Practice SL 4.10—Spearman’s rank correlation coefficient with authentic IB Mathematics Applications & Interpretation (AI) exam questions for both SL and HL students. This question bank mirrors Paper 1, 2, 3 structure, covering key topics like core principles, advanced applications, and practical problem-solving. Get instant solutions, detailed explanations, and build exam confidence with questions in the style of IB examiners.
A city council studies the relationship between the number of public transport trips per month and the monthly expenditure on transport (in dollars) for 9 residents. The data is shown below, with some residents having equal trip counts.
| Resident | Trips per Month | Expenditure () |
|---|---|---|
| A | 20 | 150 |
| B | 15 | 120 |
| C | 25 | 180 |
| D | 15 | 130 |
| E | 30 | 200 |
| F | 20 | 160 |
| G | 10 | 100 |
| H | 25 | 170 |
| I | 12 | 110 |
Assign ranks to the data, accounting for equal trip counts.
Calculate Spearman's rank correlation coefficient.
Conduct a hypothesis test at the significance level to determine if there is a correlation between trips and expenditure. State the null and alternative hypotheses, the critical value, and the conclusion.
The council adjusts the expenditure for resident A from 150 to 145 dollars. Explain whether this affects Spearman's rank correlation coefficient.
The council plots the original data (not ranks) on a scatter graph and calculates the regression line . Estimate the expenditure for 35 trips and explain why this estimate may not be reliable.
A sports coach records the sprint times (seconds) and long jump distances (meters) for 6 athletes. The data is shown below.
| Athlete | Sprint Time | Long Jump |
|---|---|---|
| A | 10.5 | 6.8 |
| B | 11.2 | 6.5 |
| C | 10.8 | 7.0 |
| D | 11.0 | 6.7 |
| E | 12.0 | 6.3 |
| F | 10.2 | 7.2 |
Calculate Spearman's rank correlation coefficient for the data.
Sketch a scatter graph of the ranks, labeling the axes appropriately.
The coach adjusts the long jump distance for athlete F from 7.2 m to 7.1 m . Explain whether this change affects the value of Spearman's rank correlation coefficient.
A teacher records the number of hours spent studying and the exam scores (out of 50) for 6 students. The data includes two students with equal exam scores.
| Student | Study Hours | Exam Score |
|---|---|---|
| A | 15 | 45 |
| B | 10 | 40 |
| C | 20 | 48 |
| D | 12 | 40 |
| E | 8 | 35 |
| F | 18 | 42 |
Assign ranks to the data, accounting for equal exam scores.
Calculate Spearman's rank correlation coefficient.
Sketch a scatter graph of the ranks.
Interpret the result from part (b).
A coach records the number of training sessions attended by 7 athletes and their performance scores in a competition. The data is shown below.
| Athlete | Training Sessions | Performance Score |
|---|---|---|
| 1 | 10 | 92 |
| 2 | 15 | 88 |
| 3 | 8 | 95 |
| 4 | 12 | 90 |
| 5 | 20 | 85 |
| 6 | 5 | 98 |
| 7 | 18 | 87 |
Explain why Spearman's rank correlation coefficient is appropriate for this data.
Calculate Spearman's rank correlation coefficient.
A librarian records the number of books borrowed and the time spent reading (in hours) by 6 library users over a month.
| User | Books Borrowed | Reading Time (hours) |
|---|---|---|
| 1 | 4 | 10 |
| 2 | 8 | 15 |
| 3 | 2 | 8 |
| 4 | 6 | 12 |
| 5 | 10 | 18 |
| 6 | 3 | 9 |
Calculate Spearman's rank correlation coefficient for the data.
State the conclusion about the relationship between books borrowed and reading time.
A librarian studies the relationship between the number of library visits and the number of books borrowed by 7 patrons over a month. The data is shown below.
| Patron | Library Visits | Books Borrowed |
|---|---|---|
| 1 | 5 | 8 |
| 2 | 10 | 12 |
| 3 | 3 | 6 |
| 4 | 8 | 10 |
| 5 | 12 | 15 |
| 6 | 7 | 9 |
| 7 | 4 | 7 |
Calculate Spearman's rank correlation coefficient for the data.
