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:
(a) a test to assess whether the swimming speeds are normally distributed.
(b) the correlation coefficient, .
(e) the meaning of in context.
(f) Conduct a one-sample -test at the significance level to if the sample mean sunlight hours differs from 6.5 hours. hypotheses and conclusion. Use the sample standard deviation and the degrees of freedom.
| 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.
(d) Find the linear regression equation of on .
(g) 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 |
Assume a linear model and test whether the sample mean sunlight hours align with a regional average of 6.5 hours.
(a) what is meant by criterion validity in this context.
(c) whether sunlight hours are a valid predictor of growth rate.
(d) Find the coefficients and .
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.
(c) why combining categories may be necessary when performing a chi-squared goodness-of-fit test.
Trays at a warehouse each contain five ceramic mugs. Each mug is either intact or cracked. It is hypothesized that the number of cracked mugs in a tray follows a binomial distribution. A sample of 120 trays was inspected with the following results.
| Number of cracked mugs per tray | 0 | 1 | 2 | 3 | 4 | 5 |
|---|---|---|---|---|---|---|
| Frequency | 22 | 43 | 34 | 16 | 4 | 1 |
(a) Find the arithmetic mean number of cracked mugs per tray.
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 |
(a) the specific term used for this method of determining reliability.
(b) a drawback associated with applying this reliability test.
(c) Pearson's product-moment correlation coefficient for the given data.
(d) Based on your result, comment on the evidence for the reliability of the quiz.
(e) the speed at .
(f) Test if the sample mean temperature differs from at the significance level. hypotheses and conclusion.
(b) Pearson's correlation coefficient, .
(a) the mean number of cracked eggs per carton and the value of .
(b) Conduct a chi-squared goodness-of-fit test at the significance level to whether a binomial distribution is an appropriate model. the hypotheses, expected frequencies, and your conclusion.
(d) Without performing further calculations, the effect on the degrees of freedom if the value of had been specified by Luis beforehand rather than estimated from the data.
(b) Thus, an estimate for , the success probability of choosing a cracked mug.
(c) Using a goodness of fit test, at the 5% significance level if the binomial model is an appropriate fit for this dataset.
| 2.8 |
