Charlotte decides to model the shape of a cupcake to calculate its volume.

From rotating a photograph of her cupcake she estimates that its cross-section passes through the points $(0, 3.5), (4, 6), (6.5, 4), (7, 3)$ and $(7.5, 0)$, where all units are in centimetres. The cross-section is symmetrical in the $x$-axis, as shown below:

She models the section from $(0, 3.5)$ to $(4, 6)$ as a straight line.

Charlotte models the section of the cupcake that passes through the points $(4, 6), (6.5, 4), (7, 3)$ and $(7.5, 0)$ with a quadratic curve.

Charlotte thinks that a quadratic with a maximum point at $(4, 6)$ and that passes through the point $(7.5, 0)$ would be a better fit.

Believing this to be a better model for her cupcake, Charlotte finds the volume of revolution about the $x$-axis to estimate the volume of the cupcake.

It is known that the weights of male Persian cats are normally distributed with mean $6.1 \text{kg}$ and variance $0.5^2 {\text{kg}}^2$.

A group of $80$ male Persian cats are drawn from this population.

The male cats are now joined by $80$ female Persian cats. The female cats are drawn from a population whose weights are normally distributed with mean $4.5 \text{kg}$ and standard deviation $0.45 \text{kg}$.

Ten female cats are chosen at random.

A city has two cable companies, X and Y. Each year 20 % of the customers using company X move to company Y and 10 % of the customers using company Y move to company X. All additional losses and gains of customers by the companies may be ignored.

Initially company X and company Y both have 1200 customers.

In a small village there are two doctors’ clinics, one owned by Doctor Black and the other owned by Doctor Green. It was noted after each year that $3.5\%$ of Doctor Black’s patients moved to Doctor Green’s clinic and $5\%$ of Doctor Green’s patients moved to Doctor Black’s clinic. All additional losses and gains of patients by the clinics may be ignored.

At the start of a particular year, it was noted that Doctor Black had $2100$ patients on their register, compared to Doctor Green’s $3500$ patients.

The masses in kilograms of melons produced by a farm can be modelled by a normal distribution with a mean of $2.6\text{\hspace{0.05em}kg}$ and a standard deviation of $0.5\text{\hspace{0.05em}kg}$.

Find the probability that two melons picked at random and independently of each other will

One year due to favourable weather conditions it is thought that the mean mass of the melons has increased.

The owner of the farm decides to take a random sample of $16$ melons to test this hypothesis at the $5\%$ significance level, assuming the standard deviation of the masses of the melons has not changed.

Unknown to the farmer the favourable weather conditions have led to all the melons having $10\%$ greater mass than the model described above.

The random variable X is thought to follow a binomial distribution B (4, $p$). In order to investigate this belief, a random sample of 100 observations on X was taken with the following results.

An automatic machine is used to fill bottles of water. The amount delivered, $Y$ ml, may be assumed to be normally distributed with mean $\mathrm{\mu <!--\; \mu \; -->}$ ml and standard deviation 8 ml. Initially, the machine is adjusted so that the value of $\mathrm{\mu <!--\; \mu \; -->}$ is 500. In order to check that the value of $\mathrm{\mu <!--\; \mu \; -->}$ remains equal to 500, a random sample of 10 bottles is selected at regular intervals, and the mean amount of water, $\stackrel{¯<!--\; ¯\; -->}{y}$, in these bottles is calculated. The following hypotheses are set up.

H₀: $\mathrm{\mu <!--\; \mu \; -->}$ = 500;  H₁: $\mathrm{\mu <!--\; \mu \; -->}$ ≠ 500

The critical region is defined to be .

A student investigating the relationship between chemical reactions and temperature finds the Arrhenius equation on the internet.

$k=A{\text{e}}^{-\frac{c}{T}}$

This equation links a variable $k$ with the temperature $T$, where $A$ and $c$ are positive constants and $T>0$.

The Arrhenius equation predicts that the graph of $\ln k$ against $\frac{1}{T}$ is a straight line.

Write down

The following data are found for a particular reaction, where $T$ is measured in Kelvin and $k$ is measured in ${\text{cm}}^3 {\text{mol}}^{-1} {\text{s}}^{-1}$:

Find an estimate of

Willow finds that she receives approximately 70 emails per working day.

She decides to model the number of emails received per working day using the random variable $X$, where $X$ follows a Poisson distribution with mean 70.

In order to test her model, Willow records the number of emails she receives per working day over a period of 6 months. The results are shown in the following table.

