestant in a talent competition is given a score out of \(20\) by a judge. The organisers suspect that the judge's scores are associated with the age of the contestant. Table \(5.1\) and the scatter diagram in Fig. \(5.2\) show the scores and ages of a random sample of \(7\) contestants.

Contestant	A	B	C	D	E	F	G
Age	66	51	39	29	9	22	14
Score	12	11	15	17	16	18	9

Table 5.1 \includegraphics{figure_1} Fig. 5.2 Contestant G did not finish her performance, so it is decided to remove her data.

1. Spearman's rank correlation coefficient between age and score, including all \(7\) contestants, is \(-0.25\). Explain why Spearman's rank correlation coefficient becomes more negative when the data for contestant G is removed. [1]
2. Calculate Spearman's rank correlation coefficient for the \(6\) remaining contestants. [3]
3. Using this value of Spearman's rank correlation coefficient, carry out a hypothesis test at the \(5\%\) level to investigate whether there is any association between age and score. [5]
4. Briefly explain why it may be inappropriate to carry out a hypothesis test based on Pearson's product moment correlation coefficient using these data. [1]