5.08e Spearman rank correlation

5 Five runners, \(A , B , C , D\) and \(E\), take part in two different races.
Spearman's rank correlation coefficient for the orders in which the runners finish is calculated and a test for positive agreement is carried out at the \(5 \%\) significance level.

1. State suitable hypotheses for the test.
2. Find the largest possible value of \(\sum d ^ { 2 }\) for which the result of the test is to reject the null hypothesis.
3. In the first race, the order in which the five runners finished was: \(A , B , C , D , E\). In the second race, three of the runners finished in the same positions as in the first race. The result of the test is to reject the null hypothesis. Find a possible order for the runners to finish in the second race.

5 The numbers of each of 9 items sold in two different supermarkets in a week are given in the following table. A researcher wants to test whether there is association between the numbers of these items sold in the two supermarkets. However, it is known that the collection of data in Supermarket \(B\) was done inaccurately and each of the numbers in the corresponding row of the table could have been in error by as much as 2 items greater or 2 items fewer.

1. Explain why Spearman's rank correlation coefficient might be preferred to the use of Pearson's product-moment correlation coefficient in this context.
2. Carry out the test at the \(5 \%\) significance level using Spearman's rank correlation coefficient.

2. A teacher believes that those of her students with strong mathematical ability may also have enhanced short-term memory. She shows a random sample of 11 students a tray of different objects for eight seconds and then asks them to write down as many of the objects as they can remember. The results, along with their percentage score in a recent mathematics test, are given in the table below.

1. Calculate Spearman's rank correlation coefficient for these data. Show your working clearly.
2. Stating your hypotheses clearly, carry out a suitable test to assess the teacher's belief. Use a \(5 \%\) level of significance and state your critical value. The teacher shows these results to her class and argues that spending more time trying to improve their short-term memory would improve their mathematical ability.
3. Explain whether or not you agree with the teacher's argument.

3. The table shows the time, in seconds, of the fastest qualifying lap for 10 different Formula One racing drivers and their finishing position in the actual race.

1. Calculate the value of Spearman's rank correlation coefficient for these data.
2. Stating your hypotheses clearly, test at the \(1 \%\) level of significance, whether or not there is evidence of a positive correlation between the fastest qualifying lap time and finishing position for these Formula One racing drivers.

2 The table shows the season's best times, \(x\) seconds, for the 8 athletes who took part in the 200 m final in the 2021 Tokyo Olympics. It also shows their finishing position in the race. Given that the fastest season's best time is ranked number 1

1. calculate the value of the Spearman's rank correlation coefficient for these data.
2. Stating your hypotheses clearly, test, at the \(1 \%\) level of significance, whether or not there is evidence of a positive correlation between the rank of the season's best time and the finishing position for these athletes. Chris suggests that it would be better to use the actual finishing time, \(y\) seconds, of these athletes rather than their finishing position. Given that $$S _ { x x } = 0.1286875 \quad S _ { y y } = 0.55275 \quad S _ { x y } = 0.225175$$
3. calculate the product moment correlation coefficient between the season's best time and the finishing time for these athletes.
   Give your answer correct to 3 decimal places.
4. Use your value of the product moment correlation coefficient to test, at the \(1 \%\) level of significance, whether or not there is evidence of a positive correlation between the season's best time and the finishing time for these athletes.

1. The table shows the annual tea consumption, \(t\) (kg/person), and population, \(p\) (millions), for a random sample of 7 European countries.

$$\text { (You may use } \mathrm { S } _ { t t } = 2.486 \quad \mathrm {~S} _ { p p } = 3026.234 \quad \mathrm {~S} _ { p t } = 83.634 \text { ) }$$ Angela suggests using the product moment correlation coefficient to calculate the correlation between annual tea consumption and population.

1. Use Angela's suggestion to test, at the \(5 \%\) level of significance, whether or not there is evidence of any correlation between annual tea consumption and population. State your hypotheses clearly and the critical value used. Johan suggests using Spearman's rank correlation coefficient to calculate the correlation between the rank of annual tea consumption and the rank of population.
2. Calculate Spearman's rank correlation coefficient between the rank of annual tea consumption and the rank of population.
3. Use Johan's suggestion to test, at the \(5 \%\) level of significance, whether or not there is evidence of a positive correlation between annual tea consumption and population.
   State your hypotheses clearly and the critical value used.

