Practice AHL 4.12—Data collection, reliability and validity tests 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.
Paper
Level
Question 1
HLPaper 2
A marine biologist, Raj, investigates whether water temperature ( ) affects the swimming speed (m/s) of a fish species. He observes 10 fish in controlled tanks and records:
| Fish | Temperature (X) | Speed (Y) |
|---|---|---|
| 1 | 15 | 0.8 |
| 2 | 16 | 0.9 |
| 3 | 18 | 1.0 |
| 4 | 20 | 1.2 |
| 5 | 22 | 1.3 |
| 6 | 24 | 1.5 |
| 7 | 26 | 1.6 |
| 8 | 28 | 1.7 |
| 9 | 30 | 1.8 |
| 10 | 32 | 1.9 |
Raj assumes a linear model and tests if the sample mean temperature aligns with an ocean average of . He also checks if speeds are normally distributed.
1.[1]
Name a test to verify normality of swimming speeds.
2.[2]
Calculate the correlation coefficient, .
3.[3]
Conduct a one-tailed test at the significance level for positive correlation. State hypotheses and conclusion.
4.[3]
(i) Find the linear regression equation. (ii) Predict the speed at .
5.[4]
Test if the sample mean temperature differs from at the significance level. State hypotheses and conclusion.
6.[1]
Suggest one way to improve the validity of Raj's study.
Question 2
HLPaper 2
A nutritionist, Clara, investigates whether daily fiber intake (grams) influences energy levels in adults. She surveys 15 adults, recording their fiber intake and energy scores (out of 100) based on a standardized questionnaire. The data is:
| Adult | Fiber Intake (X) | Energy Score (Y) |
|---|---|---|
| 1 | 10 | 60 |
| 2 | 12 | 62 |
| 3 | 15 | 65 |
| 4 | 18 | 68 |
| 5 | 20 | 70 |
| 6 | 22 | 72 |
| 7 | 25 | 75 |
| 8 | 28 | 78 |
| 9 | 30 | 80 |
| 10 | 32 | 82 |
| 11 | 35 | 85 |
| 12 | 38 | 88 |
| 13 | 40 | 90 |
| 14 | 42 | 92 |
| 15 | 45 | 95 |
Clara assumes a linear model and tests if the sample mean fiber intake aligns with a population mean of 25 grams. She also checks for normality of energy scores.
1.[1]
Name a test to verify if energy scores are normally distributed.
2.[2]
Calculate Pearson's correlation coefficient, .
3.[3]
Conduct a one-tailed test at the significance level to determine if fiber intake positively correlates with energy scores. State hypotheses and conclusion.
4.[3]
(i) Find the linear regression coefficients and . (ii) Predict the energy score for a fiber intake of 27 grams.
5.[4]
Test if the sample mean fiber intake differs from 25 grams at the significance level. State the test, hypotheses, and conclusion.
6.[1]
Suggest one method to improve the reliability of Clara's study.
Question 3
HLPaper 1
A researcher, Omar, studies whether the number of hours spent on meditation per week affects anxiety levels in employees. He collects data from 10 employees:
| Employee | Meditation Hours (X) | Anxiety Score (Y) |
|---|---|---|
| 1 | 1 | 80 |
| 2 | 2 | 75 |
| 3 | 3 | 70 |
| 4 | 4 | 65 |
| 5 | 5 | 60 |
| 6 | 6 | 55 |
| 7 | 7 | 50 |
| 8 | 8 | 45 |
| 9 | 9 | 40 |
| 10 | 10 | 35 |
Omar assumes a linear model and tests if the sample mean meditation hours align with a company average of 5 hours.
1.[2]
Define what is meant by a criterion-related validity test in this context.
2.[2]
Calculate the correlation coefficient, .
3.[2]
Justify whether meditation hours are a valid predictor of anxiety levels.
4.[3]
(i) Find the linear regression equation. (ii) Interpret in context.
5.[4]
Test if the sample mean meditation hours differ from 5 hours at the significance level. State hypotheses and conclusion.
Question 4
HLPaper 2
A farmer claims defective apples in crates of 10 follow . She inspects 100 crates:
| Defective Apples | Crates |
|---|---|
| 0 | 20 |
| 1 | 35 |
| 2 | 25 |
| 3 | 15 |
| 4 | 5 |
1.[2]
Estimate .
2.[6]
Test at significance if the data follows . State hypotheses, expected frequencies, and conclusion.
Question 5
HLPaper 2
A dairy farmer, Sofia, believes the number of defective milk bottles in a batch of 10 follows a binomial distribution . She inspects 80 batches:
| Defective Bottles | Batches |
|---|---|
| 0 | 25 |
| 1 | 30 |
| 2 | 20 |
| 3 | 5 |
1.[2]
Calculate the mean number of defective bottles and estimate .
2.[6]
Conduct a chi-squared test at the significance level to verify the binomial distribution. State hypotheses, expected frequencies, and conclusion.
