Hypothesis test for negative correlation

1 A researcher believes that there may be negative association between the quantity of fertiliser used and the percentage of the population who live in rural areas in different countries. The data below show the percentage of the population who live in rural areas and the fertiliser use measured in kg per hectare, for a random sample of 11 countries.

1. Draw a scatter diagram to illustrate the data.
2. Explain why it might not be valid to carry out a test based on the product moment correlation coefficient in this case.
3. Calculate the value of Spearman's rank correlation coefficient.
4. Carry out a hypothesis test at the \(1 \%\) significance level to investigate the researcher's belief.
5. Explain the meaning of ' \(1 \%\) significance level'.
6. In order to carry out a test based on Spearman's rank correlation coefficient, what modelling assumptions, if any, are required about the underlying distribution?

1. A random sample of 9 footballers is chosen to participate in an obstacle course. The time taken, \(y\) seconds, for each footballer to complete the obstacle course is recorded, together with the footballer's Body Mass Index, \(x\). The results are shown in the table below.

Russell claims, that for footballers, as Body Mass Index increases the time taken to complete the obstacle course tends to decrease.

1. Find, to 3 decimal places, Spearman's rank correlation coefficient between \(x\) and \(y\).
2. Use your value of Spearman's rank correlation coefficient to test Russell's claim. Use a 5\% significance level and state your hypotheses clearly. The product moment correlation coefficient for these data is - 0.5594
3. Use the value of the product moment correlation coefficient to test for evidence of a negative correlation between Body Mass Index and the time taken to complete the obstacle course. Use a 5\% significance level.
4. Using your conclusions to part (b) and part (c), describe the relationship between Body Mass Index and the time taken to complete the obstacle course.

6 A student is investigating the relationship between age and grip strength in adults. The student selects 10 people and records their ages in years and the grip strengths of their dominant hand, measured in kg. The data are shown in the table below, together with a scatter diagram to illustrate the data. The student decides to carry out a hypothesis test to investigate whether there is negative association between age and grip strength.

1. Explain why the student decides to carry out a test based on Spearman's rank correlation coefficient.
2. State what property of the sample is required in order for it to be valid to carry out a hypothesis test.
3. In this question you must show detailed reasoning. Assuming that the property in part (b) holds, carry out the test at the \(5 \%\) significance level.

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]