For a study, a researcher collected 200 leaves from oak trees. After measuring the lengths of the leaves, in cm, she produced the following cumulative frequency graph.

On $90$ journeys to his office, Isaac noted whether or not it rained. He also recorded his journey time to the office, and classified each journey as short, medium or long.

Of the $90$ journeys to the office, there were $3$ short journeys when it rained, $22$ medium journeys when it rained, and $15$ long journeys when it rained. There were also $14$ short journeys when it did not rain.

Isaac carried out a ${\chi }^2$ test at the $5\%$ level of significance on these data, looking at the weather and the types of journeys.

Each athlete on a running team recorded the distance ($M$ miles) they ran in $30$ minutes.

The median distance is $4$ miles and the interquartile range is $1.1$ miles.

This information is shown in the following box-and-whisker plot.

The distance in miles, $M$, can be converted to the distance in kilometres, $K$, using the formula $K=\frac{8}{5}M$.

The variance of the distances run by the athletes is $\frac{16}{9} {\text{km}}^2$.

The standard deviation of the distances is $b$ miles.

A total of $600$ athletes from different teams compete in a $5 \text{km}$ race. The times the $600$ athletes took to run the $5 \text{km}$ race are shown in the following cumulative frequency graph.

There were $400$ athletes who took between $22$ and $m$ minutes to complete the $5 \text{km}$ race.

Malthouse school opens at 08:00 every morning.

The daily arrival times of the 500 students at Malthouse school follow a normal distribution. The mean arrival time is 52 minutes after the school opens and the standard deviation is 5 minutes.

A food scientist measures the weights of $760$ potatoes taken from a single field and the distribution of the weights is shown by the cumulative frequency curve below.

Consider the following sets:

The universal set $U$ consists of all positive integers less than 15;
$A$ is the set of all numbers which are multiples of 3;
$B$ is the set of all even numbers.

Jim heated a liquid until it boiled. He measured the temperature of the liquid as it cooled. The following table shows its temperature, $d$ degrees Celsius, $t$ minutes after it boiled.

Jim believes that the relationship between $d$ and $t$ can be modelled by a linear regression equation.

A florist sells bouquets of roses. The florist recorded, in Table 1, the number of roses in each bouquet sold to customers.

Table 1

The roses can be arranged into bouquets of size small, medium or large. The data from Table 1 has been organized into a cumulative frequency table, Table 2.

Table 2

Deb used a thermometer to record the maximum daily temperature over ten consecutive days. Her results, in degrees Celsius ($°\text{C}$), are shown below.

$14, 15, 14, 11, 10, 9, 14, 15, 16, 13$

For this data set, find the value of

A group of $120$ students sat a history exam. The cumulative frequency graph shows the scores obtained by the students.

The students were awarded a grade from $1$ to $5$, depending on the score obtained in the exam. The number of students receiving each grade is shown in the following table.

The mean grade for these students is $3.65$.

The masses of Fuji apples are normally distributed with a mean of $163 \text{g}$ and a standard deviation of $6.83 \text{g}$.

When Fuji apples are picked, they are classified as small, medium, large or extra large depending on their mass. Large apples have a mass of between $172 \text{g}$ and $183 \text{g}$.

Approximately $68\%$ of Fuji apples have a mass within the medium-sized category, which is between $k$ and $172 \text{g}$.

Anne-Marie planted four sunflowers in order of height, from shortest to tallest.

Flower $\text{C}$ is $32 \text{cm}$ tall.

The median height of the flowers is $24 \text{cm}$.

The range of the heights is $50 \text{cm}$. The height of Flower $\text{A}$ is $p \text{cm}$ and the height of Flower $\text{D}$ is $q \text{cm}$.

The mean height of the flowers is $27 \text{cm}$.

The marks achieved by students taking a college entrance test follow a normal distribution with mean 300 and standard deviation 100.

In this test, 10 % of the students achieved a mark greater than k.

Marron College accepts only those students who achieve a mark of at least 450 on the test.

Ms Calhoun measures the heights of students in her mathematics class. She is interested to see if the mean height of male students, ${\mathrm{\mu <!--\; \mu \; -->}}_1$, is the same as the mean height of female students, ${\mathrm{\mu <!--\; \mu \; -->}}_2$. The information is recorded in the table.

At the 10 % level of significance, a $t$-test was used to compare the means of the two groups. The data is assumed to be normally distributed and the standard deviations are equal between the two groups.

In a school, students in grades 9 to 12 were asked to select their preferred drink. The choices were milk, juice and water. The data obtained are organized in the following table.

