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 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 (a) 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. mark]
State the conclusion about the relationship based on the result from part (a). mark]
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 two 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. marks]
Sketch a scatter graph of the ranks, labeling the axes appropriately.
The researcher calculates Pearson's product-moment correlation coefficient as -0.623 . Explain why this may not be appropriate for this data.
Conduct a hypothesis test at the significance level to determine if there is a 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. marks]
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.
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 university researcher studies the relationship between the number of research papers published and the citation count for 11 faculty members in two departments: Science and Humanities. The data is shown below.
| Faculty | Department | Papers Published | Citation Count |
|---|---|---|---|
| A | Science | 10 | 150 |
| B | Science | 15 | 200 |
| C | Science | 8 | 120 |
| D | Science | 12 | 180 |
| E | Science | 20 | 250 |
| F | Humanities | 5 | 80 |
| G | Humanities | 7 | 90 |
| H | Humanities | 4 | 70 |
| I | Humanities | 6 | 85 |
| J | Humanities | 3 | 60 |
| K | Science | 18 | 220 |
Calculate Spearman's rank correlation coefficient for the Science department.
Calculate Spearman's rank correlation coefficient for the Humanities department.
Conduct a hypothesis test at the significance level for the Science department. State the null and alternative hypotheses, the critical value, and the conclusion.
Explain why it may not be appropriate to calculate a single Spearman's rank correlation coefficient for all 11 faculty members. marks]
Sketch a scatter graph of the ranks for the Science department.
The citation count for faculty E is adjusted from 250 to 240. Recalculate Spearman's rank correlation coefficient for the Science department and comment on the impact.
A fitness coach investigates the relationship between the number of weekly workouts and the resting heart rate (beats per minute) for 10 clients, some with equal workout counts. The data is shown below.
| Client | Workouts | Heart Rate |
|---|---|---|
| A | 4 | 65 |
| B | 3 | 70 |
| C | 5 | 62 |
| D | 4 | 68 |
| E | 2 | 75 |
| F | 6 | 60 |
| G | 3 | 72 |
| H | 5 | 63 |
| I | 4 | 66 |
| J | 2 | 74 |
Assign ranks to the data, accounting for ties.
Calculate Spearman's rank correlation coefficient.
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 coach calculates the regression line for the original data. Estimate the heart rate for 7 workouts and explain why this estimate may not be reliable.
If the heart rate for client A is adjusted from 65 to 64, recalculate Spearman's rank correlation coefficient and comment on the impact.
In Lucy's music academy, eight students took their piano diploma examination and achieved scores out of 150. For her records, Lucy decided to record the average number of hours per week each student reported practising in the weeks prior to their examination. These results are summarized in the table below.
| Average weekly practice time (h) | 28 | 13 | 45 | 33 | 17 | 29 | 39 | 36 |
|---|---|---|---|---|---|---|---|---|
| Diploma score (D) | 115 | 82 | 120 | 116 | 79 | 101 | 110 | 121 |
Find Pearson's product-moment correlation coefficient, r, for these data.
The relationship between the variables can be modelled by the regression equation , where represents the diploma score and represents the average weekly practice time in hours. Write down the value of and the value of .
One of these eight students was disappointed with her result and wished she had practised more. Based on the given data, determine how her score could have been expected to alter had she practised an extra five hours per week.