4.15 The central limit theorem questions for IB Maths AI

Find the probability that a single randomly selected jar contains less than 450 g of honey.

Write down the distribution of $\bar{X}$, stating its parameters.

Find $\mathrm{P}(\bar{X}<450)$.

Find the probability that exactly 5 croissants are sold in a given hour.

The café makes a profit of 1.40 euros (EUR) on each croissant and 0.95 EUR on each muffin. The cost of running the pastry counter is 8 EUR per hour. The profit, $P$ EUR, from the pastry counter in one hour is therefore

(i) Find $\mathrm{E}(P)$.
(ii) Find $\mathrm{V}\mathrm{a}\mathrm{r}(P)$.

State one reason why $P$ does not follow a Poisson distribution.

Find the probability that at least 35 pastries in total are sold in a 3-hour period.

The café is open for 8 hours each day. Let $Q$ be the total profit from the pastry counter in one day.

Use the central limit theorem to estimate the probability that $Q$ is less than 20 EUR.

Aroha will measure a stratified sample of 60 trout, stratified by region.

Find the number of trout she should measure from each region.

Let $\bar{L}$ be the mean length, in cm, of a random sample of 60 trout from the lake, and let $\mu$ be the mean length of all the trout in the lake.

Explain why $\bar{L}$ can be modelled by a normal distribution, and state this distribution in terms of $\mu$.

Find the probability that $\bar{L}$ is within 1.5 cm of $\mu$.

Find the smallest sample size for which the probability that the sample mean is within 1.5 cm of $\mu$ is at least 0.98.

Aroha measures a random sample of 60 trout. Their mean length is 34.6 cm.

Find a 95 % confidence interval for $\mu$.

Ten years earlier, the mean length of trout in Lake Moana was 32.5 cm.

Comment on whether Aroha's interval suggests that the mean length has changed.

Aroha also studies trout in Lake Kōwhai, where the standard deviation of lengths is unknown and the lengths are normally distributed. She measures the lengths, in cm, of a random sample of 10 trout.

Find unbiased estimates of the mean and the variance of the lengths of trout in Lake Kōwhai.

Find a 90 % confidence interval for the mean length of trout in Lake Kōwhai.

Explain why a $t$-distribution, and not a normal distribution, is used for this interval.

Trout in Lake Kōwhai can only be fished commercially if their mean length is greater than 30 cm.

Use Aroha's sample to test, at the 5 % significance level, whether the mean length of trout in Lake Kōwhai is greater than 30 cm.

The price, $V$ euros (EUR), paid for a trout from Lake Moana of length $L$ cm is given by $V=0.9L-6$. Assume that $L$ has mean 34.6 cm and standard deviation 6.2 cm.

Find the mean and the standard deviation of $V$.

A box contains 25 trout from Lake Moana, whose lengths are independent.

Find the mean and the standard deviation of the total price paid for the box.

Twelve adults enter the lift. Let $T$ be their total mass, in kg. Find the mean and the variance of $T$.

Find the probability that the total mass of the twelve adults exceeds the maximum load.

The building manager wants the probability that the total mass of the adults in the lift exceeds 1000 kg to be less than 1%. Find the greatest number of adults that should be allowed in the lift at one time.

Write down the value of $p$.

Find
(i) $\mathrm{E}(X)$;
(ii) $\mathrm{V}\mathrm{a}\mathrm{r}(X)$.

Let $G$ be the charity's profit, in NZD, from one game, so that $G=6-X$. Find the mean and the standard deviation of $G$.

The game is played 200 times, and the games are independent.

(i) Find the mean and the standard deviation of the charity's total profit.
(ii) Use the central limit theorem to estimate the probability that the charity makes a loss.

The organiser, Tama, suggests doubling every prize and charging 12 NZD per game, so that the profit from one game is $H=12-2X$.

(i) Find $\mathrm{E}(H)$ and $\mathrm{V}\mathrm{a}\mathrm{r}(H)$.
(ii) Tama claims that this change will double the expected profit from 200 games without increasing the risk of a loss. Comment on his claim.

Find the probability that a randomly chosen tile is rejected.

Before inspection, tiles are packed in boxes of 25. The thicknesses of tiles are independent.

Find the expected number of tiles in a box that would be rejected.

Find the probability that a box contains at most one tile that would be rejected.

The manager, Farida, wants no more than 1 % of tiles to be rejected. The mean thickness stays at 9.2 mm.

Find the largest standard deviation of the thickness that achieves this.

The masses of the tiles are independent and normally distributed with mean 410 g and standard deviation 12 g. The mass of an empty box is normally distributed with mean 350 g and standard deviation 20 g.

Find the probability that a full box of 25 tiles has a total mass of more than 10.7 kg.

Every hour, an inspector measures the thickness of a random sample of 16 tiles and calculates the sample mean, $\bar{T}$. The machine is stopped if $\bar{T}$ lies outside the interval $9.2-k<\bar{T}<9.2+k$, where $k$ is chosen so that, when the mean thickness is 9.2 mm, $\mathrm{P}(9.2-k<\bar{T}<9.2+k)=0.99$.

Find the value of $k$.

This procedure is a hypothesis test about $\mu$, the mean thickness of the tiles being made.

State the null and alternative hypotheses and the significance level of the test.

One hour, the sample mean is 9.27 mm.

State, with a reason, whether the machine should be stopped.

Write down the probability that the machine is stopped when the mean thickness is in fact 9.2 mm.

A fault causes the mean thickness to drift to 9.3 mm, while the standard deviation stays at 0.15 mm.

Find the probability that the next hourly sample does not lead to the machine being stopped.

Find the proportion of tiles that would be rejected when the mean thickness is 9.3 mm.

Hence comment on the effectiveness of the hourly check.