Practice IB Mathematics Applications & Interpretation (AI) Topic AHL 4.18—T and Z Test, Type I and II Errors with authentic exam-style questions for both SL and HL students. This question bank focuses on the exact syllabus content for AHL 4.18—T and Z Test, Type I and II Errors and mirrors Paper 1, 2, 3 style where relevant.
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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) suitable null and alternative hypotheses.
(b) Find the probability of a Type I error.
(e) If the probability of committing a Type II error is , the specific value of , correct to three significant figures.
(c) If the true average is 3.5 crabs per trap, find the probability of a Type II error.
A supermarket claims that of customers use a loyalty card. A survey of 150 customers finds that 36 use the card. Use a significance level of .
(a) the hypotheses and perform the z-test.
(b) the p-value for the test.
(c) If the true proportion is , find the probability of a Type II error.
A factory claims that the mean lifespan of its batteries is hours. Battery lifespans are normally distributed with a known population standard deviation of hours. A random sample of batteries is tested at the significance level to determine whether the population mean lifespan is less than hours. Assume that the battery lifespans in the sample are independent.
(a) the hypotheses and find the critical region. Give the boundary for the sample mean correct to two decimal places.
(c) If the true mean is 190 hours, find the probability of a Type II error.
A factory seals tea in foil sachets. The mass of tea in a sachet has population mean g and variance . Independent random samples of size are taken.
The quality team suspects underfilling and tests against at a significance level of using a sample of 64 sachets.
Use the normal approximation to the sampling distribution of the sample mean where appropriate.
(c) the critical region for the quality team's hypothesis test, rounding your value to two decimal places.
(d) the likelihood that the quality team commits a Type I error.
Each of 8 microchips selected from a batch either passes a stress test or fails it independently. A supplier claims that the probability a microchip passes is 0.75. An engineer suspects this probability is lower and rejects the claim if fewer than five microchips pass.
(a) the null and alternative hypotheses for the engineer’s test.
(b) the probability of committing a Type I error.
(c) Suppose the true probability that a microchip passes the stress test is 0.5. the probability of committing a Type II error.
(b) If the sample mean is 195 hours, the -value and the conclusion.
(a) the Central Limit Theorem for the sample mean of a random sample of size drawn from a population with mean and variance .
(b) A sample of 64 sachets gives a mean mass of . a 90% confidence interval for the population mean .
