Practice IB Mathematics Applications & Interpretation (AI) Topic AHL 4.12—data Collection, Reliability and Validity Tests with authentic exam-style questions for both SL and HL students. This question bank focuses on the exact syllabus content for AHL 4.12—data Collection, Reliability and Validity Tests and mirrors Paper 1, 2, 3 style where relevant.
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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 |
Assume a linear model. Test whether the sample mean temperature equals . Raj also assesses whether the swimming speeds are normally distributed.
State a test to assess whether the swimming speeds are normally distributed.
Calculate the correlation coefficient, .
Conduct a one-tailed test at the significance level for positive correlation. State hypotheses and conclusion.
Find the linear regression equation of on .
Predict the speed at .
Test if the sample mean temperature differs from at the significance level. State hypotheses and conclusion.
Suggest one way to improve the validity of Raj's study.
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 |
Assume a linear model and test whether the sample mean sunlight hours align with a regional average of 6.5 hours.
Define what is meant by criterion validity in this context.
Calculate Pearson's correlation coefficient, .
Justify whether sunlight hours are a valid predictor of growth rate.
Find the coefficients and .
Interpret the meaning of in context.
Conduct a one-sample -test at the significance level to determine if the sample mean sunlight hours differs from 6.5 hours. State hypotheses and conclusion. Use the sample standard deviation and state the degrees of freedom.
A supermarket manager, Luis, believes the number of cracked eggs in a carton of 6 follows a binomial distribution . He inspects 100 cartons and records the results in the following table:
| Number of Cracked Eggs | Number of Cartons |
|---|---|
| 0 | 28 |
| 1 | 36 |
| 2 | 24 |
| 3 or more | 12 |
For the purpose of estimation in this question, treat “3 or more” as 3 cracked eggs.
Calculate the mean number of cracked eggs per carton and estimate the value of .
Conduct a chi-squared goodness-of-fit test at the significance level to determine whether a binomial distribution is an appropriate model. State the hypotheses, expected frequencies, and your conclusion.
Explain why combining categories may be necessary when performing a chi-squared goodness-of-fit test.
Without performing further calculations, describe the effect on the degrees of freedom if the value of had been specified by Luis beforehand rather than estimated from the data.
An online language school is trialling a listening-placement quiz for new students. Two equivalent versions were completed by the same students four days apart. Their scores out of 40 are shown.
| Student | A | B | C | D | E | F | G |
|---|---|---|---|---|---|---|---|
| Version P | 11 | 17 | 20 | 26 | 30 | 34 | 39 |
| Version Q | 12 | 21 | 18 | 30 | 26 | 34 | 29 |
Identify the specific term used for this method of determining reliability.
Identify a drawback associated with applying this reliability test.
Compute the value of Pearson’s product-moment correlation coefficient for the given data.
Based on your result, justify whether or not the survey effectively demonstrates reliability.
A survey recorded the diet type and age group of 125 people. The results are shown in the table.
| Diet Type | 11–13 | 14–15 | 16–17 | 18–19 |
|---|---|---|---|---|
| Vegetarian | 8 | 15 | 8 | 7 |
| Vegan | 20 | 10 | 8 | 6 |
| Meat | 14 | 20 | 6 | 3 |
State suitable hypotheses for a test for independence.
Explain why the last two age groups (16–17 and 18–19) need to be combined.
Conduct a test for independence, using a 5% significance level. State your conclusion in context.