A wind turbine is designed so that the rotation of the blades generates electricity. The turbine is built on horizontal ground and is made up of a vertical tower and three blades.

The point $\text{A}$ is on the base of the tower directly below point $\text{B}$ at the top of the tower. The height of the tower, $\text{AB}$, is $90 \text{m}$. The blades of the turbine are centred at $\text{B}$ and are each of length $40 \text{m}$. This is shown in the following diagram.

The end of one of the blades of the turbine is represented by point $\text{C}$ on the diagram. Let $h$ be the height of $\text{C}$ above the ground, measured in metres, where $h$ varies as the blade rotates.

Find the

The blades of the turbine complete $12$ rotations per minute under normal conditions, moving at a constant rate.

The height, $h$, of point $\text{C}$ can be modelled by the following function. Time, $t$, is measured from the instant when the blade $\text{[BC]}$ first passes $\text{[AB]}$ and is measured in seconds.

$h\left(t\right)=90-40 \cos \left(72t°\right), t\ge 0$

Looking through his window, Tim has a partial view of the rotating wind turbine. The position of his window means that he cannot see any part of the wind turbine that is more than $\mathbf{100}\mathbf{ }\mathbf{\text{m}}$ above the ground. This is illustrated in the following diagram.

The following table shows values of ln x and ln y.

The relationship between ln x and ln y can be modelled by the regression equation ln y = a ln x + b.

In the month before their IB Diploma examinations, eight male students recorded the number of hours they spent on social media.

For each student, the number of hours spent on social media ($x$) and the number of IB Diploma points obtained ($y$) are shown in the following table.

Use your graphic display calculator to find

Ten female students also recorded the number of hours they spent on social media in the month before their IB Diploma examinations. Each of these female students spent between 3 and 30 hours on social media.

The equation of the regression line y on x for these ten female students is

$$y=-<!--\; -\; -->\frac{2}{3}x+\frac{125}{3}.$$

An eleventh girl spent 34 hours on social media in the month before her IB Diploma examinations.

A medical centre is testing patients for a certain disease. This disease occurs in $5\%$ of the population.

They test every patient who comes to the centre on a particular day.

It is intended that if a patient has the disease, they test “positive”, and if a patient does not have the disease, they test “negative”.

However, the tests are not perfect, and only $99\%$ of people who have the disease test positive. Also, $2\%$ of people who do not have the disease test positive.

The tree diagram shows some of this information.

Write down the value of

Use the tree diagram to find the probability that a patient selected at random

The staff at the medical centre looked at the care received by all visiting patients on a randomly chosen day. All the patients received at least one of these services: they had medical tests ($M$), were seen by a nurse ($N$), or were seen by a doctor ($D$). It was found that:

• $78$ had medical tests,
• $45$ were seen by a nurse;
• $30$ were seen by a doctor;
• $9$ had medical tests and were seen by a doctor and a nurse;
• $18$ had medical tests and were seen by a doctor but were not seen by a nurse;
• $11$ patients were seen by a nurse and had medical tests but were not seen by a doctor;
• $2$ patients were seen by a doctor without being seen by nurse and without having medical tests.

A group of 800 students answered 40 questions on a category of their choice out of History, Science and Literature.

For each student the category and the number of correct answers, $N$, was recorded. The results obtained are represented in the following table.

A ${\mathrm{\chi <!--\; \chi \; -->}}^2$ test at the 5% significance level is carried out on the results. The critical value for this test is 12.592.

Fiona walks from her house to a bus stop where she gets a bus to school. Her time, $W$ minutes, to walk to the bus stop is normally distributed with $W~\text{N}\left(12, 3^2\right)$.

Fiona always leaves her house at 07:15. The first bus that she can get departs at 07:30.

The length of time, $B$ minutes, of the bus journey to Fiona’s school is normally distributed with $B~\text{N}\left(50, {\sigma }^2\right)$. The probability that the bus journey takes less than $60$ minutes is $0.941$.

If Fiona misses the first bus, there is a second bus which departs at 07:45. She must arrive at school by 08:30 to be on time. Fiona will not arrive on time if she misses both buses. The variables $W$ and $B$ are independent.

A company performs an experiment on the efficiency of a liquid that is used to detect a nut allergy.

A group of 60 people took part in the experiment. In this group 26 are allergic to nuts. One person from the group is chosen at random.

A second person is chosen from the group.

When the liquid is added to a person’s blood sample, it is expected to turn blue if the person is allergic to nuts and to turn red if the person is not allergic to nuts.

