5.08e Spearman rank correlation

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]

On a Welsh television game show, contestants are asked to guess the weights of a random sample of seven cows. The game show judges want to investigate whether there is positive correlation between the actual weights and the estimated weights. The results are shown below for one contestant.

Cow	A	B	C	D	E	F	G
Actual weight, kg	614	1105	718	1001	889	770	682
Estimated weight, kg	700	1500	850	1400	750	900	800

1. Calculate Spearman's rank correlation coefficient for this data set. [5]
2. Stating your hypotheses clearly, determine whether or not there is evidence at the 5% significance level of a positive association between the actual weights and the weights as estimated by this contestant. [3]
3. One of the game show judges says, "This contestant was good at guessing the weights of the cows." Comment on this statement. [1]

A class of 8 students sit examinations in History and Geography. The marks obtained by these students are given below.

Student	A	B	C	D	E	F	G	H
History mark	73	59	83	49	57	82	67	60
Geography mark	55	51	58	59	44	66	49	67

1. Calculate Spearman's rank correlation coefficient for this data set. [6]
2. Hence determine whether or not, at the 5% significance level, there is evidence of a positive association between marks in History and marks in Geography. [2]
3. Explain why it might not have been appropriate to use Pearson's product moment correlation coefficient to test association using this data set. [1]

Arlosh, Sarah and Desi are investigating the ratings given to six different films by two critics.

1. Arlosh calculates Spearman's rank correlation coefficient \(r_s\) for the critics' ratings. He calculates that \(\Sigma d^2 = 72\). Show that this value must be incorrect. [2]
2. Arlosh checks his working with Sarah, whose answer \(r_s = \frac{39}{35}\) is correct. Find the correct value of \(\Sigma d^2\). [2]
3. Carry out an appropriate two-tailed significance test of the value of \(r_s\) at the 5% significance level, stating your hypotheses clearly. [4]

Nine athletes, \(A\), \(B\), \(C\), \(D\), \(E\), \(F\), \(G\), \(H\) and \(I\), competed in both the 100m sprint and the long jump. After the two events the positions of each athlete were recorded and Spearman's rank correlation coefficient was calculated and found to be 0.85 The piece of paper the positions were recorded on was mislaid. Although some of the athletes agreed their positions, there was some disagreement between athletes \(B\), \(C\) and \(D\) over their long jump results. The table shows the results that are agreed to be correct. Given that there were no tied ranks,

1. find the correct positions of athletes \(B\), \(C\) and \(D\) in the long jump. You must show your working clearly and give reasons for your answers. [5]
2. Without recalculating the coefficient, explain how Spearman's rank correlation coefficient would change if athlete \(H\) was disqualified from both the 100m sprint and the long jump. [2]

Two music critics, \(P\) and \(Q\), give scores to seven concerts as follows.

1. Calculate Spearman's rank correlation coefficient, \(r_s\), for these scores. [4]
2. Without carrying out a hypothesis test, state what your answer tells you about the views of the two critics. [1]