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 .
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 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 .
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 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 determine whether 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 -value and state the conclusion.
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.
Define the Central Limit Theorem for the sample mean of a random sample of size drawn from a population with mean and variance .
A sample of 64 sachets gives a mean mass of . Determine a 90% confidence interval for the population mean .
Calculate the critical region for the quality team's hypothesis test, rounding your value to two decimal places.
State the likelihood that the quality team commits a Type I error.
If the probability of committing a Type II error is , determine the specific value of , correct to three significant figures.
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.
Define the null and alternative hypotheses for the engineer’s test.
Calculate the probability of committing a Type I error.
Suppose the true probability that a microchip passes the stress test is 0.5. Determine the probability of committing a Type II error.