The company claims that the probability that the test result is correct is 98% for people who are allergic to nuts and 95% for people who are not allergic to nuts.

It is known that 6 in every 1000 adults are allergic to nuts.

This information can be represented in a tree diagram.

An adult, who was not part of the original group of 60, is chosen at random and tested using this liquid.

The liquid is used in an office to identify employees who might be allergic to nuts. The liquid turned blue for 38 employees.

A survey was conducted on a group of people. The first question asked how many pets they each own. The results are summarized in the following table.

The second question asked each member of the group to state their age and preferred pet. The data obtained is organized in the following table.

A ${\mathrm{\chi <!--\; \chi \; -->}}^2$ test is carried out at the 10 % significance level.

Sila High School has 110 students. They each take exactly one language class from a choice of English, Spanish or Chinese. The following table shows the number of female and male students in the three different language classes.

A ${\mathrm{\chi <!--\; \chi \; -->}}^2$ test was carried out at the 5 % significance level to analyse the relationship between gender and student choice of language class.

Use your graphic display calculator to write down

The critical value at the 5 % significance level for this test is 5.99.

One student is chosen at random from this school.

Another student is chosen at random from this school.

There are three fair six-sided dice. Each die has two green faces, two yellow faces and two red faces.

All three dice are rolled.

Ted plays a game using these dice. The rules are:

• Having a turn means to roll all three dice.
• He wins $10 for each green face rolled and adds this to his winnings.
• After a turn Ted can either:
  
  • end the game (and keep his winnings), or
  • have another turn (and try to increase his winnings).
• If two or more red faces are rolled in a turn, all winnings are lost and the game ends.

The random variable $D$ ($) represents how much is added to his winnings after a turn.

The following table shows the distribution for $D$, where $$w$ represents his winnings in the game so far.

Don took part in a project investigating wind speed, $x {\text{km\hspace{0.05em}h}}^{-1}$, and the time, $y$ minutes, to fully charge a solar powered robot.

The investigation was carried out six times. The results are recorded in the table.

$\text{M}$ is the point with coordinates $(\overline{x}, \overline{y})$.

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.

In a company it is found that 25 % of the employees encountered traffic on their way to work. From those who encountered traffic the probability of being late for work is 80 %.

From those who did not encounter traffic, the probability of being late for work is 15 %.

The tree diagram illustrates the information.

The company investigates the different means of transport used by their employees in the past year to travel to work. It was found that the three most common means of transport used to travel to work were public transportation (P ), car (C ) and bicycle (B ).

The company finds that 20 employees travelled by car, 28 travelled by bicycle and 19 travelled by public transportation in the last year.

Some of the information is shown in the Venn diagram.

There are 54 employees in the company.

In a school, all Mathematical Studies SL students were given a test. The test contained four questions, each one on a different topic from the syllabus. The quality of each response was classified as satisfactory or not satisfactory. Each student answered only three of the four questions, each on a separate answer sheet.

The table below shows the number of satisfactory and not satisfactory responses for each question.

A ${\mathrm{\chi <!--\; \chi \; -->}}^2$ test is carried out at the 5% significance level for the data in the table.

The critical value for this test is 7.815.

Mackenzie conducted an experiment on the reaction times of teenagers. The results of the experiment are displayed in the following cumulative frequency graph.

Use the graph to estimate the

Mackenzie created the cumulative frequency graph using the following grouped frequency table.

Upon completion of the experiment, Mackenzie realized that some values were grouped incorrectly in the frequency table. Some reaction times recorded in the interval $0<t\le 0.2$ should have been recorded in the interval $0.2<t\le 0.4$.

A pharmaceutical company has developed a new drug to decrease cholesterol. The final stage of testing the new drug is to compare it to their current drug. They have 150 volunteers, all recently diagnosed with high cholesterol, from which they want to select a sample of size 18. They require as close as possible 20% of the sample to be below the age of 30, 30% to be between the ages of 30 and 50 and 50% to be over the age of 50.

Half of the 18 volunteers are given the current drug and half are given the new drug. After six months each volunteer has their cholesterol level measured and the decrease during the six months is shown in the table.

Calculate the mean decrease in cholesterol for

The company uses a t-test, at the 1% significance level, to determine if the new drug is more effective at decreasing cholesterol.

The following table shows a probability distribution for the random variable $X$, where $\text{E}(X)=1.2$.

A bag contains white and blue marbles, with at least three of each colour. Three marbles are drawn from the bag, without replacement. The number of blue marbles drawn is given by the random variable $X$.

