5.08f Hypothesis test: Spearman rank

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OCR MEI Further Statistics Major Specimen Q6
3 marks Standard +0.3
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]
\includegraphics[alt={},max width=\textwidth]{e6ee3a4a-3e76-4422-9a78-17b64b458f83-07_771_1278_340_392} \captionsetup{labelformat=empty} \caption{Fig. 6}
\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.
Edexcel S3 2015 June Q2
9 marks Standard +0.3
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.
Rank123456789
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]
Edexcel S3 Q4
11 marks Standard +0.3
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.
AthletePerformanceDedication
A8672
B6069
C7859
D5668
E8080
F6684
G5165
H5955
I7379
J4953
  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]
Edexcel S3 2002 June Q4
11 marks Standard +0.3
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.
AthletePerformanceDedication
A8672
B6069
C7859
D5668
E8080
F6684
G3165
H5955
I7379
J4953
  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]
Edexcel S3 2006 June Q4
9 marks Standard +0.3
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.
ShopDistance from tourist attraction (m)Price (£)
A501.75
B1751.20
C2702.00
D3751.05
E4250.95
F5801.25
G7100.80
H7900.75
I8901.00
J9800.85
  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]
Edexcel S3 2009 June Q3
11 marks Standard +0.3
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\(A\)\(B\)\(C\)\(D\)\(E\)\(F\)\(G\)\(H\)\(I\)\(J\)
BMI17.421.418.924.419.420.122.618.425.828.1
Finishing position35196410278
  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]
Edexcel S3 2011 June Q2
10 marks Standard +0.3
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\(A\)\(B\)\(C\)\(D\)\(E\)\(F\)\(G\)
\(h\)14201618371924
\(c\)52454342618255
  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.
Edexcel S3 2016 June Q3
Moderate -0.3
  1. Describe when you would use Spearman's rank correlation coefficient rather than the product moment correlation coefficient to measure the strength of the relationship between two variables. (1) A shop sells sunglasses and ice cream. For one week in the summer the shopkeeper ranked the daily sales of ice cream and sunglasses. The ranks are shown in the table below.
    SunMonTuesWedsThursFriSat
    Ice cream6475321
    Sunglasses6572341
  2. Calculate Spearman's rank correlation coefficient for these data. (3)
  3. Test, at the 5\% level of significance, whether or not there is a positive correlation between sales of ice cream and sales of sunglasses. State your hypotheses clearly. (4) The shopkeeper calculates the product moment correlation coefficient from his raw data and finds \(r = 0.65\)
  4. Using this new coefficient, test, at the 5\% level of significance, whether or not there is a positive correlation between sales of ice cream and sales of sunglasses. (2)
  5. Using your answers to part (c) and part (d), comment on the nature of the relationship between sales of sunglasses and sales of ice cream. (1)
Edexcel S3 Q7
16 marks Standard +0.3
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 ability8.58.69.57.56.89.19.49.2
Artistic performance6.27.58.26.76.07.28.09.1
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]
Edexcel S3 Specimen Q4
11 marks Standard +0.3
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.
Club\(A\)\(B\)\(C\)\(D\)\(E\)\(F\)\(G\)\(H\)
Position12345678
Average3738192734262232
  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]
OCR S1 2013 January Q7
7 marks Standard +0.3
  1. Two judges rank \(n\) competitors, where \(n\) is an even number. Judge 2 reverses each consecutive pair of ranks given by Judge 1, as shown.
    Competitor\(C_1\)\(C_2\)\(C_3\)\(C_4\)\(C_5\)\(C_6\)\(\ldots\)\(C_{n-1}\)\(C_n\)
    Judge 1 rank123456\(\ldots\)\(n-1\)\(n\)
    Judge 2 rank214365\(\ldots\)\(n\)\(n-1\)
    Given that the value of Spearman's coefficient of rank correlation is \(\frac{63}{65}\), find \(n\). [4]
  2. An experiment produced some data from a bivariate distribution. The product moment correlation coefficient is denoted by \(r\), and Spearman's rank correlation coefficient is denoted by \(r_s\).
    1. Explain whether the statement $$r = 1 \Rightarrow r_s = 1$$ is true or false. [1]
    2. Use a diagram to explain whether the statement $$r \neq 1 \Rightarrow r_s \neq 1$$ is true or false. [2]
OCR S1 2013 June Q2
7 marks Moderate -0.8
  1. The table shows the times, in minutes, spent by five students revising for a test, and the grades that they achieved in the test.
    StudentAnnBillCazDenEd
    Time revising0603510045
    GradeCDEBA
    Calculate Spearman's rank correlation coefficient. [5]
  2. The table below shows the ranks given by two judges to four competitors.
    CompetitorPQRS
    Judge 1 rank1234
    Judge 2 rank3214
    Spearman's rank correlation coefficient for these ranks is denoted by \(r_s\). With the same set of ranks for Judge 1, write down a different set of ranks for Judge 2 which gives the same value of \(r_s\). There is no need to find the value of \(r_s\). [2]
Edexcel S3 Q5
12 marks Standard +0.3
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.
Rank12345678910
Enthusiast\(D\)\(C\)\(J\)\(A\)\(G\)\(F\)\(B\)\(E\)\(I\)\(H\)
Price\(A\)\(C\)\(D\)\(H\)\(J\)\(B\)\(F\)\(I\)\(G\)\(E\)
  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]
Edexcel S3 Q3
10 marks Standard +0.3
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]
OCR MEI Further Statistics Minor Specimen Q5
10 marks Standard +0.3
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.
ContestantABCDEFG
Age6651392992214
Score1211151716189
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]
WJEC Further Unit 2 2018 June Q4
9 marks Standard +0.3
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.
CowABCDEFG
Actual weight, kg61411057181001889770682
Estimated weight, kg70015008501400750900800
  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]
WJEC Further Unit 2 2023 June Q5
12 marks Standard +0.3
  1. Give two circumstances where it may be more appropriate to use Spearman's rank correlation coefficient rather than Pearson's product moment correlation coefficient. [2]
  2. A farmer needs a new tractor. The tractor salesman selects 6 tractors at random to show the farmer. The farmer ranks these tractors, in order of preference, according to their ability to meet his needs on the farm. The tractor salesman makes a note of the price and power take-off (PTO) of the tractors.
    TractorFarmer's rankPTO (horsepower)Price (£1000s)
    A177·580
    B687·945
    C553·047
    D441·053
    E2112·060
    F390·061
    Spearman's rank correlation coefficient between the farmer's ranks and the price is 0·9429.
    1. Test at the 5% significance level whether there is an association between the price of a tractor and the farmer's judgement of the ability of the tractor to meet his needs on the farm. [4]
    2. Calculate Spearman's rank correlation coefficient between the farmer's rank and PTO. [4]
    3. How should the tractor salesman interpret the results in (i) and (ii)? [2]
WJEC Further Unit 2 Specimen Q3
9 marks Standard +0.3
A class of 8 students sit examinations in History and Geography. The marks obtained by these students are given below.
StudentABCDEFGH
History mark7359834957826760
Geography mark5551585944664967
  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]
SPS SPS ASFM Statistics 2021 May Q5
8 marks Moderate -0.3
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]
SPS SPS FM Statistics 2021 January Q7
7 marks Challenging +1.2
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.
Athlete\(A\)\(B\)\(C\)\(D\)\(E\)\(F\)\(G\)\(H\)\(I\)
Position in 100m sprint467928315
Position in long jump549312
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]