From the table, calculate

Archie works for a different company and knows that he receives emails according to a Poisson distribution, with a mean of $\mathrm{\lambda <!--\; \lambda \; -->}$ emails per day.

Long term experience shows that if it is sunny on a particular day in Vokram, then the probability that it will be sunny the following day is $0.8$. If it is not sunny, then the probability that it will be sunny the following day is $0.3$.

The transition matrix $\mathit{T}$ is used to model this information, where $\mathit{T}=\left(\begin{array}{cc}0.8& 0.3\\ 0.2& 0.7\end{array}\right)$.

The matrix $\mathit{T}$ can be written as a product of three matrices, $\mathit{P}\mathit{D}\mathbf{ }{\mathit{P}}^{-1}$ , where $\mathit{D}$ is a diagonal matrix.

A continuous random variable $X$ has probability density function $f$ given by

It is given that $\text{P}(X⩾<!--\; ⩾\; -->2)=0.75$.

Eight independent observations of $X$ are now taken and the random variable $Y$ is the number of observations such that $X⩾<!--\; ⩾\; -->2$.

Arianne plays a game of darts.

The distance that her darts land from the centre, $\text{O}$, of the board can be modelled by a normal distribution with mean $10 \text{cm}$ and standard deviation $3 \text{cm}$.

Find the probability that

Each of Arianne’s throws is independent of her previous throws.

In a competition a player has three darts to throw on each turn. A point is scored if a player throws all three darts to land within a central area around $\text{O}$. When Arianne throws a dart the probability that it lands within this area is $0.8143$.

In the competition Arianne has ten turns, each with three darts.

A Chocolate Shop advertises free gifts to customers that collect three vouchers. The vouchers are placed at random into 10% of all chocolate bars sold at this shop. Kati buys some of these bars and she opens them one at a time to see if they contain a voucher. Let $\text{P}(X=n)$ be the probability that Kati obtains her third voucher on the $n\text{th}$ bar opened.

(It is assumed that the probability that a chocolate bar contains a voucher stays at 10% throughout the question.)

It is given that $\text{P}(X=n)=\frac{n^2+an+b}{2000}\times <!--\; \times \; -->{0.9}^{n-<!--\; -\; -->3}$ for $n⩾<!--\; ⩾\; -->3,n\in <!--\; \in \; -->\mathbb{N}$.

Kati’s mother goes to the shop and buys $x$ chocolate bars. She takes the bars home for Kati to open.

The weights, in grams, of individual packets of coffee can be modelled by a normal distribution, with mean $102 \text{g}$ and standard deviation $8 \text{g}$.

Hank sets up a bird table in his garden to provide the local birds with some food. Hank notices that a specific bird, a large magpie, visits several times per month and he names him Bill. Hank models the number of times per month that Bill visits his garden as a Poisson distribution with mean $3.1$.

Over the course of $3$ consecutive months, find the probability that Bill visits the garden:

John likes to go sailing every day in July. To help him make a decision on whether it is safe to go sailing he classifies each day in July as windy or calm. Given that a day in July is calm, the probability that the next day is calm is 0.9. Given that a day in July is windy, the probability that the next day is calm is 0.3. The weather forecast for the 1st July predicts that the probability that it will be calm is 0.8.

The curve $y=f\left(x\right)$ is shown in the graph, for $0⩽<!--\; ⩽\; -->x⩽<!--\; ⩽\; -->10$.

The curve $y=f\left(x\right)$ passes through the following points.

It is required to find the area bounded by the curve, the $x$-axis, the $y$-axis and the line $x=10$.

One possible model for the curve $y=f\left(x\right)$ is a cubic function.

A café serves sandwiches and cakes. Each customer will choose one of the following three options; buy only a sandwich, buy only a cake or buy both a sandwich and a cake.

The probability that a customer buys a sandwich is 0.72 and the probability that a customer buys a cake is 0.45.

Find the probability that a customer chosen at random will buy

On a typical day 200 customers come to the café.

It is known that 46 % of the customers who come to the café are male, and that 80 % of these buy a sandwich.

Dana has collected some data regarding the heights $h$ (metres) of waves against a pier at $50$ randomly chosen times in a single day. This data is shown in the table below.

She wishes to perform a ${\chi }^2$-test at the $5\%$ significance level to see if the height of waves could be modelled by a normal distribution. Her null hypothesis is

${\text{H}}_0:$ The data can be modelled by a normal distribution.