A game is played in which three marbles are drawn from the bag of ten marbles, without replacement. A player wins a prize if three white marbles are drawn.

Two events A and B are such that P(A) = 0.62 and P$\left(A\cap <!--\; \cap \; -->B\right)$ = 0.18.

The weight, W, of basketball players in a tournament is found to be normally distributed with a mean of 65 kg and a standard deviation of 5 kg.

The probability that a basketball player has a weight that is within 1.5 standard deviations of the mean is q.

A basketball coach observed 60 of her players to determine whether their performance and their weight were independent of each other. Her observations were recorded as shown in the table.

She decided to conduct a χ ² test for independence at the 5% significance level.

On one day 180 flights arrived at a particular airport. The distance travelled and the arrival status for each incoming flight was recorded. The flight was then classified as on time, slightly delayed, or heavily delayed.

The results are shown in the following table.

A χ² test is carried out at the 10 % significance level to determine whether the arrival status of incoming flights is independent of the distance travelled.

The critical value for this test is 7.779.

A flight is chosen at random from the 180 recorded flights.

A transportation company owns 30 buses. The distance that each bus has travelled since being purchased by the company is recorded. The cumulative frequency curve for these data is shown.

It is known that 8 buses travelled more than m kilometres.

A discrete random variable $X$ has the following probability distribution.

The scores of the eight highest scoring countries in the $2019$ Eurovision song contest are shown in the following table.

For this data, find

Chester is investigating the relationship between the highest-scoring countries’ Eurovision score and their population size to determine whether population size can reasonably be used to predict a country’s score.

The populations of the countries, to the nearest million, are shown in the table.

Chester finds that, for this data, the Pearson’s product moment correlation coefficient is $r=0.249$.

Chester then decides to find the Spearman’s rank correlation coefficient for this data, and creates a table of ranks.

Write down the value of:

All lengths in this question are in metres.

Consider the function $f\left(x\right)=\sqrt{\frac{4-<!--\; -\; -->x^2}{8}}$, for −2 ≤ $x$ ≤ 2. In the following diagram, the shaded region is enclosed by the graph of $f$ and the $x$-axis.

A container can be modelled by rotating this region by 360˚ about the $x$-axis.

Water can flow in and out of the container.

The volume of water in the container is given by the function $g\left(t\right)$, for 0 ≤ $t$ ≤ 4 , where $t$ is measured in hours and $g\left(t\right)$ is measured in m³. The rate of change of the volume of water in the container is given by $g^{\prime }\left(t\right)=0.9-<!--\; -\; -->2.5\text{cos}\left(0.4t^2\right)$.

The volume of water in the container is increasing only when $p$ < $t$ < $q$.

Let $f(x)=-<!--\; -\; -->0.5x^4+3x^2+2x$. The following diagram shows part of the graph of $f$.

There are $x$-intercepts at $x=0$ and at $x=p$. There is a maximum at A where $x=a$, and a point of inflexion at B where $x=b$.

The marks obtained by nine Mathematical Studies SL students in their projects (x) and their final IB examination scores (y) were recorded. These data were used to determine whether the project mark is a good predictor of the examination score. The results are shown in the table.

The equation of the regression line y on x is y = mx + c.

A tenth student, Jerome, obtained a project mark of 17.

A group of $1280$ students were asked which electronic device they preferred. The results per age group are given in the following table.

A student from the group is chosen at random. Calculate the probability that the student

A ${\chi }^2$ test for independence was performed on the collected data at the $1\%$ significance level. The critical value for the test is $13.277$.

160 students attend a dual language school in which the students are taught only in Spanish or taught only in English.

A survey was conducted in order to analyse the number of students studying Biology or Mathematics. The results are shown in the Venn diagram.

Set S represents those students who are taught in Spanish.

Set B represents those students who study Biology.

Set M represents those students who study Mathematics.

A student from the school is chosen at random.

On a school excursion, $100$ students visited an amusement park. The amusement park’s main attractions are rollercoasters ($\mathit{\text{R}}$), water slides ($\mathit{\text{W}}$), and virtual reality rides ($\mathit{\text{V}}$).

The students were asked which main attractions they visited. The results are shown in the Venn diagram.

A total of $74$ students visited the rollercoasters or the water slides.

The following table shows the hand lengths and the heights of five athletes on a sports team.

The relationship between x and y can be modelled by the regression line with equation y = ax + b.

