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.
A factory, producing plastic gifts for a fast food restaurant’s Jolly meals, claims that just ofthe toys produced are faulty.
A restaurant manager wants to test this claim. A box of toys is delivered to the restaurant.The manager checks all the toys in this box and four toys are found to be faulty.
The restaurant manager performs a one-tailed hypothesis test, at the significance level,to determine whether the factory’s claim is reasonable. It is known that faults in the toysoccur independently.
Identify the type of sampling used by the restaurant manager.
Write down the null and alternative hypotheses.
Find the -value for the test.
State the conclusion of the test. Give a reason for your answer.
The proportion of students who participate in extracurricular activities is known to be 55% nationally. Alex suspects that the proportion in his school might be higher. Out of a random sample of 40 students, he finds that 28 of them participate in extracurricular activities.
Is there evidence, at the 5% significance level, that Alex’s suspicion is justified?
The proportion of people who prefer online shopping over in-store shopping is known to be 60% nationally. Maria suspects that the proportion in her city might be different. Out of a random sample of 50 people, she finds that 35 of them prefer online shopping.
Is there evidence, at the 1% significance level, that Maria’s suspicion is justified?
Linda is a farmer who grows and sells zucchinis. Interested in the weights of zucchinis produced, she records the weights of eight zucchinis and obtains the following results in kilograms.
Assume that these weights form a random sample from a distribution.
Linda claims that the mean zucchini weight is 7.5 kilograms. In order to test this claim, she sets up the null hypothesis .
Determine unbiased estimates for and .
Use a two-tailed test to determine the -value for the above results.
Interpret your -value at the 5% level of significance, justifying your conclusion.
The times taken for male runners to complete a marathon can be modelled by a normal distribution with a mean 196 minutes and a standard deviation 24 minutes.
It is found that 5% of the male runners complete the marathon in less than minutes.
The times taken for female runners to complete the marathon can be modelled by a normal distribution with a mean 210 minutes. It is found that 58% of female runners complete the marathon between 185 and 235 minutes.
Find the probability that a runner selected at random will complete the marathon in less than 3 hours.
Calculate .
Find the standard deviation of the times taken by female runners.
The number of fish that can be caught in one hour from a particular lake can be modelled bya Poisson distribution.
The owner of the lake, Emily, states in her advertising that the average number of fish caughtin an hour is three.
Tom, a keen fisherman, is not convinced and thinks it is less than three. He decides to setup the following test. Tom will fish for one hour and if he catches fewer than two fish he willreject Emily’s claim.
State a suitable null and alternative hypotheses for Tom’s test.
Find the probability of a Type I error.
The average number of fish caught in an hour is actually 2.5.
Find the probability of a Type II error.