From the table she calculates the mean of the heights in her sample to be $0.828 \text{m}$ and the standard deviation of the heights $s_n$ to be $0.257 \text{m}$.

She calculates the expected values for each interval under this null hypothesis, and some of these values are shown in the table below.

The times taken for male runners to complete a marathon can be modelled by a normal distribution with a mean 196 minutes and a standard deviation 24 minutes.

It is found that 5% of the male runners complete the marathon in less than $T_1$ minutes.

The times taken for female runners to complete the marathon can be modelled by a normal distribution with a mean 210 minutes. It is found that 58% of female runners complete the marathon between 185 and 235 minutes.

A Principal would like to compare the students in his school with a national standard. He decides to give a test to eight students made up of four boys and four girls. One of the teachers offers to find the volunteers from his class.

The marks out of $40$, for the students who took the test, are:

$25, 29, 38, 37, 12, 18, 27, 31.$

For the eight students find

The national standard mark is $25.2$ out of $40$.

Two additional students take the test at a later date and the mean mark for all ten students is $28.1$ and the standard deviation is $8.4$.

For further analysis, a standardized score out of $100$ for the ten students is obtained by multiplying the scores by $2$ and adding $20$.

For the ten students, find

The number of marathons that Audrey runs in any given year can be modelled by a Poisson distribution with mean 1.3 .

Packets of biscuits are produced by a machine. The weights $X$, in grams, of packets of biscuits can be modelled by a normal distribution where $X\sim <!--\; \sim \; -->\text{N}(\mathrm{\mu <!--\; \mu \; -->},{\mathrm{\sigma <!--\; \sigma \; -->}}^2)$. A packet of biscuits is considered to be underweight if it weighs less than 250 grams.

The manufacturer makes the decision that the probability that a packet is underweight should be 0.002. To do this $\mathrm{\mu <!--\; \mu \; -->}$ is increased and $\mathrm{\sigma <!--\; \sigma \; -->}$ remains unchanged.

The manufacturer is happy with the decision that the probability that a packet is underweight should be 0.002, but is unhappy with the way in which this was achieved. The machine is now adjusted to reduce $\mathrm{\sigma <!--\; \sigma \; -->}$ and return $\mathrm{\mu <!--\; \mu \; -->}$ to 253.

In a reforested area of pine trees, heights of trees planted in a specific year seem to follow a normal distribution. A sample of 100 such trees is selected to test the validity of this hypothesis. The results of measuring tree heights, to the nearest centimetre, are recorded in the first two columns of the table below.

Describe what is meant by

The random variable X has a normal distribution with mean μ = 50 and variance σ ² = 16 .

Loreto is a manager at the Da Vinci health centre. If the mean rate of patients arriving at the health centre exceeds $1.5$ per minute then Loreto will employ extra staff. It is assumed that the number of patients arriving in any given time period follows a Poisson distribution.

Loreto performs a hypothesis test to determine whether she should employ extra staff. She finds that $320$ patients arrived during a randomly selected $3$-hour clinic.

Loreto is also concerned about the average waiting time for patients to see a nurse. The health centre aims for at least $95\%$ of patients to see a nurse in under $20$ minutes.

Loreto assumes that the waiting times for patients are independent of each other and decides to perform a hypothesis test at a $10\%$ significance level to determine whether the health centre is meeting its target.

Loreto surveys $150$ patients and finds that $11$ of them waited more than $20$ minutes.

The marks achieved by eight students in a class test are given in the following list.

The teacher increases all the marks by 2. Write down the new value for

A random variable $X$ is normally distributed with mean $\mathrm{\mu <!--\; \mu \; -->}$ and standard deviation $\mathrm{\sigma <!--\; \sigma \; -->}$, such that and $\text{P}(X>42.52)=0.3060$.

A geneticist uses a Markov chain model to investigate changes in a specific gene in a cell as it divides. Every time the cell divides, the gene may mutate between its normal state and other states.

The model is of the form

$\left(\begin{array}{c}{X}_{n+1}\\ Z_{n+1}\end{array}\right)=\mathit{M}\left(\begin{array}{c}{X}_n\\ Z_n\end{array}\right)$

where $X_n$ is the probability of the gene being in its normal state after dividing for the $n\text{th}$ time, and $Z_n$ is the probability of it being in another state after dividing for the $n\text{th}$ time, where $n\in \mathrm{\mathbb{N}}$.

Matrix $\mathit{M}$ is found to be $\left(\begin{array}{cc}0.94 & b\\ 0.06 & 0.98\end{array}\right)$.

