Practice IB Mathematics Applications & Interpretation (AI) Topic AHL 4.15—central Limit Theorem with authentic exam-style questions for both SL and HL students. This question bank focuses on the exact syllabus content for AHL 4.15—central Limit Theorem and mirrors Paper 1, 2, 3 style where relevant.
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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.
State the distribution of the sample mean, including its mean and standard deviation.
Find the probability that the sample mean processing time is more than 13 minutes.
Determine the value such that there is a probability that the sample mean lies within minutes of the population mean.
A snack manufacturer fills granola bags whose masses are normally distributed with a mean of and a standard deviation of . A sample of bags is chosen.
Find the probability that the total mass of the bags exceeds .
The manufacturer claims that the mean mass is . If the sample mean is , perform a hypothesis test at the significance level to test this claim.
Sketch the distribution used in part 2, indicating the critical region.
The daily electricity consumption of households in a town is normally distributed with a mean of kWh and a standard deviation of kWh. A sample of households is taken.
Find the probability that the sample mean consumption is more than kWh.
Find the confidence interval for the population mean based on a sample mean of kWh.
State the assumptions made in applying the Central Limit Theorem.
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.
A courier service records delivery times for packages, normally distributed with a mean of 45 minutes and a standard deviation of 10 minutes. A random sample of 100 deliveries is taken, and the company wants to ensure that the probability of the sample mean exceeding 47 minutes is less than 0.01.
Find the probability that the sample mean delivery time exceeds 47 minutes.
Determine the minimum sample size required to meet the company's requirement.
Using the sample mean minutes, perform a one-tailed -test at the significance level to test whether the courier service’s mean delivery time is 45 minutes against the alternative that it is greater than 45 minutes. Include a sketch of the standard normal distribution with the critical region shaded.
If the true mean is 46 minutes, calculate the power of the test in part 3.