Hypothesis test of Spearman’s rank correlation coefficien

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

   	Sun	Mon	Tues	Weds	Thurs	Fri	Sat
   Ice cream	6	4	7	5	3	2	1
   Sunglasses	6	5	7	2	3	4	1

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)

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.

2 Eight runners took part in two races. The positions in which the runners finished in the two races are shown in the table. Test at the 5\% significance level whether those runners who do better in one race tend to do better in the other.

1 A researcher is investigating whether there is a relationship between the population size of cities and the average walking speed of pedestrians in the city centres. Data for the population size, \(x\) thousands, and the average walking speed of pedestrians, \(y \mathrm {~m} \mathrm {~s} ^ { - 1 }\), of eight randomly selected cities are given in the table below.

1. Calculate the value of Spearman's rank correlation coefficient.
2. Carry out a hypothesis test at the \(5 \%\) significance level to determine whether there is any association between population size and average walking speed. In another investigation, the researcher selects a random sample of six adult males of particular ages and measures their maximum walking speeds. The data are shown in the table below, where \(t\) years is the age of the adult and \(w \mathrm {~m} \mathrm {~s} ^ { - 1 }\) is the maximum walking speed. Also shown are summary statistics and a scatter diagram on which the regression line of \(w\) on \(t\) is drawn.

   \(t\)	20	30	40	50	60	70
   \(w\)	2.49	2.41	2.38	2.14	1.97	2.03

   $$n = 6 \quad \Sigma t = 270 \quad \Sigma w = 13.42 \quad \Sigma t ^ { 2 } = 13900 \quad \Sigma w ^ { 2 } = 30.254 \quad \Sigma t w = 584.6$$ \includegraphics[max width=\textwidth, alt={}, center]{77b97142-afb6-41d6-8fec-e982b7a7501b-2_728_1091_1379_529}
3. Calculate the equation of the regression line of \(w\) on \(t\).
4. (A) Use this equation to calculate an estimate of maximum walking speed of an 80 -year-old male.
   (B) Explain why it might not be appropriate to use the equation to calculate an estimate of maximum walking speed of a 10 -year-old male.

1 Nine long-distance runners are starting an exercise programme to improve their strength. During the first session, each of them has to do a 100 metre run and to do as many push-ups as possible in one minute. The times taken for the run, together with the number of push-ups each runner achieves, are shown in the table.

1. Draw a scatter diagram to illustrate the data.
2. Calculate the value of Spearman's rank correlation coefficient.
3. Carry out a hypothesis test at the \(5 \%\) significance level to examine whether there is any association between time taken for the run and number of push-ups achieved.
4. Under what circumstances is it appropriate to carry out a hypothesis test based on the product moment correlation coefficient? State, with a reason, which test is more appropriate for these data.

A class of 8 students sit examinations in History and Geography. The marks obtained by these students are given below.

Student	A	B	C	D	E	F	G	H
History mark	73	59	83	49	57	82	67	60
Geography mark	55	51	58	59	44	66	49	67

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]

3 An expert tested the quality of the wines produced by a vineyard in 9 particular years. He placed them in the following order, starting with the best. $$\begin{array} { l l l l l l l l l } 1980 & 1983 & 1981 & 1982 & 1984 & 1985 & 1987 & 1986 & 1988 \end{array}$$

1. Calculate Spearman's rank correlation coefficient, \(r _ { s }\), between the year of production and the quality of these wines. The years should be ranked from the earliest (1) to the latest (9).
2. State what this value of \(r _ { s }\) shows in this context.

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]

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]

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]

1

1. Calculate the value of Spearman's rank correlation coefficient between the two sets of rankings, \(A\) and \(B\), shown in Table 1. \begin{table}[h]

   \(A\)	1	2	3	4	5
   \(B\)	4	1	3	2	5

   \captionsetup{labelformat=empty} \caption{Table 1}
   \end{table}
2. The value of Spearman's rank correlation coefficient between the set of rankings \(B\) and a third set of rankings, \(C\), is known to be - 1 . Copy and complete Table 2 showing the set of rankings \(C\). \begin{table}[h]

   \(B\)	4	1	3	2	5
   \(C\)

   \captionsetup{labelformat=empty} \caption{Table 2}
   \end{table}

4

1. The table gives the heights and masses of 5 people.

   Person	\(A\)	\(B\)	\(C\)	\(D\)	\(E\)
   Height (m)	1.72	1.63	1.77	1.68	1.74
   Mass (kg)	75	62	64	60	70

   Calculate Spearman's rank correlation coefficient.
2. In an art competition the value of Spearman's rank correlation coefficient, \(r _ { s }\), calculated from two judges' rankings was 0.75 . A late entry for the competition was received and both judges ranked this entry lower than all the others. By considering the formula for \(r _ { s }\), explain whether the new value of \(r _ { s }\) will be less than 0.75 , equal to 0.75 , or greater than 0.75 .

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.

3

1. Shula calculates the value of Spearman's rank correlation coefficient \(r _ { s }\) for 9 pairs of rankings.
   Find the largest possible value of \(r _ { s }\) that Shula can obtain that is less than 1 .
2. A set of bivariate data consists of 5 pairs of values. It is known that for this data the value of Spearman's rank correlation coefficient is - 1 but the value of Pearson's product-moment correlation coefficient is not - 1 . Sketch a possible scatter diagram illustrating the data.

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}

4 Three tutors each marked the coursework of five students. The marks are given in the table.

1. Calculate Spearman's rank correlation coefficient, \(r _ { \mathrm { s } }\), between the marks for tutors 1 and 2 .
2. The values of \(r _ { \mathrm { s } }\) for the other pairs of tutors, are as follows. $$\begin{array} { c c } \text { Tutors } 1 \text { and 3: } & r _ { \mathrm { s } } = - 0.9 \\ \text { Tutors } 2 \text { and 3: } & r _ { \mathrm { s } } = 0.3 \end{array}$$ State which two tutors differ most widely in their judgements. Give your reason.

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. 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]

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.

   Tractor	Farmer's rank	PTO (horsepower)	Price (£1000s)
   A	1	77·5	80
   B	6	87·9	45
   C	5	53·0	47
   D	4	41·0	53
   E	2	112·0	60
   F	3	90·0	61

   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]