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
A six-sided dice is rolled 60 times, with the following results.
Outcome: 1, 2, 3, 4, 5, 6.
Frequency: 12, 15, 13, 7, 6, 7
Is there evidence, at the 5% significance level, that the dice is not fair?
1.[1]
State the null and alternative hypotheses for the chi-squared test.
2.[1]
Calculate the expected frequency for each outcome if the dice is fair.
3.[2]
Calculate the chi-squared and p statistic for the observed frequencies.
4.[2]
Determine the critical value for the chi-squared test at the 10% significance level.
Question 8
HLPaper 2
1.[5]
A six-sided dice is rolled 37 times, with the following results.
Outcome 1, 2, 3, 4, 5, 6
Frequency 8, 3, 7, 9, 5, 7
Is there evidence, at the 10% significance level, that the dice is not fair?
Question 9
HLPaper 1
A factory, producing plastic gifts for a fast food restaurant’s Jolly meals, claims that just ofthe toys produced are faulty.
A restaurant manager wants to test this claim. A box of toys is delivered to the restaurant.The manager checks all the toys in this box and four toys are found to be faulty.
The restaurant manager performs a one-tailed hypothesis test, at the significance level,to determine whether the factory’s claim is reasonable. It is known that faults in the toysoccur independently.
1.[1]
Identify the type of sampling used by the restaurant manager.
2.[2]
Write down the null and alternative hypotheses.
3.[2]
Find the -value for the test.
4.[2]
State the conclusion of the test. Give a reason for your answer.
Question 10
HLPaper 2
| Diet Type | 11-13 | 14-15 | 16-17 | 17-18 |
|---|---|---|---|---|
| Vegetarian | 8 | 15 | 8 | 7 |
| Vegan | 20 | 10 | 8 | 6 |
| Meat | 14 | 20 | 6 | 3 |
1.[2]
State suitable hypotheses for a 2 test for independence.
2.[1]
Explain why the last two columns of the table need to be combined.
3.[5]
Conduct a 2 test for independence, using a 5% significance level. State your conclusion in context.