IB Mathematics Applications & Interpretation Topic…

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.

the distribution of the sample mean, including its mean and standard deviation.

the value $k$ such that there is a $95 \%$ probability that the sample mean lies within $k$ minutes of the population mean.

A snack manufacturer fills granola bags whose masses are normally distributed with a mean of $600\text{ g}$ and a standard deviation of $40\text{ g}$ . A sample of $64$ bags is chosen.

Find the probability that the total mass of the $64$ bags exceeds $39.04\text{ kg}$ .

The manufacturer claims that the mean mass is $600\text{ g}$ . If the sample mean is $592\text{ g}$ , perform a hypothesis test at the $10\%$ significance level to test this claim.

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 $15$ 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 $90\%$ confidence interval for the population mean based on a sample mean of $15.5$ kWh.

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 $\mu$ g and variance $1.44$ . Independent random samples of size $n=64$ are taken.

The quality team suspects underfilling and tests $H_0: \mu=7.5$ against $H_1: \mu<7.5$ at a significance level of $2.5\%$ using a sample of 64 sachets.

the Central Limit Theorem for the sample mean of a random sample of size $n$ drawn from a population with mean $\mu$ and variance $\sigma^2$ .

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.

the minimum sample size required to meet the company's requirement.

Using the sample mean $\bar{x}=46$ minutes, perform a one-tailed $z$ -test at the $5\%$ 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 of the standard normal distribution with the critical region shaded.

If the true mean is 46 minutes, the power of the test in part 3.

the critical region for the quality team's hypothesis test, rounding your value to two decimal places.

the likelihood that the quality team commits a Type I error.

If the probability of committing a Type II error is $0.10$ , the specific value of $\mu$ , correct to three significant figures.

IB Mathematics Applications & Interpretation AHL 4.15—central Limit Theorem Questionbank

IB Mathematics Applications & Interpretation AHL 4.15—central Limit Theorem Questionbank

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IB Mathematics Applications & Interpretation AHL 4.15—central Limit Theorem Questionbank

IB Mathematics Applications & Interpretation AHL 4.15—central Limit Theorem Questionbank

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IB Mathematics Applications & Interpretation (AI) Topic AHL 4.15—central Limit Theorem Questionbank