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.
A fishing company claims that the average number of crabs caught per trap is 4, modeled by a Poisson distribution. To test , the researcher rejects if .
A city council studies the relationship between the number of public transport trips per month and the monthly expenditure on transport (in dollars) for 9 residents. The data is shown below, with some residents having equal trip counts.
A sports scientist compares sprint times (in seconds) of athletes using two training programs, A and B. Assume the data are approximately normally distributed with equal variances. Sample statistics:
Elena is modeling the decay of a radioactive substance. The mass, (in grams), remaining after days is given by:
Practice IB Mathematics Applications & Interpretation (AI) Topic Statistics and Probability with authentic exam-style questions for both SL and HL students. This question bank focuses on the exact syllabus content for Statistics and Probability and mirrors Paper 1, 2, 3 style where relevant.
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Find the linear regression equation of on .
the speed at .
Test if the sample mean temperature differs from at the significance level. hypotheses and conclusion.
one way to improve the validity of Raj's study.
suitable null and alternative hypotheses.
Find the probability of a Type I error.
If the true average is 3.5 crabs per trap, find the probability of a Type II error.
| Resident | Trips per Month | Expenditure ($) |
|---|
| A | 20 | 150 |
| B | 15 | 120 |
| C | 25 | 180 |
| D | 15 | 130 |
| E | 30 | 200 |
| F | 20 | 160 |
| G | 10 | 100 |
| H | 25 | 170 |
| I | 12 | 110 |
Assign ranks to the data, accounting for equal trip counts.
Spearman's rank correlation coefficient.
what your value of Spearman's rank correlation coefficient indicates about the relationship between trips and expenditure.
A t-test is conducted at the significance level. The scientist also examines the combined data for normality.
the null and alternative hypotheses, and why a t-test is appropriate.
the probability that an athlete from Program A runs faster than seconds.
Compute the t-test statistic and find the p-value.
Combine the samples and assess whether the combined times are normally distributed using a normal probability plot. the mean and standard deviation (in seconds), and the expected frequency for . Intervals: , . Observed frequencies: .
| 0 | 5 | 10 | 15 |
|---|
| 100 | 60.7 | 36.8 | 22.3 |
She proposes a model of the form , where is in grams and is measured in .
Show that the model satisfies the differential equation .