4. In a survey 10 randomly selected men had their systolic blood pressure, \(x\), and weight, \(w\), measured. Their results are as follows

1. Calculate the value of Spearman's rank correlation coefficient between \(x\) and \(w\).
2. Stating your hypotheses clearly, test at the \(5 \%\) level of significance, whether or not there is evidence of a positive correlation between systolic blood pressure and weight. The product moment correlation coefficient for these data is 0.5114
3. Use the value of the product moment correlation coefficient to test, at the \(5 \%\) level of significance, whether or not there is evidence of a positive correlation between systolic blood pressure and weight.
4. Using your conclusions to part (b) and part (c), describe the relationship between systolic blood pressure and weight.

1. The table below shows the distance travelled by car and the amount of commission earned by each of 8 salespersons in 2015

1. Find Spearman's rank correlation coefficient for these data.
2. Stating your hypotheses clearly, test, at the \(5 \%\) level of significance, whether or not there is evidence of a positive correlation between the distance travelled by car and the amount of commission earned.
   

1. The ages, in years, of a random sample of 8 parrots are shown in the table below.

A parrot breeder does not know the ages of these 8 parrots. She examines each of these 8 parrots and is asked to put them in order of decreasing age. She puts them in the order $$\begin{array} { l l l l l l l l } D & G & H & C & A & B & F & E \end{array}$$

1. Find, to 3 decimal places, Spearman's rank correlation coefficient between the breeder's order and the actual order.
   (5)
2. Use your value of Spearman's rank correlation coefficient to test for evidence of the breeder's ability to order parrots correctly, by their age, after examining them. Use a \(1 \%\) significance level and state your hypotheses clearly.

1. A random sample of 9 footballers is chosen to participate in an obstacle course. The time taken, \(y\) seconds, for each footballer to complete the obstacle course is recorded, together with the footballer's Body Mass Index, \(x\). The results are shown in the table below.

Russell claims, that for footballers, as Body Mass Index increases the time taken to complete the obstacle course tends to decrease.

1. Find, to 3 decimal places, Spearman's rank correlation coefficient between \(x\) and \(y\).
2. Use your value of Spearman's rank correlation coefficient to test Russell's claim. Use a 5\% significance level and state your hypotheses clearly. The product moment correlation coefficient for these data is - 0.5594
3. Use the value of the product moment correlation coefficient to test for evidence of a negative correlation between Body Mass Index and the time taken to complete the obstacle course. Use a 5\% significance level.
4. Using your conclusions to part (b) and part (c), describe the relationship between Body Mass Index and the time taken to complete the obstacle course.

1. A plant biologist claims that as the percentage moisture content of the soil in a field increases, so does the percentage plant coverage. He splits the field into equal areas labelled \(A , B , C , D\) and \(E\) and measures the percentage plant coverage and the percentage moisture content for each area. The results are shown in the table below.

1. Calculate Spearman's rank correlation coefficient for these data.
2. Stating your hypotheses clearly, test at the \(5 \%\) level of significance, whether or not these data provide support for the plant biologist's claim.

1. The table below shows the number of televised tournaments won and the total number of tournaments won by the top 10 ranked darts players in 2020

Michael did not want to calculate Spearman's rank correlation coefficient between player's rank and the rank in televised tournaments won because there would be tied ranks.

1. Explain how Michael could have dealt with these tied ranks. Given that the largest number of total tournaments won is ranked number 1
2. calculate the value of Spearman's rank correlation coefficient between player's rank and the rank in total tournaments won.
3. Stating your hypotheses and critical value clearly, test at the \(5 \%\) level of significance, whether or not there is evidence of a positive correlation between player's rank and the rank in total tournaments won for these darts players. Michael does not believe that there is a positive correlation between player's rank and the rank in total number of tournaments won.
4. Find the largest level of significance, that is given in the tables provided, which could be used to support Michael's claim.
   You must state your critical value.

1. Aarush is asked to estimate the price of 7 kettles and rank them in order of decreasing price.

Aarush's order of decreasing price is \(D A F C B G E\) The actual prices of the 7 kettles are shown in the table below.