The table below shows the distribution of test grades for 50 IB students at Greendale School.

A student is chosen at random from these 50 students.

A second student is chosen at random from these 50 students.

The number of minutes that the 50 students spent preparing for the test was normally distributed with a mean of 105 minutes and a standard deviation of 20 minutes.

A dice manufacturer claims that for a novelty die he produces the probability of scoring the numbers $1$ to $5$ are all equal, and the probability of a $6$ is two times the probability of scoring any of the other numbers.

To test the manufacture’s claim one of the novelty dice is rolled $350$ times and the numbers scored on the die are shown in the table below.

A ${\chi }^2$ goodness of fit test is to be used with a $5\%$ significance level.

The stopping distances for bicycles travelling at $20 {\text{km\hspace{0.05em}h}}^{-1}$ are assumed to follow a normal distribution with mean $6.76 \text{m}$ and standard deviation $0.12 \text{m}$.

Under this assumption, find, correct to four decimal places, the probability that a bicycle chosen at random travelling at $20 {\text{km\hspace{0.05em}h}}^{-1}$ manages to stop

$1000$ randomly selected bicycles are tested and their stopping distances when travelling at $20 {\text{km\hspace{0.05em}h}}^{-1}$ are measured.

Find, correct to four significant figures, the expected number of bicycles tested that stop between

The measured stopping distances of the $1000$ bicycles are given in the table.

It is decided to perform a ${\chi }^2$ goodness of fit test at the $5\%$ level of significance to decide whether the stopping distances of bicycles travelling at $20 {\text{km\hspace{0.05em}h}}^{-1}$ can be modelled by a normal distribution with mean $6.76 \text{m}$ and standard deviation $0.12 \text{m}$.

The maximum temperature $T$, in degrees Celsius, in a park on six randomly selected days is shown in the following table. The table also shows the number of visitors, $N$, to the park on each of those six days.

The relationship between the variables can be modelled by the regression equation $N=aT+b$.

A water container is made in the shape of a cylinder with internal height $h$ cm and internal base radius $r$ cm.

The water container has no top. The inner surfaces of the container are to be coated with a water-resistant material.

The volume of the water container is $0.5{\text{m}}^3$.

The water container is designed so that the area to be coated is minimized.

One can of water-resistant material coats a surface area of $2000\text{ c}{\text{m}}^2$.

A group of 66 people went on holiday to Hawaii. During their stay, three trips were arranged: a boat trip ($B$), a coach trip ($C$) and a helicopter trip ($H$).

From this group of people:

One person in the group is selected at random.

As part of his mathematics exploration about classic books, Jason investigated the time taken by students in his school to read the book The Old Man and the Sea. He collected his data by stopping and asking students in the school corridor, until he reached his target of $10$ students from each of the literature classes in his school.

Jason constructed the following box and whisker diagram to show the number of hours students in the sample took to read this book.

Mackenzie, a member of the sample, took $25$ hours to read the novel. Jason believes Mackenzie’s time is not an outlier.

For each student interviewed, Jason recorded the time taken to read The Old Man and the Sea $\left(x\right)$, measured in hours, and paired this with their percentage score on the final exam $\left(y\right)$. These data are represented on the scatter diagram.

Jason correctly calculates the equation of the regression line $y$ on $x$ for these students to be

$y=-1.54x+98.8$.

He uses the equation to estimate the percentage score on the final exam for a student who read the book in $1.5$ hours.

Jason found a website that rated the ‘top $50$’ classic books. He randomly chose eight of these classic books and recorded the number of pages. For example, Book $\text{H}$ is rated $44\text{th}$ and has $281$ pages. These data are shown in the table.

Jason intends to analyse the data using Spearman’s rank correlation coefficient, $r_s$.

Emlyn plays many games of basketball for his school team. The number of minutes he plays in each game follows a normal distribution with mean $m$ minutes.

In any game there is a $30 \%$ chance he will play less than $13.6 \text{minutes}$.

In any game there is a $70 \%$ chance he will play less than $17.8 \text{minutes}$.

The standard deviation of the number of minutes Emlyn plays in any game is $4$.

There is a $60 \%$ chance Emlyn plays less than $x$ minutes in a game.

Emlyn will play in two basketball games today.

Emlyn and his teammate Johan each practise shooting the basketball multiple times from a point $X$. A record of their performance over the weekend is shown in the table below.

On Monday, Emlyn and Johan will practise and each will shoot $200$ times from point $X$.