The gene is in its normal state when $n=0$. Calculate the probability of it being in its normal state

Scientists have developed a type of corn whose protein quality may help chickens gain weight faster than the present type used. To test this new type, 20 one-day-old chicks were fed a ration that contained the new corn while another control group of 20 chicks was fed the ordinary corn. The data below gives the weight gains in grams, for each group after three weeks.

The random variable X has a binomial distribution with parameters n and p.
It is given that E(X) = 3.5.

A survey of British holidaymakers found that $15 \%$ of those surveyed took a holiday in the Lake District in $2019$.

A random sample of $16$ British holidaymakers was taken. The number of people in the sample who took a holiday in the Lake District in $2019$ can be modelled by a binomial distribution.

A discrete random variable $X$ follows a Poisson distribution $\text{Po}(\mathrm{\mu <!--\; \mu \; -->})$.

It is known that 56 % of Infiglow batteries have a life of less than 16 hours, and 94 % have a life less than 17 hours. It can be assumed that battery life is modelled by the normal distribution $\text{N}\left(\mathrm{\mu <!--\; \mu \; -->},{\mathrm{\sigma <!--\; \sigma \; -->}}^2\right)$.

The continuous random variable X has probability density function $f$ given by

Steffi the stray cat often visits Will’s house in search of food. Let $X$ be the discrete random variable “the number of times per day that Steffi visits Will’s house”.

The random variable $X$ can be modelled by a Poisson distribution with mean 2.1.

Let Y be the discrete random variable “the number of times per day that Steffi is fed at Will’s house”. Steffi is only fed on the first four occasions that she visits each day.

Timmy owns a shop. His daily income from selling his goods can be modelled as a normal distribution, with a mean daily income of $820, and a standard deviation of $230. To make a profit, Timmy’s daily income needs to be greater than $1000.

The number of bananas that Lucca eats during any particular day follows a Poisson distribution with mean 0.2.

Events $A$ and $B$ are such that $\text{P}(A\cup <!--\; \cup \; -->B)=0.95,\text{ P}(A\cap <!--\; \cap \; -->B)=0.6$ and $\text{P}(A|B)=0.75$.

Iqbal attempts three practice papers in mathematics. The probability that he passes the first paper is 0.6. Whenever he gains a pass in a paper, his confidence increases so that the probability of him passing the next paper increases by 0.1. Whenever he fails a paper the probability of him passing the next paper is 0.6.

There are 75 players in a golf club who take part in a golf tournament. The scores obtained on the 18th hole are as shown in the following table.

Jorge is carefully observing the rise in sales of a new app he has created.

The number of sales in the first four months is shown in the table below.

Jorge believes that the increase is exponential and proposes to model the number of sales $N$ in month $t$ with the equation

$N=A{\text{e}}^{rt}, A, r\in \mathrm{ℝ}$

Jorge plans to adapt Euler’s method to find an approximate value for $r$.

With a step length of one month the solution to the differential equation can be approximated using Euler’s method where

$N\left(n+1\right)\approx N\left(n\right)+1\times N\text{'}\left(n\right), n\in \mathrm{\mathbb{N}}$

Jorge decides to take the mean of these values as the approximation of $r$ for his model. He also decides the graph of the model should pass through the point $(2, 52)$.

The sum of the square residuals for these points for the least squares regression model is approximately $6.555$.

A random variable $X$ has a probability distribution given in the following table.

When carpet is manufactured, small faults occur at random. The number of faults in Premium carpets can be modelled by a Poisson distribution with mean 0.5 faults per 20$$m². Mr Jones chooses Premium carpets to replace the carpets in his office building. The office building has 10 rooms, each with the area of 80$$m².

The hens on a farm lay either white or brown eggs. The eggs are put into boxes of six. The farmer claims that the number of brown eggs in a box can be modelled by the binomial distribution, B(6, $p$). By inspecting the contents of 150 boxes of eggs she obtains the following data.

The number of taxis arriving at Cardiff Central railway station can be modelled by a Poisson distribution. During busy periods of the day, taxis arrive at a mean rate of 5.3 taxis every 10 minutes. Let T represent a random 10 minute busy period.

The age, L, in years, of a wolf can be modelled by the normal distribution L ~ N(8, 5).

Consider two events $A$ and $B$ such that $\text{P}(A)=k,\text{ P}(B)=3k,\text{ P}(A\cap <!--\; \cap \; -->B)=k^2$ and $\text{P}(A\cup <!--\; \cup \; -->B)=0.5$.