1. Calculate Spearman's rank correlation coefficient between Aarush's order and the actual order. Use a rank of 1 for the highest priced kettle.
   Show your working clearly.
2. Using a \(5 \%\) level of significance, test whether or not there is evidence to suggest that Aarush is able to rank kettles in order of decreasing price. You should state your hypotheses and critical value.
3. Explain why Aarush did not use the product moment correlation coefficient in this situation. Aarush discovered that kettle A's price was recorded incorrectly and should have been \(\pounds 49.99\) rather than \(\pounds 99.99\)
4. Explain what effect this has on the rankings for the price.

3. Each of 7 athletes competed in a 200 metre race and a 400 metre race. The table shows the time, in seconds, taken by each athlete to complete the 200 metre race. The finishing order in the 400 metre race is shown below, with athlete \(A\) finishing in the fastest time. \(\begin{array} { l l l l l l l } A & B & G & C & D & F & E \end{array}\)

1. Calculate the Spearman's rank correlation coefficient between the finishing order in the 200 metre race and the finishing order in the 400 metre race.
2. Stating your hypotheses clearly, test whether or not there is evidence of a positive correlation between the finishing order in the 200 metre race and the finishing order in the 400 metre race. Use a \(5 \%\) level of significance. The 7 athletes also competed in a long jump competition with the following results.

   Athlete	\(A\)	\(B\)	\(C\)	\(D\)	\(E\)	\(F\)	\(G\)
   Long jump (metres)	6.50	6.47	6.12	6.12	6.48	6.38	6.47

   Yuliya wants to calculate the Spearman's rank correlation coefficient between the finishing order in the 200 metre race and the finishing order in the long jump for these athletes.
3. Without carrying out any further calculations, explain how Yuliya should do this.

3. A cafe owner wishes to know whether the price of strawberry jam is related to the taste of the jam. He finds a website that lists the price per 100 grams and a mark for the taste, out of 100, awarded by a judge, for 9 different strawberry jams \(A , B , C , D , E , F , G , H\) and \(I\). He then ranks the marks for taste and the prices. The ranks are shown in the table below.

1. Calculate Spearman's rank correlation coefficient for these data.
2. Test, at the \(5 \%\) level of significance, whether or not there is a relationship between the price and the taste of these strawberry jams. State your hypotheses clearly. A friend suggests that it would be better to use the price per 100 grams, \(c\), and the mark for the taste, \(m\), for each strawberry jam rather than rank them. Given that $$\mathrm { S } _ { c c } = 2.0455 \quad \mathrm {~S} _ { m m } = 243.5556 \quad \mathrm {~S} _ { c m } = 16.4943$$
3. calculate the product moment correlation coefficient between the price and the mark for taste of these strawberry jams, giving your answer correct to 3 decimal places.
4. Use your value of the product moment correlation coefficient to test, at the \(5 \%\) level of significance, whether or not there is evidence of a positive correlation between the price and the mark for taste of these 9 strawberry jams. State your hypotheses clearly.
5. State which of the tests in parts (b) and (d) is more appropriate for the cafe owner to use. Give a reason for your answer.

2. Nine dancers, Adilzhan \(( A )\), Bianca \(( B )\), Chantelle \(( C )\), Lee \(( L )\), Nikki \(( N )\), Ranjit \(( R )\), Sergei \(( S )\), Thuy \(( T )\) and Yana \(( Y )\), perform in a dancing competition. Two judges rank each dancer according to how well they perform. The table below shows the rankings of each judge starting from the dancer with the strongest performance.

1. Calculate Spearman's rank correlation coefficient for these data.
2. Stating your hypotheses clearly, test at the 1\% level of significance, whether or not the two judges are generally in agreement.

   Rank	1	2	3	4	5	6	7	8	9
   Judge 1	\(S\)	\(N\)	\(B\)	\(C\)	\(T\)	\(A\)	\(Y\)	\(R\)	\(L\)
   Judge 2	\(S\)	\(T\)	\(N\)	\(B\)	\(C\)	\(Y\)	\(L\)	\(A\)	\(R\)

   1. Calculate Spearman's rank correlation coefficient for these data.

1. A researcher claims that, at a river bend, the water gradually gets deeper as the distance from the inner bank increases. He measures the distance from the inner bank, \(b \mathrm {~cm}\), and the depth of a river, \(s \mathrm {~cm}\), at seven positions. The results are shown in the table below.

