5.08f Hypothesis test: Spearman rank

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.

2. A random sample of 8 students sat examinations in Geography and Statistics. The product moment correlation coefficient between their results was 0.572 and the Spearman rank correlation coefficient was 0.655 .

1. Test both of these values for positive correlation. Use a \(5 \%\) level of significance.
2. Comment on your results.

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).

2. The table below shows the number of students per member of staff and the student satisfaction scores for 7 universities.

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 there is evidence of a correlation between the number of students per member of staff and the student satisfaction score.

1. A journalist is investigating factors which influence people when they buy a new car. One possible factor is fuel efficiency. The journalist randomly selects 8 car models. Each model's annual sales and fuel efficiency, in km/litre, are shown in the table below.

1. Calculate Spearman's rank correlation coefficient for these data. The journalist believes that car models with higher fuel efficiency will achieve higher sales.
2. Stating your hypotheses clearly, test whether or not the data support the journalist's belief. Use a \(5 \%\) level of significance.
3. State the assumption necessary for a product moment correlation coefficient to be valid in this case.
4. The mean and median fuel efficiencies of the car models in the random sample are 14.5 km /litre and 15.65 km /litre respectively. Considering these statistics, as well as the distribution of the fuel efficiency data, state whether or not the data suggest that the assumption in part (c) might be true in this case. Give a reason for your answer. (No further calculations are required.)
   

8. The heights, in metres, and weights, in kilograms, of a random sample of 9 men are shown in the table below

1. Given that \(\mathrm { S } _ { x x } = 0.0632 , \mathrm {~S} _ { y y } = 1957.5556\) and \(\mathrm { S } _ { x y } = 9.3433\) calculate, to 3 decimal places, the product moment correlation coefficient between height and weight for these men.
2. Use your value of the product moment correlation coefficient to test whether or not there is evidence of a positive correlation between the height and weight of men. Use a \(5 \%\) significance level. State your hypotheses clearly. Peter does not know the heights or weights of the 9 men. He is given photographs of them and asked to put them in order of increasing weight. He puts them in the order $$A C E B G D I F H$$
3. Find, to 3 decimal places, Spearman's rank correlation coefficient between Peter's order and the actual order.
4. Use your value of Spearman's rank correlation coefficient to test for evidence of Peter's ability to correctly order men, by their weight, from their photographs. Use a 5\% significance level and state your hypotheses clearly.

A mobile library has 160 books for children on its records. The librarian believes that books with fewer pages are borrowed more often. He takes a random sample of 10 books for children.

1. Explain how the librarian should select this random sample.
   (2) The librarian ranked the 10 books according to how often they had been borrowed, with 1 for the book borrowed the most and 10 for the book borrowed the least. He also recorded the number of pages in each book. The results are in the table below.

   Book	\(A\)	\(B\)	\(C\)	\(D\)	\(E\)	\(F\)	\(G\)	\(H\)	\(I\)	\(J\)
   Borrowing rank	1	2	3	4	5	6	7	8	9	10
   Number of pages	50	212	115	80	301	90	356	283	152	317

2. Calculate Spearman's rank correlation coefficient for these data.
3. Test the librarian's belief using a \(5 \%\) level of significance. State your hypotheses clearly.
   

1. A junior judge is being trained by a senior judge to learn how to assess ice skaters. After the training, the judges each assess 6 ice skaters \(A , B , C , D , E\) and \(F\). They each list them in order of preference with the best ice skater first. The results 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 evidence of a positive correlation between the rankings of the junior judge and the senior judge. State your hypotheses clearly.
3. Comment on the effectiveness of the training delivered by the senior judge.

1. Phil measures the concentration of a radioactive element, \(c\), and the amount of dissolved solids, \(a\), of 8 random samples of groundwater. His results are shown in the table below.

Given that $$\mathrm { S } _ { c c } = 34787.5 \quad \mathrm {~S} _ { a a } = 0.2172875 \quad \mathrm {~S} _ { c a } = 47.7625$$

1. calculate, to 3 decimal places, the product moment correlation coefficient between the concentration of the radioactive element and the amount of dissolved solids for these groundwater samples.
2. Use your value of the product moment correlation coefficient to test whether or not there is evidence of a positive correlation between the concentration of this radioactive element and the amount of dissolved solids in groundwater. Use a \(5 \%\) significance level. State your hypotheses clearly.
3. Calculate, to 3 decimal places, Spearman's rank correlation coefficient between the concentration of the radioactive element and the amount of dissolved solids.
4. Use your value of Spearman's rank correlation coefficient to test for evidence of a positive correlation between the concentration of the radioactive element and the amount of dissolved solids. Use a \(5 \%\) significance level. State your hypotheses clearly.
5. Using your conclusions in part (b) and part (d), comment on the possible relationship between these variables.

5. A marathon runner believes that she is more likely to win a medal at her national championships the higher the temperature is on the day of the race. She records the temperature at the start of each of eight races against fields of a similar standard and her finishing position in each race. Her results are shown in the table below.

1. Calculate Spearman's rank correlation coefficient for these data.
2. Using a 5\% level of significance and stating your hypotheses clearly, interpret your result. Another runner suggests that she should use her time in each race instead of her finishing position and calculate the product moment correlation coefficient for the data.
3. Comment on this suggestion.