Hypothesis test for positive correlation

3 A researcher is investigating the relationship between temperature and levels of the air pollutant nitrous oxide at a particular site. The researcher believes that there will be a positive correlation between the daily maximum temperature, \(x\), and nitrous oxide level, \(y\). Data are collected for 10 randomly selected days. The data, measured in suitable units, are given in the table and illustrated on the scatter diagram.

1. Calculate the value of Spearman's rank correlation coefficient for these data.
2. Perform a hypothesis test at the \(5 \%\) level to check the researcher's belief, stating your hypotheses clearly.
3. It is suggested that it would be preferable to carry out a test based on the product moment correlation coefficient. State the distributional assumption required for such a test to be valid. Explain how a scatter diagram can be used to check whether the distributional assumption is likely to be valid and comment on the validity in this case.
4. A statistician investigates data over a much longer period and finds that the assumptions for the use of the product moment correlation coefficient are in fact valid. Give the critical region for the test at the \(1 \%\) level, based on a sample of 60 days.
5. In a different research project, into the correlation between daily temperature and ozone pollution levels, a positive correlation is found. It is argued that this shows that high temperatures cause increased ozone levels. Comment on this claim.

2 A medical student is trying to estimate the birth weight of babies using pre-natal scan images. The actual weights, \(x \mathrm {~kg}\), and the estimated weights, \(y \mathrm {~kg}\), of ten randomly selected babies are given in the table below.

1. Calculate the value of Spearman's rank correlation coefficient.
2. Carry out a hypothesis test at the \(5 \%\) level to determine whether there is positive association between the student's estimates and the actual birth weights of babies in the underlying population.
3. Calculate the value of the product moment correlation coefficient of the sample. You may use the following summary statistics in your calculations: $$\Sigma x = 31.63 , \quad \Sigma y = 33.1 , \quad \Sigma x ^ { 2 } = 101.92 , \quad \Sigma y ^ { 2 } = 112.61 , \quad \Sigma x y = 106.51 .$$
4. Explain why, if the underlying population has a bivariate Normal distribution, it would be preferable to carry out a hypothesis test based on the product moment correlation coefficient. Comment briefly on the significance of the product moment correlation coefficient in relation to that of Spearman's rank correlation coefficient.

1 Two celebrities judge a talent contest. Each celebrity gives a score out of 20 to each of a random sample of 8 contestants. The scores, \(x\) and \(y\), given by the celebrities to each contestant are shown 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 positive association between the scores allocated by the two celebrities.
3. State the distributional assumption required for a test based on the product moment correlation coefficient. Sketch a scatter diagram of the scores above, and discuss whether it appears that the assumption is likely to be valid.

1 A medical student is investigating the claim that young adults with high diastolic blood pressure tend to have high systolic blood pressure. The student measures the diastolic and systolic blood pressures of a random sample of ten young adults. The data are shown in the table and illustrated in the scatter diagram. \includegraphics[max width=\textwidth, alt={}, center]{17e474c4-f5be-4ca1-b7c3-e444b46c3bec-2_865_809_593_628}

1. Calculate the value of Spearman's rank correlation coefficient for these data.
2. Carry out a hypothesis test at the \(5 \%\) significance level to examine whether there is positive association between diastolic blood pressure and systolic blood pressure in the population of young adults.
3. Explain why, in the light of the scatter diagram, it might not be valid to carry out a test based on the product moment correlation coefficient. The product moment correlation coefficient between the diastolic and systolic blood pressures of a random sample of 10 athletes is 0.707 .
4. Carry out a hypothesis test at the \(1 \%\) significance level to investigate whether there appears to be positive correlation between these two variables in the population of athletes. You may assume that in this case such a test is valid.

5 In a fashion competition, two judges gave marks to a large number of contestants. The value of Spearman's rank correlation coefficient, \(\mathrm { r } _ { \mathrm { s } }\), between the marks given to 7 randomly chosen contestants is \(\frac { 27 } { 28 }\).

1. An excerpt from the table of critical values of \(\mathrm { r } _ { \mathrm { s } }\) is shown below. \section*{Critical values of Spearman's rank correlation coefficient}

   	1-tail test	5\%	2.5\%	1\%	0.5\%
   	2-tail test	10\%	5\%	2\%	1\%
   \multirow{3}{*}{\(n\)}	6	0.8286	0.8857	0.9429	1.0000
   	7	0.7143	0.7857	0.8929	0.9286
   	8	0.6429	0.7381	0.8333	0.8810

   Test whether there is evidence, at the 1\% significance level, that the judges agree with each another. The marks given by the two judges to the 7 randomly chosen contestants were as follows, where \(x\) is an integer.

