4.8 The binomial distribution questions for IB Maths AI

Find the probability that a randomly chosen egg is large.

Eggs are packed at random into boxes of 12. Find the probability that a box contains at least three large eggs.

Write down the interquartile range.

Determine whether the lightest strawberry is an outlier.

Explain why the diagram suggests that the masses could be modelled by a normal distribution.

The farm models the mass of a strawberry, $M$ grams, by $M\sim \mathrm{N}(18,{4.4}^2)$.

(i) Find $\mathrm{P}(M>25)$.

(ii) Hence estimate the number of the 150 strawberries that have a mass of more than 25 g.

Show that, according to the model, the lower quartile of the masses is 15.0 g, correct to three significant figures.

The heaviest 70% of strawberries are sold as Class I. Find the minimum mass of a Class I strawberry.

A punnet contains 10 strawberries chosen at random. Find the probability that at least 8 of them are Class I.

Ingrid wants to model the number of boats arriving in one hour by a Poisson distribution.

State two conditions that must hold for this model to be appropriate.

Calculate the mean and the variance of the number of boats arriving in one hour, according to Ingrid's data.

Explain how your answers to part (b)(i) support the use of a Poisson model.

Ingrid models the number of boats arriving in one hour by a Poisson distribution with mean 3.2.

Find the probability that exactly 5 boats arrive in a one-hour period.

Given that at least 3 boats arrive in a two-hour period, find the probability that fewer than 6 boats arrive in that period.

The lock operates in 30-minute cycles and can take at most 4 boats in each cycle. Any extra boats must wait for the next cycle.

Find the probability that more than 4 boats arrive during one 30-minute cycle.

On one day the lock operates 20 cycles. The numbers of boats arriving in different cycles are independent.

Find the probability that more than 4 boats arrive in at least two of the cycles on that day.

The following summer a new marina opens upstream. Ingrid believes that the mean number of boats arriving per hour has increased. She will count the number of boats, $T$, arriving during one randomly chosen five-hour period and test her belief at the 5 % significance level.

State the null and alternative hypotheses for this test.

Find the critical region for this test.

During the five-hour period, 25 boats arrive at the lock.

State the conclusion of the test, in context. Give a reason for your answer.

Find the probability of a Type I error for this test.

In fact, after the marina opens, the mean number of boats arriving is 4.4 per hour.

Find the probability of a Type II error for this test.

Suggest how Ingrid could reduce the probability of a Type II error without increasing the significance level of the test.

Find the probability that there are at least 2 accidents at the junction in a randomly chosen week.

Find the probability that, in a period of 12 weeks, there are exactly 2 weeks in which at least 2 accidents occur at the junction.

Find the value of $p$ and the value of $q$.

Find the standard deviation of the number of faulty components per box.

The inspector models the number of faulty components in a box, $X$, by $X\sim \mathrm{B}(20,\theta )$. Show that the mean of the data gives $\theta =0.045$.

Using the model $X\sim \mathrm{B}(20,0.045)$,

(i) find $\mathrm{P}(X=0)$;

(ii) find the expected number of the 100 boxes with no faulty components;

(iii) find the expected number of the 100 boxes with more than 2 faulty components.

Comment on how well the model fits the data, referring to your answers to part (d).

State one assumption about the components that is needed for the binomial model to be valid.

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.