A ${\mathrm{\chi <!--\; \chi \; -->}}^2$ test is carried out at the 5% significance level with hypotheses:

$$\begin{array}{l}{\text{H}}_{\text{0}}\text{: the preferred drink is independent of the grade}\\ {\text{H}}_{\text{1}}\text{: the preferred drink is not independent of the grade}\end{array}$$

The ${\mathrm{\chi <!--\; \chi \; -->}}^2$ critical value for this test is 12.6.

A disc is divided into $9$ sectors, number $1$ to $9$. The angles at the centre of each of the sectors $u_n$ form an arithmetic sequence, with $u_1$ being the largest angle.

It is given that $u_9=\frac{1}{3}{u}_1$.

A scientist measures the concentration of dissolved oxygen, in milligrams per litre (y) , in a river. She takes 10 readings at different temperatures, measured in degrees Celsius (x).

The results are shown in the table.

It is believed that the concentration of dissolved oxygen in the river varies linearly with the temperature.

A city hired 160 employees to work at a festival. The following cumulative frequency curve shows the number of hours employees worked during the festival.

The city paid each of the employees £8 per hour for the first 40 hours worked, and £10 per hour for each hour they worked after the first 40 hours.

Events $A$ and $B$ are independent with $\text{P}(A\cap <!--\; \cap \; -->B)=0.2$ and $\text{P}(A^{\prime }\cap <!--\; \cap \; -->B)=0.6$.

Sara regularly flies from Geneva to London. She takes either a direct flight or a non-directflight that goes via Amsterdam.

If she takes a direct flight, the probability that her baggage does not arrive in London is 0.01.
If she takes a non-direct flight the probability that her baggage arrives in London is 0.95.

The probability that she takes a non-direct flight is 0.2.

Consider the following Venn diagrams.

The following Venn diagram shows the events $A$ and $B$, where $\text{P}\left(A\right)=0.3$. The values shown are probabilities.

The random variable $X$ is normally distributed with a mean of 100. The following diagram shows the normal curve for $X$.

Let $R$ be the shaded region under the curve, to the right of 107. The area of $R$ is 0.24.

A bag contains 5 red and 3 blue discs, all identical except for the colour. First, Priyanka takes a disc at random from the bag and then Jorgé takes a disc at random from the bag.

Srinivasa places the nine labelled balls shown below into a box.

Srinivasa then chooses two balls at random, one at a time, from the box. The first ball is not replaced before he chooses the second.

A school consists of $740$ students divided into $5$ grade levels. The numbers of students in each grade are shown in the table below.

The Principal of the school wishes to select a sample of $25$ students. She wishes to ensure that, as closely as possible, the proportion of the students from each grade in the sample is the same as the proportions in the school.

The water temperature $\left(T\right)$ in Lake Windermere is measured on the first day of eight consecutive months $\left(m\right)$ from January to August (months $1$ to $8$) and the results are shown below. The value for May (month $5$) has been accidently deleted.

Arriane has geese on her farm. She claims the mean weight of eggs from her black geese is less than the mean weight of eggs from her white geese.

She recorded the weights of eggs, in grams, from a random selection of geese. The data is shown in the table.

In order to test her claim, Arriane performs a $t$-test at a $10\%$ level of significance. It is assumed that the weights of eggs are normally distributed and the samples have equal variances.

Andre will play in the semi-final of a tennis tournament.

If Andre wins the semi-final he will progress to the final. If Andre loses the semi-final, he will not progress to the final.

If Andre wins the final, he will be the champion.

The probability that Andre will win the semi-final is $p$. If Andre wins the semi-final, then the probability he will be the champion is $0.6$.

The probability that Andre will not be the champion is $0.58$.

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 $9.49$ and the hypotheses are

${\text{H}}_0:$ The data satisfies the model.
${\text{H}}_1:$ The data does not satisfy the model.

In a city, $32\%$ of people have blue eyes. If someone has blue eyes, the probability that they also have fair hair is $58\%$. This information is represented in the following tree diagram.

It is known that $41\%$ of people in this city have fair hair.

Calculate the value of

A bag contains $n$ marbles, two of which are blue. Hayley plays a game in which she randomly draws marbles out of the bag, one after another, without replacement. The game ends when Hayley draws a blue marble.

 Let $n$ = 5. Find the probability that the game will end on her

Karl has three brown socks and four black socks in his drawer. He takes two socks at random from the drawer.

Charles wants to measure the strength of the relationship between the price of a house and its distance from the city centre where he lives. He chooses houses of a similar size and plots a graph of price, $P$ (in thousands of dollars) against distance from the city centre, $d$ (km).

