DP Mathematics: Applications and Interpretation Questionbank

during the first and third month only.

Find the probability that over a $12$-month period, there will be exactly $3$ months when Bill does not visit the garden.

on exactly $12$ occasions.

After the first year, a number of baby magpies start to visit Hank’s garden. It may be assumed that each of these baby magpies visits the garden randomly and independently, and that the number of times each baby magpie visits the garden per month is modelled by a Poisson distribution with mean $2.1$.

Determine the least number of magpies required, including Bill, in order that the probability of Hank’s garden having at least $30$ magpie visits per month is greater than $0.2$.

Calculate the value of $r$, the product moment correlation coefficient of the sample.

Assuming that the distribution of $X, Y$ is bivariate normal with product moment correlation coefficient $\rho$, calculate the $p$-value of your result when testing the hypotheses $H_0 : \rho =0 ; H_1 : \rho >0$.

State whether your $p$-value suggests that $X$ and $Y$ are independent.

Given a further value $x=5.2$ from the distribution of $X$, $Y$, predict the corresponding value of $y$. Give your answer to one decimal place.

Jenna is a keen book reader. The number of books she reads during one week can be modelled by a Poisson distribution with mean $2.6$.

Determine the expected number of weeks in one year, of $52$ weeks, during which Jenna reads at least four books.

State two assumptions made in order for this model to be valid.

Find the probability that at least three people from the sample took a holiday in the Lake District in $2019$.

From a random sample of $n$ holidaymakers, the probability that at least one of them took a holiday in the Lake District in $2019$ is greater than $0.999$.

Determine the least possible value of $n$.

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 $95 \text{g}$.

Find the probability there will be space for him on the $9.01$ car.

Find the probability that more than $7$ people arrive at the ride before Shunsuke.

George goes fishing. From experience he knows that the mean number of fish he catches per hour is $1.1$. It is assumed that the number of fish he catches can be modelled by a Poisson distribution.

On a day in which George spends $8$ hours fishing, find the probability that he will catch more than $9$ fish.

Find the probability that Sheila will make a type I error in her test conclusion.

State the null and alternative hypotheses for the test.

Sheila finds $126$ coffees were sold during the $5$-hour period.

State Sheila’s conclusion to the test. Justify your answer.

Write down null and alternative hypotheses for Loreto’s test.

Using the data from Loreto’s sample, perform the hypothesis test at a $5\%$ significance level to determine if Loreto should employ extra staff.

Write down null and alternative hypotheses for this test.

Perform the test, clearly stating the conclusion in context.

Find $\text{P}(Y>100)$.

Assuming the null hypothesis to be true, state the distribution of $X$.

Find the probability of a Type I error.

Find the mean and variance for the sample data given in the table.

Hence state why Aimmika believes her data follows a Poisson distribution.

Find the value of $a$, of $b$, and of $c$. Give your answers to $3$ decimal places.

State one assumption that Aimmika needs to make about the sales of bags of rice to support her belief that it follows a Poisson distribution.

The number of cars passing a certain point in a road was recorded during 80 equal time intervals and summarized in the table below.

Carry out a ${\chi }^2$ goodness of fit test at the 5% significance level to decide if the above data can be modelled by a Poisson distribution.

Write down appropriate hypotheses.

Find the exact value of the mean of this distribution.

Calculate the mean number of eggs laid by these birds.

Test, at the 5% level of significance, whether or not the data can be modelled by a Poisson distribution.

Carry out a test at the 1% significance level, and state your conclusion.

Find the probability that Lucca eats at least one banana in a particular day.

Find the expected number of weeks in the year in which Lucca eats no bananas.

Find the probability that on a randomly selected day, Steffi does not visit Will’s house.

Copy and complete the probability distribution table for Y.

Hence find the expected number of times per day that Steffi is fed at Will’s house.

In any given year of 365 days, the probability that Steffi does not visit Will for at most $n$ days in total is 0.5 (to one decimal place). Find the value of $n$.

Show that the expected number of occasions per year on which Steffi visits Will’s house and is not fed is at least 30.

Find the probability that exactly 4 taxis arrive during T.

Find the most likely number of taxis that would arrive during T.

Given that more than 5 taxis arrive during T, find the probability that exactly 7 taxis arrive during T.

During quiet periods of the day, taxis arrive at a mean rate of 1.3 taxis every 10 minutes.

Find the probability that during a period of 15 minutes, of which the first 10 minutes is busy and the next 5 minutes is quiet, that exactly 2 taxis arrive.

Find the remainder when $p\left(x\right)$ is divided by $\left(x-2\right)$.

Find the remainder when $p\left(x\right)$ is divided by $\left(x-3\right)$.

Prove that $p\left(x\right)$ has only one real zero.

Write down the transformation that will transform the graph of $y=p\left(x\right)$ onto the graph of $y=8x^3-12x^2+16x-24$.

The random variable $X$ follows a Poisson distribution with a mean of $\lambda$ and $6\text{P}\left(X=3\right)=3\text{P}\left(X=2\right)-2\text{P}\left(X=1\right)+3\text{P}\left(X=0\right)$.

Find the value of $\lambda$.

The mean number of squirrels in a certain area is known to be 3.2 squirrels per hectare of woodland. Within this area, there is a 56 hectare woodland nature reserve. It is known that there are currently at least 168 squirrels in this reserve.

Assuming the population of squirrels follow a Poisson distribution, calculate the probability that there are more than 190 squirrels in the reserve.

Using this distribution model, find .

Using this distribution model, find the standard deviation of $X$.

an estimate for the mean number of emails received per working day.

an estimate for the standard deviation of the number of emails received per working day.

Give one piece of evidence that suggests Willow’s Poisson distribution model is not a good fit.

Suppose that the probability of Archie receiving more than 10 emails in total on any one day is 0.99. Find the value of λ.

Now suppose that Archie received exactly 20 emails in total in a consecutive two day period. Show that the probability that he received exactly 10 of them on the first day is independent of λ.

Show that $\text{P}(X=x+1)=\frac{\mu }{x+1}\times \text{P}(X=x),x\in \mathbb{N}$.

Given that $\text{P}(X=2)=0.241667$ and $\text{P}(X=3)=0.112777$, use part (a) to find the value of $\mu$.

Calculate the probability that Audrey will run at least two marathons in a particular year.

Find the probability that she will run at least two marathons in exactly four out of the following five years.

The random variable $X$ has the Poisson distribution $\text{Po}(m)$. Given that $\text{P}(X>0)=\frac{3}{4}$, find the value of $m$ in the form $\ln a$ where $a$ is an integer.

Find the probability that exactly seven rooms will have fewer than three faults in the carpet.

Find the probability that the carpet laid in the first room has fewer than three faults.

The random variable $Y$ has the Poisson distribution $\text{Po}(2m)$. Find $\text{P}(Y>1)$ in the form $\frac{b-\ln c}{c}$ where $b$ and $c$ are integers.