1. Calculate Spearman's rank correlation coefficient between \(b\) and \(s\).
2. Stating your hypotheses clearly, test whether or not the data provides support for the researcher's claim. Use a \(1 \%\) level of significance.
   

7. The numbers of deaths from pneumoconiosis and lung cancer in a developing country are given in the table. The correlation between the number of deaths in the different age groups for each disease is to be investigated.

1. Give one reason why Spearman's rank correlation coefficient should be used.
2. Calculate Spearman's rank correlation coefficient for these data.
3. Use a suitable test, at the \(5 \%\) significance level, to interpret your result. State your hypotheses clearly.
   (5)

6. Two judges ranked 8 ice skaters in a competition according to the table below.

1. Evaluate Spearman's rank correlation coefficient between the ranks of the two judges.
2. Use a suitable test, at the \(5 \%\) level of significance, to interpret this result.

1. During a village show, two judges, \(P\) and \(Q\), had to award a mark out of 30 to some flower displays. The marks they awarded to a random sample of 8 displays were as follows:

1. Calculate Spearman's rank correlation coefficient for the marks awarded by the two judges. After the show, one competitor complained about the judges. She claimed that there was no positive correlation between their marks.
2. Stating your hypotheses clearly, test whether or not this sample provides support for the competitor's claim. Use a \(5 \%\) level of significance.

The product moment correlation coefficient is denoted by \(r\) and Spearman's rank correlation coefficient is denoted by \(r _ { s }\).

1. Sketch separate scatter diagrams, with five points on each diagram, to show
   1. \(r = 1\),
   2. \(r _ { s } = - 1\) but \(r > - 1\). Two judges rank seven collie dogs in a competition. The collie dogs are labelled \(A\) to \(G\) and the rankings are as follows

      Rank	1	2	3	4	5	6	7
      Judge 1	\(A\)	\(C\)	\(D\)	\(B\)	\(E\)	\(F\)	\(G\)
      Judge 2	\(A\)	\(B\)	\(D\)	\(C\)	\(E\)	\(G\)	\(F\)

   (b)
   1. Calculate Spearman's rank correlation coefficient for these data.
   2. Stating your hypotheses clearly, test, at the \(5 \%\) level of significance, whether or not the judges are generally in agreement.

1. A researcher claims that, at a river bend, the water gradually gets deeper as the distance from the inner bank increases. He measures the distance from the inner bank, \(b \mathrm {~cm}\), and the depth of a river, \(s \mathrm {~cm}\), at seven positions. The results are shown in the table below.

1. Calculate Spearman's rank correlation coefficient between \(b\) and \(s\).
2. Stating your hypotheses clearly, test whether or not the data provides support for the researcher's claim. Use a \(1 \%\) level of significance.

1. Interviews for a job are carried out by two managers. Candidates are given a score by each manager and the results for a random sample of 8 candidates are shown in the table below.

1. Calculate Spearman's rank correlation coefficient for these data.
2. Test, at the \(5 \%\) level of significance, whether there is agreement between the rankings awarded by each manager. State your hypotheses clearly. Manager \(Y\) later discovered he had miscopied his score for candidate \(D\) and it should be 54 .
3. Without carrying out any further calculations, explain how you would calculate Spearman's rank correlation in this case.
   

3. The table below shows the population and the number of council employees for different towns and villages.

1. Find, to 3 decimal places, Spearman's rank correlation coefficient between the population and the number of council employees.
2. Use your value of Spearman's rank correlation coefficient to test for evidence of a positive correlation between the population and the number of council employees. Use a \(2.5 \%\) significance level. State your hypotheses clearly. It is suggested that a product moment correlation coefficient would be a more suitable calculation in this case. The product moment correlation coefficient for these data is 0.627 to 3 decimal places.
3. Use the value of the product moment correlation coefficient to test for evidence of a positive correlation between the population and the number of council employees. Use a \(2.5 \%\) significance level.
4. Interpret and comment on your results from part(b) and part(c).