The data from the graph is shown in the table.

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 set of data comprises of five numbers $x_1\text{,}{x}_2\text{,}{x}_3\text{,}{x}_4\text{,}{x}_5$ which have been placed in ascending order. 

A calculator generates a random sequence of digits. A sample of 200 digits is randomly selected from the first 100 000 digits of the sequence. The following table gives the number of times each digit occurs in this sample.

It is claimed that all digits have the same probability of appearing in the sequence.

Each month the number of days of rain in Cardiff is recorded.
The following data was collected over a period of 10 months.

11    13    8    11    8    7    8    14    x    15

For these data the median number of days of rain per month is 10.

On a work day, the probability that Mr Van Winkel wakes up early is $\frac{4}{5}$.

If he wakes up early, the probability that he is on time for work is $p$.

If he wakes up late, the probability that he is on time for work is $\frac{1}{4}$.

The probability that Mr Van Winkel arrives on time for work is $\frac{3}{5}$.

A sample of 120 oranges was tested for Vitamin C content. The cumulative frequency curve below represents the Vitamin C content, in milligrams, of these oranges.

The minimum level of Vitamin C content of an orange in the sample was 30.1 milligrams. The maximum level of Vitamin C content of an orange in the sample was 35.0 milligrams.

University students were surveyed and asked how many hours, $h$ , they worked each month. The results are shown in the following table.

Use the table to find the following values.

The first five class intervals, indicated in the table, have been used to draw part of a cumulative frequency curve as shown.

Applicants for a job had to complete a mathematics test. The time they took to complete the test is normally distributed with a mean of 53 minutes and a standard deviation of 16.3. One of the applicants is chosen at random.

For 11% of the applicants it took longer than $k$ minutes to complete the test.

There were 400 applicants for the job.

Eduardo believes that there is a linear relationship between the age of a male runner and the time it takes them to run $5000$ metres.

To test this, he recorded the age, $x$ years, and the time, $t$ minutes, for eight males in a single $5000 \text{m}$ race. His results are presented in the following table and scatter diagram.

Eduardo looked in a sports science text book. He found that the following information about $r$ was appropriate for athletic performance.

Chicken eggs are classified by grade ($4$, $5$, $6$, $7$ or $8$), based on weight. A mixed carton contains $12$ eggs and could include eggs from any grade. As part of the science project, Rocky buys $9$ mixed cartons and sorts the eggs according to their weight.

Leo is investigating whether a six-sided die is fair. He rolls the die $60$ times and records the observed frequencies in the following table:

Leo carries out a ${\chi }^2$ goodness of fit test at a $5\%$ significance level.

Rosewood College has 120 students. The students can join the sports club ($S$) and the music club ($M$).

For a student chosen at random from these 120, the probability that they joined both clubs is $\frac{1}{4}$ and the probability that they joined the music club is$\frac{1}{3}$.

There are 20 students that did not join either club.

A random variable Z is normally distributed with mean 0 and standard deviation 1. It is known that P($z$ < −1.6) = $a$ and P($z$ > 2.4) = $b$. This is shown in the following diagram.

A second random variable $X$ is normally distributed with mean $m$ and standard deviation $s$.

It is known that P($x$ < 1) = $a$.

The following scatter diagram shows the scores obtained by seven students in their mathematics test, m, and their physics test, p.

The mean point, M, for these data is (40, 16).

Sungwon plays a game where she rolls a fair $6$-sided die and spins a fair spinner with $4$ equal sectors. During each turn in the game, the die is rolled once and the spinner is spun once. The score for each turn is the sum of the two results. For example, $1$ on the die and $2$ on the spinner would receive a score of $3$.

The following diagram represents the sample space.

Sungwon takes a second turn.

In an international competition, participants can answer questions in only one of the three following languages: Portuguese, Mandarin or Hindi. 80 participants took part in the competition. The number of participants answering in Portuguese, Mandarin or Hindi is shown in the table.

A boy is chosen at random.

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 $1300$.

Paula will also spend her lunchtime outside.

Let the universal set, $U$, be the set of all integers $x$ such that $1$ ≤ $x$ < $11$.
$A$, $B$ and $C$ are subsets of $U$.

$A=\left\{1\text{, }2\text{, }3\text{, }4\text{, }6\text{, }8\right\}$
$B=\left\{2\text{, }3\text{, }5\text{, }7\right\}$
$C=\left\{1\text{, }3\text{, }5\text{, }7\text{, }9\right\}$

A factory produces bags of sugar with a labelled weight of $500 \text{g}$. The weights of the bags are normally distributed with a mean of $500 \text{g}$ and a standard deviation of $3 \text{g}$.

