Hypothesis test for positive correlation

1. The table below shows the heights cleared, in metres, for each of 6 competitors in a high jump competition.

These 6 competitors also took part in a long jump competition and finished in the following order, with C jumping the furthest.
C
A
F
D
B
E

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 a positive correlation between results in the high jump and results in the long jump. The product moment correlation coefficient between the height of the high jump and the length of the long jump for each competitor is found to be 0.678
3. Use this value to test, at the \(5 \%\) level of significance, for evidence of positive correlation between results in the high jump and results in the long jump.
4. State the condition required for the test in part (c) to be valid.
5. Explain what your conclusions in part (b) and part (c) suggest about the relationship between results in the high jump and results in the long jump.

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1. Bara is investigating whether or not the two judges of a skating competition are in agreement. The two judges gave a score to each of the 8 skaters in the competition as shown in the table below.

Bara decided to calculate Spearman's rank correlation coefficient for these data.

1. Calculate Spearman's rank correlation coefficient between the ranks of the two judges.
2. Test, at the \(1 \%\) level of significance, whether or not the two judges are in agreement. Judge 1 accidentally swapped the scores for skaters \(D\) and \(E\). The score for skater \(D\) should be 63 and the score for skater \(E\) should be 62
3. Without carrying out any further calculations, explain how Spearman's rank correlation coefficient will change. Give a reason for your answer.

1. Abena and Meghan are both given the same list of 10 films.

Each of them ranks the 10 films from most favourite to least favourite.
For the differences, \(d\), between their ranks for these 10 films, \(\sum d ^ { 2 } = 84\)

1. Calculate Spearman's rank correlation coefficient between Abena's ranks and Meghan's ranks. A test is carried out at the 5\% level of significance to see if there is agreement between their ranks for the films. The hypotheses for the test are $$\mathrm { H } _ { 0 } : \rho _ { \mathrm { S } } = 0 \quad \mathrm { H } _ { 1 } : \rho _ { \mathrm { S } } > 0$$
2. 1. Find the critical region for the test.
   2. State the conclusion of the test. An 11th film is added to the list. Abena and Meghan both agree that this film is their least favourite. A new test is carried out at the \(5 \%\) level of significance using the same hypotheses.
3. Determine the conclusion of this test. You should state the test statistic and the critical value used.

1. Every applicant for a job at Donala is given three different tasks, \(P , Q\) and \(R\).

For each task the applicant is awarded a score.
The scores awarded to 9 of the applicants, for the tasks \(P\) and \(Q\), are given below.

1. Calculate Spearman's rank correlation coefficient for the scores awarded for the tasks \(P\) and \(Q\).
2. Test, at the \(1 \%\) level of significance, whether or not there is evidence for a positive correlation between the ranks of scores for tasks \(P\) and \(Q\). You should state your hypotheses and critical value clearly. The Spearman's rank correlation coefficient for \(P\) and \(R\) is 0.290 and for \(Q\) and \(R\) is 0.795 The manager of Donala wishes to reduce the number of tasks given to job applicants from three to two.
3. Giving a reason for your answer, state which 2 tasks you would recommend the manager uses.

1. In a gymnastics competition, two judges scored each of 8 competitors on the vault.

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.
3. Give a reason to support the use of Spearman's rank correlation coefficient in this case. The judges also scored the competitors on the beam.
   Spearman's rank correlation coefficient for their ranks on the beam was found to be 0.952
4. Compare the judges' ranks on the vault with their ranks on the beam.

1. An estate agent asks customers to rank 7 features of a house, \(A , B , C , D , E , F\) and \(G\), in order of importance. The responses for two randomly selected customers are in the table below.

1. Calculate Spearman's rank correlation coefficient for these data.
2. Stating your hypotheses and critical value clearly, test at the \(5 \%\) level of significance, whether or not the two customers 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 7 positions. The results are shown in the table below.

The Spearman's rank correlation coefficient between \(b\) and \(s\) is \(\frac { 6 } { 7 }\)

1. Stating your hypotheses clearly, test whether or not the data provides support for the researcher's claim. Use a \(1 \%\) level of significance.
2. Without re-calculating the correlation coefficient, explain how the Spearman's rank correlation coefficient would change if
   1. the depth for G is 109 instead of 104
   2. an extra value H with distance from the inner bank of 800 cm and depth 130 cm is included. The researcher decided to collect extra data and found that there were now many tied ranks.
3. Describe how you would find the correlation with many tied ranks.

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.

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. [5]
2. Stating your hypotheses clearly, test at the 1\% level of significance, whether or not the two judges are generally in agreement. [4]

A doctor is interested in the relationship between a person's Body Mass Index (BMI) and their level of fitness. She believes that a lower BMI leads to a greater level of fitness. She randomly selects 10 female 18 year-olds and calculates each individual's BMI. The females then run a race and the doctor records their finishing positions. The results are shown in the table.

Individual	\(A\)	\(B\)	\(C\)	\(D\)	\(E\)	\(F\)	\(G\)	\(H\)	\(I\)	\(J\)
BMI	17.4	21.4	18.9	24.4	19.4	20.1	22.6	18.4	25.8	28.1
Finishing position	3	5	1	9	6	4	10	2	7	8

1. Calculate Spearman's rank correlation coefficient for these data. [5]
2. Stating your hypotheses clearly and using a one tailed test with a 5\% level of significance, interpret your rank correlation coefficient. [5]
3. Give a reason to support the use of the rank correlation coefficient rather than the product moment correlation coefficient with these data. [1]

A county councillor is investigating the level of hardship, \(h\), of a town and the number of calls per 100 people to the emergency services, \(c\). He collects data for 7 randomly selected towns in the county. The results are shown in the table below.

Town	\(A\)	\(B\)	\(C\)	\(D\)	\(E\)	\(F\)	\(G\)
\(h\)	14	20	16	18	37	19	24
\(c\)	52	45	43	42	61	82	55

1. Calculate the Spearman's rank correlation coefficient between \(h\) and \(c\). [6]
2. Test, at the 5\% level of significance, the councillor's claim. State your hypotheses clearly. [4]

After collecting the data, the councillor thinks there is no correlation between hardship and the number of calls to the emergency services.

A newly promoted manager is present when an experienced manager interviews six candidates, \(A\), \(B\), \(C\), \(D\), \(E\) and \(F\) for a job. Both managers rank the candidates in order of preference, starting with the best candidate, giving the following lists: Experienced Manager: \(B\) \(F\) \(A\) \(C\) \(E\) \(D\) New Manager: \(F\) \(C\) \(B\) \(D\) \(E\) \(A\)

1. Calculate Spearman's rank correlation coefficient for these data. [5]
2. Stating your hypotheses clearly, test at the 5\% level of significance whether or not there is evidence of positive correlation. [4]
3. Comment on whether the new manager needs training in the assessment of candidates at interview. [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]