5.08e Spearman rank correlation

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.

6 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. Arlosh checks his working with Sarah, whose answer \(r _ { s } = \frac { 29 } { 35 }\) is correct. Find the correct value of \(\Sigma d ^ { 2 }\).
3. Carry out an appropriate two-tailed significance test of the value of \(r _ { s }\) at the \(5 \%\) significance level, stating your hypotheses clearly. Each critic gives a score out of 100 to each film. Desi uses these scores to calculate Pearson's product-moment correlation coefficient. She carries out a two-tailed significance test of this value at the \(5 \%\) significance level.
4. Explain with a reason whether you would expect the conclusion of Desi's test to be the same as the result of the test in part (iii).

8 At a wine-tasting competition, two judges give marks out of 100 to 7 wines as follows.

1. A spectator claims that there is a high level of agreement between the rank orders of the marks given by the two judges. Test the spectator's claim at the \(1 \%\) significance level.
2. A competitor ranks the wines in a random order. The value of Spearman's rank correlation coefficient between the competitor and Judge I is \(r _ { s }\).
   1. Find the probability that \(r _ { s } = 1\).
   2. Show that \(r _ { s }\) cannot take the value \(\frac { 55 } { 56 }\).

5 The speed \(v \mathrm {~ms} ^ { - 1 }\) of a car at time \(t\) seconds after it starts to accelerate was measured at 1 -second intervals. The results are shown in the following diagram. \includegraphics[max width=\textwidth, alt={}, center]{d5843350-52f9-4fed-adf4-86ceb958033f-3_661_1186_1078_443}

1. State whether \(t\) or \(v\) or neither is a controlled variable. The value of the product moment correlation coefficient \(r\) for the data is 0.987 correct to 3 significant figures.
2. The speed of the car is converted to miles per hour and the time to minutes. State the value of \(r\) for the converted data.
3. State the value of Spearman's rank correlation coefficient \(r _ { s }\) for the data.
4. What information does \(r\) give about the data that is not given by \(r _ { s }\) ?

8 In a competition, entrants have to give ranks from 1 to 7 to each of seven resorts. The correct ranks for the resorts are decided by an expert.

1. One competitor chooses his ranks randomly. By considering all the possible rankings, find the probability that the value of Spearman's rank correlation coefficient \(r _ { s }\) between the competitor's ranks and the expert's ranks is at least \(\frac { 27 } { 28 }\).
2. Another competitor ranks the seven resorts. A significance test is carried out to test whether there is evidence that this competitor is merely guessing the rank order of the seven resorts. The critical region is \(r _ { s } \geqslant \frac { 27 } { 28 }\). State the significance level of the test. \section*{END OF QUESTION PAPER}

7 The table shows the values of 5 observations of bivariate data \(( x , y )\). $$n = 5 , \Sigma x = 33.1 , \Sigma y = 56.6 , \Sigma x ^ { 2 } = 227.95 , \Sigma y ^ { 2 } = 664.26 , \Sigma x y = 362.37$$

1. Calculate Pearson's product-moment correlation coefficient \(r\) for the data.
2. State what this value of \(r\) tells you about a scatter diagram illustrating the data.
3. Test at the \(5 \%\) significance level whether there is association between \(x\) and \(y\).
4. State the value of Spearman's rank correlation coefficient \(r _ { s }\) for the data.
5. State whether \(r , r _ { s }\), or both or neither is changed when the values of \(x\) are replaced by
   1. \(3 x - 2\),
   2. \(\sqrt { x }\).

