Hypothesis test for correlation

Phil measures the concentration of a radioactive element, c, and the amount of dissolved solids, a, of 8 random samples of groundwater. His results are shown in the table below.

Given that

calculate, to 3 decimal places, the product moment correlation coefficient between the concentration of the radioactive element and the amount of dissolved solids for these groundwater samples.

Use your value of the product moment correlation coefficient to test whether or not there is evidence of a positive correlation between the concentration of this radioactive element and the amount of dissolved solids in groundwater. Use a 5% significance level. State your hypotheses clearly.

Calculate, to 3 decimal places, Spearman’s rank correlation coefficient between the concentration of the radioactive element and the amount of dissolved solids.

Use your value of Spearman’s rank correlation coefficient to test for evidence of a positive correlation between the concentration of the radioactive element and the amount of dissolved solids. Use a 5% significance level. State your hypotheses clearly.

Using your conclusions in part (b) and part (d), comment on the possible relationship between these variables.

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

Calculate Spearman’s rank correlation coefficient for these data.

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.

Comment on the effectiveness of the training delivered by the senior judge.

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.

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.

Calculate Spearman’s rank correlation coefficient for these data.

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.

The shopkeeper calculates the product moment correlation coefficient from his raw data and finds r = 0.65

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.

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.

A mobile library has 160 books for children on its records. The librarian believes that books with fewer pages are borrowed more often. He takes a random sample of 10 books for children.

Explain how the librarian should select this random sample.

The librarian ranked the 10 books according to how often they had been borrowed, with 1 for the book borrowed the most and 10 for the book borrowed the least. He also recorded the number of pages in each book. The results are in the table below.

Calculate Spearman’s rank correlation coefficient for these data.

Test the librarian’s belief using a 5% level of significance. State your hypotheses clearly.

The table below shows the number of students per member of staff and the student satisfaction scores for 7 universities.

Calculate Spearman’s rank correlation coefficient for these data.

Stating your hypotheses clearly test, at the 5% level of significance, whether or not there is evidence of a correlation between the number of students per member of staff and the student satisfaction score.

The table below shows the population and the number of council employees for different towns and villages.

Find, to 3 decimal places, Spearman’s rank correlation coefficient between the population and the number of council employees.

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.

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.

Interpret and comment on your results from part (b) and part (c).

Interviews for a job are carried out by two managers. Candidates are given a score by each manager and the results for a random sample of 8 candidates are shown in the table below.

Calculate Spearman’s rank correlation coefficient for these data. (5)

Test, at the 5% level of significance, whether there is agreement between the rankings awarded by each manager. State your hypotheses clearly. (5) Manager Y later discovered he had miscopied his score for candidate D and it should be 54.

Without carrying out any further calculations, explain how you would calculate Spearman’s rank correlation in this case. (2)

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.

Calculate the Spearman’s rank correlation coefficient between h and c.

After collecting the data, the councillor thinks there is no correlation between hardship and the number of calls to the emergency services.

Test, at the 5% level of significance, the councillor’s claim. State your hypotheses clearly.

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.

Calculate Spearman’s rank correlation coefficient for these data.

Stating your hypotheses clearly and using a one tailed test with a 5% level of significance, interpret your rank correlation coefficient.

Give a reason to support the use of the rank correlation coefficient rather than the product moment correlation coefficient with these data.