   Contestant	A	B	C	D	\(E\)	\(F\)	G
   Judge 1	64	65	67	78	79	80	86
   Judge 2	61	63	78	80	81	90	\(x\)

2. Use the value \(\mathrm { r } _ { \mathrm { s } } = \frac { 27 } { 28 }\) to determine the range of possible values of \(x\).
3. Give a reason why it might be preferable to use the product moment correlation coefficient rather than Spearman's rank correlation coefficient in this context.

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.

4 Two magazines give numerical ratings to hi-fi systems. Li wishes to test whether there is agreement between the opinions of the magazines. Li chooses a random sample of 5 hi -fi systems and looks up the ratings given by the two magazines. The results are shown in the table.

1. Give a reason why Li might choose to use a test based on Spearman's rank correlation coefficient rather than on Pearson's product-moment correlation coefficient.
2. Calculate the value of Spearman's rank correlation coefficient for the data.
3. Use your answer to part (b) to carry out a hypothesis test at the \(5 \%\) significance level.
4. The value of Spearman's rank correlation coefficient between the ratings given by magazine I and by a third magazine, magazine III, has the same numerical value as the answer to part (b) but with the sign changed. In the Printed Answer Booklet, complete the table showing the rankings given by magazine III.

2. A teacher believes that those of her students with strong mathematical ability may also have enhanced short-term memory. She shows a random sample of 11 students a tray of different objects for eight seconds and then asks them to write down as many of the objects as they can remember. The results, along with their percentage score in a recent mathematics test, are given in the table below.

1. Calculate Spearman's rank correlation coefficient for these data. Show your working clearly.
2. Stating your hypotheses clearly, carry out a suitable test to assess the teacher's belief. Use a \(5 \%\) level of significance and state your critical value. The teacher shows these results to her class and argues that spending more time trying to improve their short-term memory would improve their mathematical ability.
3. Explain whether or not you agree with the teacher's argument.

3. The table shows the time, in seconds, of the fastest qualifying lap for 10 different Formula One racing drivers and their finishing position in the actual race.

1. Calculate the value of Spearman's rank correlation coefficient for these data.
2. Stating your hypotheses clearly, test at the \(1 \%\) level of significance, whether or not there is evidence of a positive correlation between the fastest qualifying lap time and finishing position for these Formula One racing drivers.

2 The table shows the season's best times, \(x\) seconds, for the 8 athletes who took part in the 200 m final in the 2021 Tokyo Olympics. It also shows their finishing position in the race. Given that the fastest season's best time is ranked number 1

1. calculate the value of the Spearman's rank correlation coefficient for these data.
2. Stating your hypotheses clearly, test, at the \(1 \%\) level of significance, whether or not there is evidence of a positive correlation between the rank of the season's best time and the finishing position for these athletes. Chris suggests that it would be better to use the actual finishing time, \(y\) seconds, of these athletes rather than their finishing position. Given that $$S _ { x x } = 0.1286875 \quad S _ { y y } = 0.55275 \quad S _ { x y } = 0.225175$$
3. calculate the product moment correlation coefficient between the season's best time and the finishing time for these athletes.
   Give your answer correct to 3 decimal places.
4. Use your value of the product moment correlation coefficient to test, at the \(1 \%\) level of significance, whether or not there is evidence of a positive correlation between the season's best time and the finishing time for these athletes.

4. In a survey 10 randomly selected men had their systolic blood pressure, \(x\), and weight, \(w\), measured. Their results are as follows

1. Calculate the value of Spearman's rank correlation coefficient between \(x\) and \(w\).
2. Stating your hypotheses clearly, test at the \(5 \%\) level of significance, whether or not there is evidence of a positive correlation between systolic blood pressure and weight. The product moment correlation coefficient for these data is 0.5114
3. Use the 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 systolic blood pressure and weight.
4. Using your conclusions to part (b) and part (c), describe the relationship between systolic blood pressure and weight.

1. The table below shows the distance travelled by car and the amount of commission earned by each of 8 salespersons in 2015

1. Find 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 positive correlation between the distance travelled by car and the amount of commission earned.

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 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. 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.

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.

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.

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.

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).

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.)

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.