8 The table shows the population, \(x\) million, of each of nine countries in Western Europe together with the population, \(y\) million, of its capital city. $$\left[ n = 9 , \Sigma x = 337.5 , \Sigma x ^ { 2 } = 18959.11 , \Sigma y = 28.3 , \Sigma y ^ { 2 } = 161.65 , \Sigma x y = 1533.76 . \right]$$

1. (a) Calculate Spearman's rank correlation coefficient, \(r _ { s }\).
   (b) Explain what your answer indicates about the populations of these countries and their capital cities.
2. Calculate the product moment correlation coefficient, \(r\). The data are illustrated in the scatter diagram. \includegraphics[max width=\textwidth, alt={}, center]{11316ea6-3999-4003-b77d-bee8b547c1da-09_936_881_1162_632}
3. By considering the diagram, state the effect on the value of the product moment correlation coefficient, \(r\), if the data for France and the United Kingdom were removed from the calculation.
4. In a certain country in Africa, most people live in remote areas and hence the population of the country is unknown. However, the population of the capital city is known to be approximately 1 million. An official suggests that the population of this country could be estimated by using a regression line drawn on the above scatter diagram.
   (a) State, with a reason, whether the regression line of \(y\) on \(x\) or the regression line of \(x\) on \(y\) would need to be used.
   (b) Comment on the reliability of such an estimate in this situation. 1 Some observations of bivariate data were made and the equations of the two regression lines were found to be as follows. $$\begin{array} { c c } y \text { on } x : & y = - 0.6 x + 13.0 \\ x \text { on } y : & x = - 1.6 y + 21.0 \end{array}$$

1. State, with a reason, whether the correlation between \(x\) and \(y\) is negative or positive.
2. Neither variable is controlled. Calculate an estimate of the value of \(x\) when \(y = 7.0\).
3. Find the values of \(\bar { x }\) and \(\bar { y }\). 2 A bag contains 5 black discs and 3 red discs. A disc is selected at random from the bag. If it is red it is replaced in the bag. If it is black, it is not replaced. A second disc is now selected at random from the bag. Find the probability that

1. the second disc is black, given that the first disc was black,
2. the second disc is black,
3. the two discs are of different colours. 3 Each of the 7 letters in the word DIVIDED is printed on a separate card. The cards are arranged in a row.

1. How many different arrangements of the letters are possible?
2. In how many of these arrangements are all three Ds together? The 7 cards are now shuffled and 2 cards are selected at random, without replacement.
3. Find the probability that at least one of these 2 cards has D printed on it.

4

1. The random variable \(X\) has the distribution \(\mathrm { B } ( 25,0.2 )\). Using the tables of cumulative binomial probabilities, or otherwise, find \(\mathrm { P } ( X \geqslant 5 )\).
2. The random variable \(Y\) has the distribution \(\mathrm { B } ( 10,0.27 )\). Find \(\mathrm { P } ( Y = 3 )\).
3. The random variable \(Z\) has the distribution \(B ( n , 0.27 )\). Find the smallest value of \(n\) such that \(\mathrm { P } ( Z \geqslant 1 ) > 0.95\). 5 The probability distribution of a discrete random variable, \(X\), is given in the table.

   \(x\)	0	1	2	3
   \(\mathrm { P } ( X = x )\)	\(\frac { 1 } { 3 }\)	\(\frac { 1 } { 4 }\)	\(p\)	\(q\)

   It is given that the expectation, \(\mathrm { E } ( X )\), is \(1 \frac { 1 } { 4 }\).

1. Calculate the values of \(p\) and \(q\).
2. Calculate the standard deviation of \(X\).

6 Fig. 6 shows the wages earned in the last 12 months by each of a random sample of American males aged between 16 and 65 . \begin{figure}[h] \end{figure} A researcher wishes to test whether the sample provides evidence of a tendency for higher wages to be earned by older men in the age range 16 to 65 in America.

1. The researcher needs to decide whether to use a test based on Pearson's product moment correlation coefficient or Spearman's rank correlation coefficient. Use the information in Fig. 6 to decide which test is more appropriate.
2. Should it be a one-tail or a two-tail test? Justify your answer.

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]

At the end of a season an athletics coach graded a random sample of ten athletes according to their performances throughout the season and their dedication to training. The results, expressed as percentages, are shown in the table below.