A random variable $X$ is normally distributed with mean, $\mathrm{\mu <!--\; \mu \; -->}$. In the following diagram, the shaded region between 9 and $\mathrm{\mu <!--\; \mu \; -->}$ represents 30% of the distribution.

The standard deviation of $X$ is 2.1.

The random variable $Y$ is normally distributed with mean $\mathrm{\lambda <!--\; \lambda \; -->}$ and standard deviation 3.5. The events $X>9$ and $Y>9$ are independent, and $P\left((X>9)\cap <!--\; \cap \; -->(Y>9)\right)=0.4$.

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 group of 7 adult men wanted to see if there was a relationship between their Body Mass Index (BMI) and their waist size. Their waist sizes, in centimetres, were recorded and their BMI calculated. The following table shows the results.

The relationship between $x$ and $y$ can be modelled by the regression equation $y=ax+b$.

Let $f\left(x\right)=\frac{16}{x}$. The line $L$ is tangent to the graph of $f$ at $x=8$.

$L$ can be expressed in the form r $=\left(\begin{array}{c}8\\ 2\end{array}\right)+t$u.

The direction vector of $y=x$ is $\left(\begin{array}{c}1\\ 1\end{array}\right)$.

The weights, in grams, of oranges grown in an orchard, are normally distributed with a mean of 297 g. It is known that 79 % of the oranges weigh more than 289 g and 9.5 % of the oranges weigh more than 310 g.

The weights of the oranges have a standard deviation of σ.

The grocer at a local grocery store will buy the oranges whose weights exceed the 35th percentile.

The orchard packs oranges in boxes of 36.

The following table shows a probability distribution for the random variable $X$, where $\text{E}(X)=1.2$.

A bag contains white and blue marbles, with at least three of each colour. Three marbles are drawn from the bag, without replacement. The number of blue marbles drawn is given by the random variable $X$.

A game is played in which three marbles are drawn from the bag of ten marbles, without replacement. A player wins a prize if three white marbles are drawn.

Ten students were surveyed about the number of hours, $x$, they spent browsing the Internet during week 1 of the school year. The results of the survey are given below.

$$\underset{i=1}{\overset{10}{\sum <!--\; \sum \; -->}}{x}_i=252,\mathrm{\sigma <!--\; \sigma \; -->}=5\text{ and median}=27.$$

During week 4, the survey was extended to all 200 students in the school. The results are shown in the cumulative frequency graph:

The following table shows the mean weight, y kg , of children who are x years old.

The relationship between the variables is modelled by the regression line with equation $y=ax+b$.

A healthy human body temperature is 37.0 °C. Eight people were medically examined and the difference in their body temperature (°C), from 37.0 °C, was recorded. Their heartbeat (beats per minute) was also recorded.

A teacher is concerned about the amount of lesson time lost by $8$ students through arriving late at school. Over a period of $2$ weeks he records the total number of minutes they are late. He also asks them how far they live from school. The results are shown in the table below.

Adam is a beekeeper who collected data about monthly honey production in his bee hives. The data for six of his hives is shown in the following table.

The relationship between the variables is modelled by the regression line with equation $P=aN+b$.

Adam has 200 hives in total. He collects data on the monthly honey production of all the hives. This data is shown in the following cumulative frequency graph.

Adam’s hives are labelled as low, regular or high production, as defined in the following table.

Adam knows that 128 of his hives have a regular production.

The manager of a folder factory recorded the number of folders produced by the factory (in thousands) and the production costs (in thousand Euros), for six consecutive months.

Every month the factory sells all the folders produced. Each folder is sold for 2.99 Euros.

In a large university the probability that a student is left handed is 0.08. A sample of 150 students is randomly selected from the university. Let $k$ be the expected number of left-handed students in this sample.

Contestants in a TV gameshow try to get through three walls by passing through doors without falling into a trap. Contestants choose doors at random.
If they avoid a trap they progress to the next wall.
If a contestant falls into a trap they exit the game before the next contestant plays.
Contestants are not allowed to watch each other attempt the game.

The first wall has four doors with a trap behind one door.

Ayako is a contestant.

Natsuko is the second contestant.

The second wall has five doors with a trap behind two of the doors.

The third wall has six doors with a trap behind three of the doors.

The following diagram shows the branches of a probability tree diagram for a contestant in the game.

A manufacturer produces 1500 boxes of breakfast cereal every day.

The weights of these boxes are normally distributed with a mean of 502 grams and a standard deviation of 2 grams.

