4.9 The normal distribution questions for IB Maths AI

State which curve represents the distribution of $\bar{X}$. Give a reason for your answer.

Find
(i) $\mathrm{P}(X>45)$;
(ii) $\mathrm{P}(\bar{X}>45)$.

The biologist says that the answer to part (b)(ii) is exact and does not rely on the central limit theorem. Explain why she is correct.

(i) Name this sampling technique.

(ii) Write down the number of bottles in the sample.

The volumes, $v$ mL, of the bottles in the sample are shown in the following table.

Write down the modal class.

Find an estimate of

(i) the mean volume;

(ii) the standard deviation of the volumes.

The plant manager models the volume of water in a bottle, $V$ mL, by $V\sim \mathrm{N}(500,{2.4}^2)$. Use this model to find $\mathrm{P}(V<496)$.

The law requires every bottle labelled 500 mL to contain at least 495 mL. Use the model to estimate the number of the 2000 bottles filled during the shift that do not meet this requirement.

The machine has 8 filling nozzles which fill the bottles in turn, so that bottles 1, 9, 17, … are filled by the first nozzle, bottles 2, 10, 18, … by the second nozzle, and so on. Explain why the inspector's sample may give a misleading picture of the machine's performance.

Suggest a different sampling technique that would avoid this problem.

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.

Find the probability that a randomly chosen capsule is underweight.

An inspector now uses a new rule: six capsules are selected at random from a batch, and the batch fails the inspection if the mean mass of coffee in the six capsules is less than 9.8 g.

Find the probability that a batch fails the inspection.

The company wants the probability that a batch fails this inspection to be only 0.01. The standard deviation stays the same. Find the mean mass to which the filling machine should be set.

Find an expression for $(s\circ p)(r)$.

Find the scaled score of Chiara, whose raw mark is 52.

Find an expression for $(s\circ p{)}^{-1}(x)$.

A grade 7 is awarded for a scaled score of 75 or more. Hence find the minimum raw mark needed for a grade 7.

Find $(s\circ p)(80)$ and comment on your answer.

The raw marks of the candidates are modelled by a normal distribution with mean 48 and standard deviation 9. Find the probability that a randomly chosen candidate is awarded a grade 7.

Find the mean and the standard deviation of the scaled scores.

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.