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 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 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 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 researcher is studying the effect of a new drug on blood pressure. The population standard deviation of blood pressure reduction is known to be . A sample of patients is treated with the drug, and the mean reduction in blood pressure is found to be .
Calculate a confidence interval for the mean reduction in blood pressure.
The researcher is concerned that the blood pressure reductions may not be normally distributed. State, with a reason, whether the calculation of the confidence interval in Part 1 is still valid.
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 weights, in grams, of individual packets of coffee can be modelled by a normal distribution, with mean $102 \text{ g}$ and standard deviation $8 \text{ g}$.
Find the probability that a randomly selected packet has a weight less than $100 \text{ g}$.
The probability that a randomly selected packet has a weight greater than $w$ grams is $0.444$. Find the value of $w$.
A packet is randomly selected. Given that the packet has a weight greater than $105 \text{ g}$, find the probability that it has a weight greater than $110 \text{ g}$.
From a random sample of $500$ packets, determine the number of packets that would be expected to have a weight lying within $1.5$ standard deviations of the mean.
Packets are delivered to supermarkets in batches of $80$. Determine the probability that at least $20$ packets from a randomly selected batch have a weight less than $95 \text{ g}$.
Let be a random variable which follows a normal distribution with mean . Given that , find
.
.
Packets of cookies are produced by a device. The weights , in grams, of packets of cookies can be modelled by a normal distribution where . A packet of cookies is considered to be underweight if it weighs less than grams.
Given that and , find the probability that a randomly chosen packet of cookies is underweight.
The producer makes the decision that the probability that a packet is underweight should be . To do this, is increased and remains unchanged.
Calculate the new value of , giving your answer correct to two decimal places.
The producer is happy with the decision that the probability that a packet is underweight should be , but is unhappy with the way in which this was achieved. The device is now adjusted to reduce and return to .
Calculate the new value of .
John rings a church bell 120 times. The time interval, $T_i$, between two successive rings is a random variable with mean of 2 seconds and variance of $\frac{1}{9} \text{ s}^2$.
Each time interval, $T_i$, is independent of the other time intervals. Let $X = \sum_{i = 1}^{119} T_i$ be the total time between the first ring and the last ring.
The church vicar subsequently becomes suspicious that John has stopped coming to ring the bell and that he is letting his friend Ray do it. When Ray rings the bell the time interval, $T_i$, has a mean of 2 seconds and variance of $\frac{1}{25} \text{ s}^2$.
The church vicar makes the following hypotheses:
$H_0$: Ray is ringing the bell; $H_1$: John is ringing the bell.
He records four values of $X$. He decides on the following decision rule:
If $236 \leqslant X \leqslant 240$ for all four values of $X$ he accepts $H_0$, otherwise he accepts $H_1$.
Assuming that John is ringing the bell, find:
(i) $E(X)$;
(ii) $\text{Var}(X)$.
Explain why a normal distribution can be used to give an approximate model for $X$.
Use this model to find the values of $A$ and $B$ such that $P(A < X < B) = 0.9$, where $A$ and $B$ are symmetrical about the mean of $X$.
Calculate the probability that he makes a Type II error.
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 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 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 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 researcher is studying the effect of a new drug on blood pressure. The population standard deviation of blood pressure reduction is known to be . A sample of patients is treated with the drug, and the mean reduction in blood pressure is found to be .
Calculate a confidence interval for the mean reduction in blood pressure.
The researcher is concerned that the blood pressure reductions may not be normally distributed. State, with a reason, whether the calculation of the confidence interval in Part 1 is still valid.
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 weights, in grams, of individual packets of coffee can be modelled by a normal distribution, with mean $102 \text{ g}$ and standard deviation $8 \text{ g}$.
Find the probability that a randomly selected packet has a weight less than $100 \text{ g}$.
The probability that a randomly selected packet has a weight greater than $w$ grams is $0.444$. Find the value of $w$.
A packet is randomly selected. Given that the packet has a weight greater than $105 \text{ g}$, find the probability that it has a weight greater than $110 \text{ g}$.
From a random sample of $500$ packets, determine the number of packets that would be expected to have a weight lying within $1.5$ standard deviations of the mean.
Packets are delivered to supermarkets in batches of $80$. Determine the probability that at least $20$ packets from a randomly selected batch have a weight less than $95 \text{ g}$.
Let be a random variable which follows a normal distribution with mean . Given that , find
.
.
Packets of cookies are produced by a device. The weights , in grams, of packets of cookies can be modelled by a normal distribution where . A packet of cookies is considered to be underweight if it weighs less than grams.
Given that and , find the probability that a randomly chosen packet of cookies is underweight.
The producer makes the decision that the probability that a packet is underweight should be . To do this, is increased and remains unchanged.
Calculate the new value of , giving your answer correct to two decimal places.
The producer is happy with the decision that the probability that a packet is underweight should be , but is unhappy with the way in which this was achieved. The device is now adjusted to reduce and return to .
Calculate the new value of .
John rings a church bell 120 times. The time interval, $T_i$, between two successive rings is a random variable with mean of 2 seconds and variance of $\frac{1}{9} \text{ s}^2$.
Each time interval, $T_i$, is independent of the other time intervals. Let $X = \sum_{i = 1}^{119} T_i$ be the total time between the first ring and the last ring.
The church vicar subsequently becomes suspicious that John has stopped coming to ring the bell and that he is letting his friend Ray do it. When Ray rings the bell the time interval, $T_i$, has a mean of 2 seconds and variance of $\frac{1}{25} \text{ s}^2$.
The church vicar makes the following hypotheses:
$H_0$: Ray is ringing the bell; $H_1$: John is ringing the bell.
He records four values of $X$. He decides on the following decision rule:
If $236 \leqslant X \leqslant 240$ for all four values of $X$ he accepts $H_0$, otherwise he accepts $H_1$.
Assuming that John is ringing the bell, find:
(i) $E(X)$;
(ii) $\text{Var}(X)$.
Explain why a normal distribution can be used to give an approximate model for $X$.
Use this model to find the values of $A$ and $B$ such that $P(A < X < B) = 0.9$, where $A$ and $B$ are symmetrical about the mean of $X$.
Calculate the probability that he makes a Type II error.