1. Calculate the Spearman rank correlation coefficient between performance and dedication. [5]
2. Stating clearly your hypotheses and using a 10\% 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]

At the end of a season an athletics coach graded a random sample of ten athletes according to their performances throughout the season and their dedication to training. The results, expressed as percentages, are shown in the table below.

1. Calculate the Spearman rank correlation coefficient between performance and dedication. [5]
2. Stating clearly your hypotheses and using a 10\% 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]

The table below shows the price of an ice cream and the distance of the shop where it was purchased from a particular tourist attraction.

1. Find, to 3 decimal places, the Spearman rank correlation coefficient between the distance of the shop from the tourist attraction and the price of an ice cream. [5]
2. Stating your hypotheses clearly and using a 5\% one-tailed test, interpret your rank correlation coefficient. [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.

For one of the activities at a gymnastics competition, 8 gymnasts were awarded marks out of 10 for each of artistic performance and technical ability. The results were as follows.

Gymnast	\(A\)	\(B\)	\(C\)	\(D\)	\(E\)	\(F\)	\(G\)	\(H\)
Technical ability	8.5	8.6	9.5	7.5	6.8	9.1	9.4	9.2
Artistic performance	6.2	7.5	8.2	6.7	6.0	7.2	8.0	9.1

The value of the product moment correlation coefficient for these data is 0.774.

1. Stating your hypotheses clearly and using a 1% level of significance, interpret this value. [5]
2. Calculate the value of the rank correlation coefficient for these data. [6]
3. Stating your hypotheses clearly and using a 1% level of significance, interpret this coefficient. [3]
4. Explain why the rank correlation coefficient might be the better one to use with these data. [2]

At the end of a season a league of eight ice hockey clubs produced the following table showing the position of each club in the league and the average attendances (in hundreds) at home matches.

1. Calculate the Spearman rank correlation coefficient between position in the league and average home attendance. [5]
2. Stating clearly your hypotheses and using a 5\% two-tailed test, interpret your rank correlation coefficient. [4]

Many sets of data include tied ranks.

1. Explain briefly how tied ranks can be dealt with. [2]

Two judges placed 7 dancers in rank order. Both judges placed dancers A and B in the first two places, but in opposite orders. The judges agreed about the ranks for all the other 5 dancers. Calculate the value of Spearman's rank correlation coefficient. [4]

Three skaters, \(A\), \(B\) and \(C\), are placed in rank order by four judges. Judge \(P\) ranks skater \(A\) in 1st place, skater \(B\) in 2nd place and skater \(C\) in 3rd place.

1. Without carrying out any calculation, state the value of Spearman's rank correlation coefficient for the following ranks. Give a reason for your answer. [1]

   Skater	\(A\)	\(B\)	\(C\)
   Judge \(P\)	1	2	3
   Judge \(Q\)	3	2	1

2. Calculate the value of Spearman's rank correlation coefficient for the following ranks. [3]

   Skater	\(A\)	\(B\)	\(C\)
   Judge \(P\)	1	2	3
   Judge \(R\)	3	1	2

3. Judge \(S\) ranks the skaters at random. Find the probability that the value of Spearman's rank correlation coefficient between the ranks of judge \(P\) and judge \(S\) is 1. [3]

In a competition, a wine-enthusiast has to rank ten bottles of wine, \(A\) to \(J\), in order starting with the one he thinks is the most expensive. The table below shows his rankings and the actual order according to price.

1. Calculate Spearman's rank correlation coefficient for these data. [6]
2. Stating your hypotheses clearly, test at the 5% level of significance whether or not there is evidence of positive correlation. [4]
3. Explain briefly how you would have been able to carry out the test if bottles \(B\) and \(F\) had the same price. [2]

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]

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

Concert	1	2	3	4	5	6	7
Score by critic \(P\)	12	11	6	13	17	16	14
Score by critic \(Q\)	9	13	8	14	18	16	20

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]