The function $p\left(x\right)$ is defined by $p\left(x\right)=x^3-3x^2+8x-24$ where $x\in <!--\; \in \; -->\mathbb{R}$.

The following table shows the time, in days, from December $1\text{st}$ and the percentage of Christmas trees in stock at a shop on the beginning of that day.

The following table shows the natural logarithm of both $d$ and $x$ on these days to $2$ decimal places.

The weights of apples from Tony’s farm follow a normal distribution with mean $158\text{\hspace{0.05em}g}$ and standard deviation $13\text{\hspace{0.05em}g}$. The apples are sold in bags that contain six apples.

A manufacturer of chocolates produces them in individual packets, claiming to have an average of $85$ chocolates per packet.

Talha bought $30$ of these packets in order to check the manufacturer’s claim.

Given that the number of individual chocolates is $x$, Talha found that, from his $30$ packets, $\text{Σ}x=2506$ and $\text{Σ}{x}^2=209 738$.

A factory, producing plastic gifts for a fast food restaurant’s Jolly meals, claims that just $1\%$ of the toys produced are faulty.

A restaurant manager wants to test this claim. A box of $200$ toys is delivered to the restaurant. The manager checks all the toys in this box and four toys are found to be faulty.

The restaurant manager performs a one-tailed hypothesis test, at the $10\%$ significance level, to determine whether the factory’s claim is reasonable. It is known that faults in the toys occur independently.

Saloni wants to find a model for the temperature of a bottle of water after she removes it from the fridge. She uses a temperature probe to record the temperature of the water, every 5 minutes.

After graphing the data, Saloni believes a suitable model will be

$T=28-<!--\; -\; -->a{b}^t$, where $a\text{,}b\in <!--\; \in \; -->{\mathbb{R}}^{+}$.

A newspaper vendor in Singapore is trying to predict how many copies of The Straits Times they will sell. The vendor forms a model to predict the number of copies sold each weekday. According to this model, they expect the same number of copies will be sold each day.

To test the model, they record the number of copies sold each weekday during a particular week. This data is shown in the table.

A goodness of fit test at the $5\%$ significance level is used on this data to determine whether the vendor’s model is suitable. The critical value for the test is $9.49$.

Two unbiased tetrahedral (four-sided) dice with faces labelled 1, 2, 3, 4 are thrown and the scores recorded. Let the random variable T be the maximum of these two scores.

The probability distribution of T is given in the following table.

The cars for a fairground ride hold four people. They arrive at the platform for loading and unloading every $30$ seconds.

During the hour from $9$ am the arrival of people at the ride in any interval of $t$ minutes can be modelled by a Poisson distribution with a mean of $9t \left(0<t<60\right)$.

When the $9$ am car leaves there is no one in the queue to get on the ride.

Shunsuke arrives at $9.01$ am.

Chloe and Selena play a game where each have four cards showing capital letters A, B, C and D.
Chloe lays her cards face up on the table in order A, B, C, D as shown in the following diagram.

Selena shuffles her cards and lays them face down on the table. She then turns them over one by one to see if her card matches with Chloe’s card directly above.
Chloe wins if no matches occur; otherwise Selena wins.

Chloe and Selena repeat their game so that they play a total of 50 times.
Suppose the discrete random variable X represents the number of times Chloe wins.

A company produces bags of sugar with a labelled weight of $1 \text{kg}$. The weights of the bags are normally distributed with a mean of $1 \text{kg}$ and a standard deviation of $100 \text{g}$. In an inspection, if the weight of a randomly chosen bag is less than $950 \text{g}$ then the company fails the inspection.

The discrete random variable X has the following probability distribution, where p is a constant.

The sex of cuttlefish is difficult to determine visually, so it is often found by weighing the cuttlefish.

The weights of adult male cuttlefish are known to be normally distributed with mean $10 \text{kg}$ and standard deviation $0.5 \text{kg}$.

The weights of adult female cuttlefish are known to be normally distributed with mean $12 \text{kg}$ and standard deviation $1 \text{kg}$.

A zoologist uses the null hypothesis that in the absence of information a cuttlefish is male.

If the weight is found to be above $11.5 \text{kg}$ the cuttlefish is classified as female.

$90\%$ of adult cuttlefish are male.

Consider two events, $A$ and $B$, such that $\text{P}\left(A\right)=\text{P}\left(A^{\prime }\cap <!--\; \cap \; -->B\right)=0.4$ and $\text{P}\left(A\cap <!--\; \cap \; -->B\right)=0.1$.