All boxes of cereal with a weight between 497.5 grams and 505 grams are sold. The manufacturer’s income from the sale of each box of cereal is $2.00.

The manufacturer recycles any box of cereal with a weight not between 497.5 grams and 505 grams. The manufacturer’s recycling cost is $0.16 per box.

A different manufacturer produces boxes of cereal with weights that are normally distributed with a mean of 350 grams and a standard deviation of 1.8 grams.

This manufacturer sells all boxes of cereal that are above a minimum weight, $w$.

They sell 97% of the cereal boxes produced.

Jim writes a computer program to generate 500 values of a variable Z. He obtains the following table from his results.

In this situation, state briefly what is meant by

Casanova restaurant offers a set menu where a customer chooses one of the following meals: pasta, fish or shrimp.

The manager surveyed $150$ customers and recorded the customer’s age and chosen meal. The data is shown in the following table.

A ${\chi }^2$ test was performed at the $10\%$ significance level. The critical value for this test is $4.605$.

Write down

A customer is selected at random.

A nationwide study on reaction time is conducted on participants in two age groups. The participants in Group X are less than 40 years old. Their reaction times are normally distributed with mean 0.489 seconds and standard deviation 0.07 seconds.

The participants in Group Y are 40 years or older. Their reaction times are normally distributed with mean 0.592 seconds and standard deviation σ seconds.

In the study, 38 % of the participants are in Group X.

At Penna Airport the probability, P(A), that all passengers arrive on time for a flight is 0.70. The probability, P(D), that a flight departs on time is 0.85. The probability that all passengers arrive on time for a flight and it departs on time is 0.65.

The number of hours that pilots fly per week is normally distributed with a mean of 25 hours and a standard deviation $\mathrm{\sigma <!--\; \sigma \; -->}$. 90 % of pilots fly less than 28 hours in a week.

The following table shows the average body weight, $x$, and the average weight of the brain, $y$, of seven species of mammal. Both measured in kilograms (kg).

The average body weight of grey wolves is 36 kg.

In fact, the average weight of the brain of grey wolves is 0.120 kg.

A biased four-sided die is rolled. The following table gives the probability of each score.

The Malvern Aquatic Center hosted a 3 metre spring board diving event. The judges, Stan and Minsun awarded 8 competitors a score out of 10. The raw data is collated in the following table.

The Commissioner for the event would like to find the Spearman’s rank correlation coefficient.

Lucy sells hot chocolate drinks at her snack bar and has noticed that she sells more hot chocolates on cooler days. On six different days, she records the maximum daily temperature, $T$, measured in degrees centigrade, and the number of hot chocolates sold, $H$. The results are shown in the following table.

The relationship between $H$ and $T$ can be modelled by the regression line with equation $H=aT+b$.

The aircraft for a particular flight has $72$ seats. The airline’s records show that historically for this flight only $90\%$ of the people who purchase a ticket arrive to board the flight. They assume this trend will continue and decide to sell extra tickets and hope that no more than $72$ passengers will arrive.

The number of passengers that arrive to board this flight is assumed to follow a binomial distribution with a probability of $0.9$.

Each passenger pays $\$150$ for a ticket. If too many passengers arrive, then the airline will give $\$300$ in compensation to each passenger that cannot board.

The mass M of apples in grams is normally distributed with mean μ. The following table shows probabilities for values of M.

The apples are packed in bags of ten.

Any apples with a mass less than 95 g are classified as small.

Consider the function $f\left(x\right)=x^2{\text{e}}^{3x}$,  $x\in <!--\; \in \; -->\mathbb{R}$.

The weights, $W$, of newborn babies in Australia are normally distributed with a mean 3.41 kg and standard deviation 0.57 kg. A newborn baby has a low birth weight if it weighs less than $w$ kg.

A discrete random variable $X$ has the following probability distribution.

The following diagram shows the graph of $f(x)=a\sin <!--\; \; -->bx+c$, for $0⩽<!--\; ⩽\; -->x⩽<!--\; ⩽\; -->12$.

The graph of $f$ has a minimum point at $(3,5)$ and a maximum point at $(9,17)$.

The graph of $g$ is obtained from the graph of $f$ by a translation of $\left(\begin{array}{c}k\\ 0\end{array}\right)$. The maximum point on the graph of $g$ has coordinates $(11.5,17)$.

The graph of $g$ changes from concave-up to concave-down when $x=w$.

A jar contains 5 red discs, 10 blue discs and $m$ green discs. A disc is selected at random and replaced. This process is performed four times.