Practice Statistics and Probability 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 1
The time taken to process an online order at a warehouse follows a normal distribution with a mean of 12 minutes and a standard deviation of 3 minutes. A sample of 25 orders is randomly selected.
1.[2]
State the distribution of the sample mean, including its mean and standard deviation.
2.[3]
Find the probability that the sample mean processing time is more than 13 minutes.
3.[4]
Determine the value such that there is a probability that the sample mean lies within minutes of the population mean.
Question 2
HLPaper 1
A system transitions between states , and with transition matrix:
1.[3]
Find an eigenvector corresponding to the eigenvalue .
2.[2]
Using part (a), determine the long-term probability of being in state Y.
Question 3
HLPaper 1
A random variable is normally distributed with mean 10 and variance 4 . A linear transformation is defined as , where and are independent observations of .
1.[2]
Find the expected value of .
2.[2]
Calculate the variance of .
3.[3]
Find the probability that is between 45 and 55 .
4.[3]
Sketch the distribution of , shading the region where .
5.[2]
Show that the estimator is unbiased for the mean of .
Question 4
HLPaper 2
A factory produces batteries with a claimed mean lifespan of 200 hours and a standard deviation of 20 hours. A sample of 50 batteries is tested at the significance level to see if the mean lifespan is less than 200 hours.
1.[4]
State the hypotheses and find the critical region.
2.[4]
If the sample mean is 195 hours, calculate the p-value and state the conclusion.
3.[3]
If the true mean is 190 hours, find the probability of a Type II error.
Question 5
HLPaper 2
A network of paths connects four nodes, A, B, C, and D, as shown:
Dashed lines indicate paths that are twice as likely to be chosen.
1.[3]
Construct the transition matrix for a particle moving between the nodes.
2.[3]
Find the steady-state probability distribution using a graphic display calculator.
3.[2]
Explain one limitation of this model in the context of path selection.
Question 6
HLPaper 2
A zoo has three habitats, H, I, and J, connected as shown:
An animal moves between habitats with equal probability for each available path.
1.[3]
Write down the transition matrix.
2.[3]
If the animal starts in habitat H , find the probability it is in habitat J after three moves.
3.[2]
Find the long-term probability of the animal being in habitat I.
Question 7
HLPaper 2
A quality control officer tests the resistance (in ohms) of resistors produced in a factory, which follows a normal distribution with mean 100 ohms and variance 16 ohms . A sample of 12 resistors is taken, with ohms and ohms .
1.[3]
Calculate an unbiased estimate for the population variance.
2.[3]
Construct a confidence interval for the population mean resistance, using the sample standard deviation.
3.[2]
The factory claims the mean resistance is 98 ohms. Comment on the validity of this claim using the confidence interval.
4.[3]
If the total resistance of three resistors is measured, find the probability that it exceeds 305 ohms.
Question 8
HLPaper 1
The daily electricity consumption of households in a town is normally distributed with a mean of 15 kWh and a standard deviation of 3 kWh . A sample of 50 households is taken.
1.[3]
Find the probability that the sample mean consumption is more than 16 kWh .
2.[4]
Find the confidence interval for the population mean based on a sample mean of 15.5 kWh .
3.[2]
State the assumptions made in applying the Central Limit Theorem.
Question 9
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 10
HLPaper 2
A sample of 9 households records their monthly water usage (in liters): 200, 210, 220, 230, 240, 250, 260, 270, 280.
1.[2]
Calculate the sample mean, , and the sample sample standard deviation, .
2.[3]
Find the confidence interval for the population mean, , assuming the population standard deviation is unknown.
3.[2]
Sketch the confidence interval on a number line.
4.[1]
Interpret the confidence interval in the context of the problem.