It is believed that the power $P$ of a signal at a point $d$ km from an antenna is inversely proportional to $d^n$ where $n\in {\mathrm{\mathbb{Z}}}^{+}$.

The value of $P$ is recorded at distances of $1 \mathrm{m}$ to $5 \mathrm{m}$ and the values of ${\log }_{10} d$ and ${\log }_{10} P$ are plotted on the graph below.

The values of ${\log }_{10} d$ and ${\log }_{10} P$ are shown in the table below.

The number of fish that can be caught in one hour from a particular lake can be modelled by a Poisson distribution.

The owner of the lake, Emily, states in her advertising that the average number of fish caught in an hour is three.

Tom, a keen fisherman, is not convinced and thinks it is less than three. He decides to set up the following test. Tom will fish for one hour and if he catches fewer than two fish he will reject Emily’s claim.

A zoologist believes that the number of eggs laid in the Spring by female birds of a certain breed follows a Poisson law. She observes 100 birds during this period and she produces the following table.

The zoologist wishes to determine whether or not a Poisson law provides a suitable model.

Mr Burke teaches a mathematics class with 15 students. In this class there are 6 female students and 9 male students.

Each day Mr Burke randomly chooses one student to answer a homework question.

In the first month, Mr Burke will teach his class 20 times.

A company sends a group of employees on a training course. Afterwards, they survey these employees to gather data on the effectiveness of the training. In order to test the reliability of the survey, they design two sets of similar questions, which are given to the employees one week apart.

The questions in the survey were grouped in different sections. The mean scores of the employees on the first section of each survey are given in the table.

On Paul’s farm, potatoes are packed in sacks labelled $50 \text{kg}$. The weights of the sacks of potatoes can be modelled by a normal distribution with mean weight $49.8 \text{kg}$ and standard deviation $0.9 \text{kg}$.

Observations on $12$ pairs of values of the random variables $X, Y$ yielded the following results.

$\sum x=76.3, \sum x^2=563.7, \sum y=72.2, \sum y^2=460.1, \sum xy=495.4$

The continuous random variable X has a probability density function given by

.

The number of coffees sold per hour at an independent coffee shop is modelled by a Poisson distribution with a mean of $22$ coffees per hour.

Sheila, the shop’s owner wants to increase the number of coffees sold in the shop. She decides to offer a discount to customers who buy more than one coffee.

To test how successful this strategy is, Sheila records the number of coffees sold over a single $5$-hour period. Sheila decides to use a $5\%$ level of significance in her test.

The heights, $x$ metres, of the 241 new entrants to a men’s college were measured and the following statistics calculated.

$$\sum <!--\; \sum \; -->x=412.11,\sum <!--\; \sum \; -->x^2=705.5721$$

The Head of Mathematics decided to use a ${\mathrm{\chi <!--\; \chi \; -->}}^2$ test to determine whether or not these heights could be modelled by a normal distribution. He therefore divided the data into classes as follows.

The graph below shows a small maze, in the form of a network of directed routes. The vertices $\text{A}$ to $\text{F}$ show junctions in the maze and the edges show the possible paths available from one vertex to another.

A mouse is placed at vertex $\text{A}$ and left to wander the maze freely. The routes shown by dashed lines indicate paths sprinkled with sugar.

When the mouse reaches any junction, she rests for a constant time before continuing.

At any junction, it may also be assumed that

•  the mouse chooses any available normal path with equal probability
•  if the junction includes a path sprinkled with sugar, the probability of choosing this path is twice that of a normal path.

Eggs at a farm are sold in boxes of six. Each egg is either brown or white. The owner believes that the number of brown eggs in a box can be modelled by a binomial distribution. He examines 100 boxes and obtains the following data.

Adesh wants to model the cooling of a metal rod. He heats the rod and records its temperature as it cools.

He believes the temperature can be modeled by $T\left(t\right)=a{\text{e}}^{bt}+25$, where $a\text{,}b\in <!--\; \in \; -->\mathbb{R}$.

Hence

Katie likes to cycle to work as much as possible. If Katie cycles to work one day then she has a probability of $0.2$ of not cycling to work on the next work day. If she does not cycle to work one day then she has a probability of $0.4$ of not cycling to work on the next work day.

A psychologist records the number of digits ($d$) of $\mathrm{\pi }$ that a sample of IB Mathematics higher level candidates could recall.

The psychologist has read that in the general population people can remember an average of $4.4$ digits of $\mathrm{\pi }$. The psychologist wants to perform a statistical test to see if IB Mathematics higher level candidates can remember more digits than the general population.

