Practice AHL 4.18—T and Z test, type I and II errors 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.
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.
State the hypotheses and find the critical region.
If the sample mean is 195 hours, calculate the p-value and state the conclusion.
If the true mean is 190 hours, find the probability of a Type II error.
A fishing company claims that the average number of crabs caught per trap is 4 , modeled by a Poisson distribution. A researcher tests whether the average is less than 4 by setting a rule: if fewer than 3 crabs are caught in a trap, the claim is rejected.
State 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.
A bakery claims that only of their bread loaves are underweight. A quality inspector examines a random sample of 250 loaves and finds 8 underweight. Perform a one-tailed hypothesis test at the 10% significance level to determine if the proportion of underweight loaves exceeds .
Identify the type of sampling used by the inspector.
State the null and alternative hypotheses.
Find the p -value for the test.
State the conclusion of the test with a reason.
A coffee shop claims that their espresso shots have a mean caffeine content of 80 mg with a variance of . A researcher takes a sample of 25 shots to test if the mean is different from 80 mg at the 10% significance level.
Perform the hypothesis test if the sample mean is 83.5 mg .
Calculate the p-value for the test.
If the true mean is 82 mg , find the probability of a Type II error.
Illustrate the test with a graph showing the null distribution and critical regions.
The weights of male penguins in a colony are normally distributed with a mean of 5 kg and a standard deviation of 0.4 kg , while female penguin weights are normally distributed with a mean of 5.8 kg and a standard deviation of 0.6 kg . A biologist classifies a penguin as female if its weight exceeds 5.5 kg , assuming it is male otherwise. Assume of penguins in the colony are male.
Find the probability of a Type I error when weighing a male penguin.
Find the probability of a Type II error when weighing a female penguin.
Calculate the overall probability of misclassification.
Sketch the distributions for male and female penguin weights, indicating the classification threshold.
A supermarket claims that of customers use a loyalty card. A survey of 150 customers finds 36 using the card. Test if the proportion is greater than at the significance level.
State the hypotheses and perform the z-test.
Calculate the p-value for the test.
If the true proportion is , find the probability of a Type II error.
A school claims that students' test scores follow a normal distribution with mean 75 and variance 100. A sample of 16 students is taken to test if the mean is greater than 75 at the significance level.
Find the critical region for the test.
If the sample mean is 79 , perform the hypothesis test.
If the true mean is 78 , calculate the probability of a Type II error.
Sketch the null and alternative distributions, showing the critical region.
A company claims that their product's defect rate is . A quality control team samples 200 items and finds 10 defects. Test if the defect rate is higher than at the significance level.
State the hypotheses and perform the z-test.
If the true defect rate is , find the probability of a Type II error.
Illustrate the test with a graph of the sampling distribution.
A university claims that scores on a programming exam follow a normal distribution with a mean of 65 and a variance of 144 . A random sample of 100 scores is taken, with a sample mean of 67.2 and a sample variance of 160 . The top of scores receive a distinction.
Find unbiased estimates for the population mean and variance.
Perform a z-test at the significance level to determine if the mean score differs from 65.
Use the normal distribution to find the score required for a distinction.
If the true mean is 66, find the probability of a Type II error.
A machine fills coffee capsules with a mean weight of 10 grams and a standard deviation of 0.5 grams, assumed normally distributed. To check if the mean weight remains 10 grams, a sample of 9 capsules is taken, and a two-tailed test is conducted at the significance level. The critical region is defined as or .
Find the significance level of this test.
If the true mean weight is 10.3 grams, find the probability of a Type II error.
Sketch the sampling distribution under the null hypothesis, indicating the critical regions.