3.[2]
Explain why combining categories may be necessary in this test.
4.[2]
Without further calculations, describe how to test for with unspecified .
Question 6
HLPaper 2
A forester, Mia, hypothesizes that the diameters (cm) of trees in a forest follow a normal distribution. She measures 150 trees, with summary statistics: 141750. The data is grouped:
| Diameter | Frequency |
|---|---|
| 10 | |
| 35 | |
| 50 | |
| 40 | |
| 15 |
1.[3]
Calculate unbiased estimates of the population mean and variance.
2.[1]
State hypotheses for a chi-squared goodness of fit test.
3.[5]
Calculate expected frequencies for the normal distribution and the chi-squared statistic. Conclude at the significance level.
4.[2]
Describe the purpose of a goodness of fit test in this context.
Question 7
HLPaper 2
Emma examines if daily water intake (liters) affects cognitive performance in office workers. She collects data from 10 workers:
| Worker | Water Intake (X) | Cognitive Score (Y) |
|---|---|---|
| 1 | 1.0 | 60 |
| 2 | 1.5 | 65 |
| 3 | 2.0 | 70 |
| 4 | 2.5 | 75 |
| 5 | 3.0 | 80 |
| 6 | 3.5 | 85 |
| 7 | 4.0 | 88 |
| 8 | 4.5 | 90 |
| 9 | 5.0 | 92 |
| 10 | 5.5 | 94 |
Emma uses a linear model and tests if the sample mean water intake matches a population mean of 2.5 liters. She checks normality of cognitive scores.
1.[1]
Name a test for normality of cognitive scores.
2.[2]
Calculate .
3.[3]
Test at significance for positive correlation between water intake and cognitive scores. State hypotheses and conclusion.
4.[3]
(i) Find the linear regression equation. (ii) Predict the cognitive score for 3.2 liters.
5.[4]
Test if the sample mean water intake differs from 2.5 liters at significance.
Question 8
HLPaper 1
A botanist, Elena, studies whether the daily sunlight exposure (hours) of a certain plant species affects its growth rate (cm per week). She randomly selects 12 plants and records their sunlight exposure and growth rate. The data is:
| Plant | Sunlight Hours (X) | Growth Rate (Y) |
|---|---|---|
| 1 | 4.0 | 1.2 |
| 2 | 4.5 | 1.3 |
| 3 | 5.0 | 1.5 |
| 4 | 5.5 | 1.6 |
| 5 | 6.0 | 1.8 |
| 6 | 6.5 | 2.0 |
| 7 | 7.0 | 2.1 |
| 8 | 7.5 | 2.3 |
| 9 | 8.0 | 2.4 |
| 10 | 8.5 | 2.6 |
| 11 | 9.0 | 2.7 |
| 12 | 9.5 | 2.8 |
Elena assumes a linear model and tests whether the sample mean sunlight hours align with a regional average of 6.5 hours.
1.[2]
Define what is meant by a criterion-related validity test in this context.
2.[2]
Calculate Pearson's correlation coefficient, .
3.[2]
Determine, with justification, whether sunlight hours are a valid predictor of growth rate.
4.[3]
(i) Find the coefficients and . (ii) Interpret the meaning of in context.
5.[4]
Conduct a -test at the significance level to determine if the sample mean sunlight hours differ from 6.5 hours. State hypotheses and conclusion.
Question 9
HLPaper 2
A farmer, Aisha, claims that the number of defective tomatoes in a basket of 8 follows a binomial distribution . She inspects 120 baskets and records:
| Defective Tomatoes | Baskets |
|---|---|
| 0 | 30 |
| 1 | 40 |
| 2 | 30 |
| 3 | 15 |
| 4 | 5 |
1.[2]
Calculate the mean number of defective tomatoes per basket and estimate .
2.[6]
Conduct a chi-squared test at the significance level to verify the binomial distribution claim. State hypotheses, expected frequencies, and conclusion.
3.[2]
Explain why combining categories might be necessary in the chi-squared test.
Question 10
HLPaper 1
Noah studies if biology study hours affect exam scores for 10 students:
| Student | Study Hours (X) | Exam Score (Y) |
|---|---|---|
| 1 | 5 | 60 |
| 2 | 7 | 65 |
| 3 | 10 | 70 |
| 4 | 12 | 75 |
| 5 | 15 | 80 |
| 6 | 18 | 85 |
| 7 | 20 | 88 |
| 8 | 22 | 90 |
| 9 | 25 | 92 |
| 10 | 30 | 95 |
Noah tests if study hours predict scores and if the sample mean study hours match a population mean of 15 hours.
1.[1]
Name the validity test type.
2.[2]
Calculate .
3.[2]
Justify if study hours predict scores.
4.[2]
Find the linear regression equation.
5.[4]
Test if the sample mean study hours differ from 15 hours at significance.