${\text{H}}_0 : \mu =4.4$ is the null hypothesis for this test.

A manager wishes to check the mean mass of flour put into bags in his factory. He randomly samples 10 bags and finds the mean mass is 1.478 kg and the standard deviation of the sample is 0.0196 kg.

As part of the selection process for an engineering course at a particular university, applicants are given an exam in mathematics. This year the university has produced a new exam and they want to test if it is a valid indicator of future performance, before giving it to applicants. They randomly select 8 students in their first year of the engineering course and give them the exam. They compare the exam scores with their results in the engineering course.

The results of the 8 students are shown in the table.

A $2\times <!--\; \times \; -->2$ transition matrix for a Markov chain will have the form .

A robot moves around the maze shown below.

Whenever it leaves a room it is equally likely to take any of the exits.

The time interval between the robot entering and leaving a room is the same for all transitions.

The number of cars arriving at a junction in a particular town in any given minute between $9:00 \text{am}$ and $10:00 \text{am}$ is historically known to follow a Poisson distribution with a mean of $5.4$ cars per minute.

A new road is built near the town. It is claimed that the new road has decreased the number of cars arriving at the junction.

To test the claim, the number of cars, $X$, arriving at the junction between $9:00 \text{am}$ and $10:00 \text{am}$ on a particular day will be recorded. The test will have the following hypotheses:

${\text{H}}_0 :$ the mean number of cars arriving at the junction has not changed,
${\text{H}}_1 :$ the mean number of cars arriving at the junction has decreased.

The alternative hypothesis will be accepted if $X\le 300$.

Consider two events $A$ and $A$ defined in the same sample space.

Given that $\text{P}(A\cup <!--\; \cup \; -->B)=\frac{4}{9},\text{ P}(B|A)=\frac{1}{3}$ and $\text{P}(B|A^{\prime })=\frac{1}{6}$,

Sue sometimes goes out for lunch. If she goes out for lunch on a particular day then the probability that she will go out for lunch on the following day is 0.4. If she does not go out for lunch on a particular day then the probability she will go out for lunch on the following day is 0.3.

The number of telephone calls received by a helpline over 80 one-minute periods are summarized in the table below.

The diagram below shows part of the screen from a weather forecasting website showing the data for town $\text{A}$. The percentages on the bottom row represent the likelihood of some rain during the hour leading up to the time given. For example there is a $69\%$ chance (a probability of $0.69$) of rain falling on any point in town $\text{A}$ between $0900$ and $1000$.

Paula works at a building site in the area covered by this page of the website from $0900$ to $1700$. She has lunch from $1300$ to $1400$.

In the following parts you may assume all probabilities are independent.

Paula needs to work outside between $1000$ and $1300$ and will also spend her lunchtime outside.

Let $X$ be a random variable which follows a normal distribution with mean $\mathrm{\mu <!--\; \mu \; -->}$. Given that  , find

The matrix $\mathit{M}=\left(\begin{array}{cc}0.2 & 0.7\\ 0.8 & 0.3\end{array}\right)$ has eigenvalues $-5$ and $1$.

A switch has two states, $\text{A}$ and $\text{B}$. Each second it either remains in the same state or moves according to the following rule: If it is in state $\text{A}$ it will move to state $\text{B}$ with a probability of $0.8$ and if it is in state $\text{B}$ it will move to state $\text{A}$ with a probability of $0.7$.

The faces of a fair six-sided die are numbered 1, 2, 2, 4, 4, 6. Let $X$ be the discrete random variable that models the score obtained when this die is rolled.

The strength of earthquakes is measured on the Richter magnitude scale, with values typically between $0$ and $8$ where $8$ is the most severe.

The Gutenberg–Richter equation gives the average number of earthquakes per year, $N$, which have a magnitude of at least $M$. For a particular region the equation is

${\log }_{10} N=a-M$, for some $a\in \mathrm{ℝ}$.

This region has an average of $100$ earthquakes per year with a magnitude of at least $3$.

The equation for this region can also be written as $N=\frac{b}{{10}^M}$.

Within this region the most severe earthquake recorded had a magnitude of $7.2$.

The number of earthquakes in a given year with a magnitude of at least $7.2$ can be modelled by a Poisson distribution, with mean $N$. The number of earthquakes in one year is independent of the number of earthquakes in any other year.

Let $Y$ be the number of years between the earthquake of magnitude $7.2$ and the next earthquake of at least this magnitude.