Explain why Spearman's rank correlation coefficient is appropriate for this data.
Interpret the result from part 1 in the context of the data.
A researcher examines the relationship between the average daily temperature () and the number of ice creams sold at a beach kiosk over 7 days. The data is shown below.
| Day | Temperature () | Ice Creams Sold |
|---|---|---|
| 1 | 22 | 50 |
| 2 | 25 | 60 |
| 3 | 20 | 45 |
| 4 | 28 | 70 |
| 5 | 24 | 55 |
| 6 | 30 | 75 |
| 7 | 26 | 65 |
Calculate Spearman's rank correlation coefficient for the data.
The researcher initially considered using Pearson's product-moment correlation coefficient. Explain why this may not be appropriate.
State the conclusion about the relationship based on the result from part 1.
A researcher is analyzing the relationship between the number of hours employees spend on professional development courses and their annual performance ratings (out of 100) at a tech company. The data for 10 employees is shown below, with some employees having identical performance ratings.
| Employee | Hours of Training () | Performance Rating () |
|---|---|---|
| A | 20 | 85 |
| B | 15 | 90 |
| C | 25 | 80 |
| D | 10 | 95 |
| E | 30 | 75 |
| F | 18 | 88 |
| G | 22 | 85 |
| H | 12 | 92 |
| I | 28 | 78 |
| J | 16 | 90 |
The researcher also collects data on the employees' job satisfaction scores (out of 10) and wishes to compare these with their performance ratings.
Assign ranks to the hours of training and performance ratings, accounting for equal ratings.
Calculate Spearman's rank correlation coefficient for hours of training and performance ratings.
Sketch a scatter graph of the ranks, labeling the axes appropriately.
The researcher calculates Pearson's product-moment correlation coefficient for the original data as . Explain why Spearman's rank correlation coefficient might be preferred in some research scenarios over Pearson's.
Conduct a hypothesis test at the significance level to determine if there is a negative correlation between hours of training and performance ratings, using the Spearman's rank correlation coefficient. State the null and alternative hypotheses, the critical value, and the conclusion.
The job satisfaction scores for employees A to J are , respectively. Calculate Spearman's rank correlation coefficient for job satisfaction and performance ratings.
A school conducts a study on the relationship between the number of extracurricular activities students participate in and their average weekly study hours. The data for 12 students is shown below.
| Student | Activities | Study Hours |
|---|---|---|
| A | 3 | 15 |
| B | 5 | 12 |
| C | 2 | 18 |
| D | 4 | 14 |
| E | 6 | 10 |
| F | 1 | 20 |
| G | 3 | 16 |
| H | 5 | 13 |
| I | 2 | 19 |
| J | 4 | 15 |
| K | 6 | 11 |
| L | 1 | 21 |
Calculate Spearman's rank correlation coefficient, accounting for any ties.
Sketch a scatter graph of the ranks, labeling the axes appropriately.
Conduct a hypothesis test at the significance level to determine if there is a correlation. State the null and alternative hypotheses, the critical value, and the conclusion.
The school considers using Pearson's correlation coefficient. Explain why this may not be appropriate and why Spearman's is preferred.
The study hours for student L are adjusted from 21 to 20 hours. Recalculate Spearman's rank correlation coefficient and comment on the impact of this change.
A student investigates the relationship between the price of used cars (in thousands of dollars) and their age (in years) for 5 cars. The data is given below.
| Car | Age (years) | Price (thousands) |
|---|---|---|
| A | 2 | 20 |
| B | 5 | 15 |
| C | 3 | 18 |
| D | 1 | 22 |
| E | 4 | 16 |
Explain why Pearson's product moment correlation coefficient may not be appropriate.
Calculate Spearman's rank correlation coefficient for the data.
Sketch a scatter graph of the ranks.