Practice AHL 4.15—Central limit theorem 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.
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.
The daily electricity consumption of households in a town is normally distributed with a mean of 15 kWh and a standard deviation of 3 kWh . A sample of 50 households is taken.
Find the probability that the sample mean consumption is more than 16 kWh .
Find the confidence interval for the population mean based on a sample mean of 15.5 kWh .
State the assumptions made in applying the Central Limit Theorem.
A factory produces resistors with resistances normally distributed with mean ohms and variance . A random sample of 81 resistors is taken to test against at the significance level. The sample mean is 99.2 ohms.
State the Central Limit Theorem as applied to the sample mean of this distribution.
Find the critical region for the hypothesis test, correct to two decimal places.
Calculate the -value for the test and state the conclusion.
If the true mean resistance is 98 ohms, find the probability of a Type II error.
Determine the sample size required to ensure the probability of a Type II error is less than 0.1 when the true mean is 98 ohms, with the test conducted at the significance level.
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 for the current sample.
Determine the minimum sample size required to meet the company's requirement.
A competitor claims their mean delivery time is 43 minutes. Using the sample mean of 46 minutes, perform a hypothesis test at the significance level to test if the courier's mean is 45 minutes against the alternative that it is greater than 45 minutes. Include a sketch of the distribution with the critical region shaded.
If the true mean is 46 minutes, calculate the power of the test in part (c).
A factory produces light bulbs with lifetimes normally distributed with a mean of 1200 hours and a standard deviation of 150 hours. A random sample of 36 bulbs is tested.
Find the probability that the sample mean lifetime is less than 1150 hours.
The manufacturer claims the mean lifetime is 1200 hours. Perform a hypothesis test at the significance level to test this claim if the sample mean is 1150 hours. State the null and alternative hypotheses and the conclusion.
The volumes of juice cartons produced by a machine are normally distributed with a mean of 1.02 liters and a standard deviation of 0.05 liters. A random sample of 49 cartons is taken.
State why the Central Limit Theorem applies to the sample mean.
Find the probability that the sample mean volume is greater than 1.03 liters.
Find the sample size required so that the probability that the sample mean is within 0.01 liters of the population mean is at least 0.99.
A machine fills bottles with a mean volume of 500 ml and a standard deviation of 20 ml , normally distributed. A sample of 100 bottles is inspected.
Find the probability that the sample mean volume is less than 498 ml .
Perform a hypothesis test at the significance level to test if the mean volume is 500 ml if the sample mean is 498 ml .
Sketch the distribution of the sample mean, shading the critical region for part (b).
A coffee machine dispenses coffee with volumes normally distributed with a mean of 200 ml and a standard deviation of 15 ml . A random sample of 36 cups is taken.
Find the probability that the sample mean volume is within 5 ml of the true mean.
If the sample mean is 195 ml , test at the significance level whether the machine is dispensing the correct mean volume.
Sketch the distribution for part (b), indicating the critical region.
A machine fills jars with jam, with weights normally distributed with mean and standard deviation 12 g . A random sample of 36 jars has a mean weight of 502 g . The manufacturer wants to adjust the machine so that the probability of a jar weighing less than 480 g is 0.001 .
Find the probability that a single jar weighs less than 480 g , assuming .
Determine the new mean required to achieve the manufacturer's goal, keeping the standard deviation at 12 g .
Using the sample mean of 502 g , perform a hypothesis test at the significance level to test if the mean weight is 500 g against the alternative that it is greater than 500 g . Include a sketch of the distribution with the critical region shaded.
If the true mean is 503 g , calculate the probability of a Type II error for the test in part (c).
Find the minimum sample size needed to detect a true mean of 503 g with a power of at least 0.9 in the test from part (c).
The breaking strength of cables produced by a manufacturer is normally distributed with a mean of 1200 N and a standard deviation of 80 N . A random sample of 64 cables is tested, and the sample data is summarized as and , where is the breaking strength of a cable in newtons.
Find unbiased estimates for the population mean and variance of the breaking strength.
Using the Central Limit Theorem, state the distribution of the sample mean, including its mean and variance.
Construct a confidence interval for the population mean based on the sample data.
The manufacturer claims the mean breaking strength is 1200 N . Perform a hypothesis test at the significance level to test this claim, using the sample mean.
If the true mean breaking strength is 1190 N , calculate the probability of a Type II error for the test in part (d).