A bag that weighs less than $495 \text{g}$ is rejected by the factory for being underweight.

The following box-and-whisker plot shows the number of text messages sent by students in a school on a particular day.

A school café sells three flavours of smoothies: mango ($M$), kiwi fruit ($K$) and banana ($B$).
85 students were surveyed about which of these three flavours they like.

35 students liked mango, 37 liked banana, and 26 liked kiwi fruit
2 liked all three flavours
20 liked both mango and banana
14 liked mango and kiwi fruit
3 liked banana and kiwi fruit

Hafizah harvested $49$ mangoes from her farm. The weights of the mangoes, $w$, in grams, are shown in the following grouped frequency table.

Give your answers to four significant figures.

A die is thrown 120 times with the following results.

The histogram shows the time, t, in minutes, that it takes the customers of a restaurant to eat their lunch on one particular day. Each customer took less than 25 minutes.

The histogram is incomplete, and only shows data for 0 ≤ t < 20.

The mean time it took all customers to eat their lunch was estimated to be 12 minutes.

It was found that k customers took between 20 and 25 minutes to eat their lunch.

Consider the following graphs of normal distributions.

At an airport, the weights of suitcases (in kg) were measured. The weights are normally distributed with a mean of 20 kg and standard deviation of 3.5 kg.

A tetrahedral (four-sided) die has written on it the numbers 1, 2, 3 and 4. The die is rolled many times and the scores are noted. The table below shows the resulting frequency distribution.

The die was rolled a total of 100 times.

The mean score is 2.71.

The lengths of trout in a fisherman’s catch were recorded over one month, and are represented in the following histogram.

A study was conducted to investigate whether the mean reaction time of drivers who are talking on mobile phones is the same as the mean reaction time of drivers who are talking to passengers in the vehicle. Two independent groups were randomly selected for the study.

To gather data, each driver was put in a car simulator and asked to either talk on a mobile phone or talk to a passenger. Each driver was instructed to apply the brakes as soon as they saw a red light appear in front of the car. The reaction times of the drivers, in seconds, were recorded, as shown in the following table.

At the $10\%$ level of significance, a $t$-test was used to compare the mean reaction times of the two groups. Each data set is assumed to be normally distributed, and the population variances are assumed to be the same.

Let ${\mu }_1$ and ${\mu }_2$ be the population means for the two groups. The null hypothesis for this test is ${\text{H}}_0 : {\mu }_1-{\mu }_2=0$.

The Home Shine factory produces light bulbs, 7% of which are found to be defective.

Francesco buys two light bulbs produced by Home Shine.

The Bright Light factory also produces light bulbs. The probability that a light bulb produced by Bright Light is not defective is $a$.

Deborah buys three light bulbs produced by Bright Light.

A quadratic function $f$ can be written in the form $f(x)=a(x-<!--\; -\; -->p)(x-<!--\; -\; -->3)$. The graph of $f$ has axis of symmetry $x=2.5$ and $y$-intercept at $(0,-<!--\; -\; -->6)$

The following diagram shows the graph of $f^{\prime }$, the derivative of $f$.

The graph of $f^{\prime }$ has a local minimum at A, a local maximum at B and passes through $(4,-<!--\; -\; -->2)$.

The point $\text{P}(4,3)$ lies on the graph of the function, $f$.

Galois Airways has flights from Hong Kong International Airport to different destinations. The following table shows the distance, $x$ kilometres, between Hong Kong and the different destinations and the corresponding airfare, $y$, in Hong Kong dollars (HKD).

The Pearson’s product–moment correlation coefficient for this data is $0.948$, correct to three significant figures.

The distance from Hong Kong to Tokyo is $2900$ km.

The weights of apples on a tree can be modelled by a normal distribution with a mean of $85$ grams and a standard deviation of $7.5$ grams.

A sample of apples are taken from $2$ trees, $\text{A}$ and $\text{B}$, in different parts of the orchard.

The data is shown in the table below.

The owner of the orchard wants to know whether the mean weight of the apples from tree $\text{A}\left({\mu }_A\right)$ is greater than the mean weight of the apples from tree $\text{B}\left({\mu }_B\right)$ so sets up the following test:

${\text{H}}_0: {\mu }_A={\mu }_B$ and ${\text{H}}_1: {\mu }_A>{\mu }_B$

A hospital collected data from 1000 patients in four hospital wards to review the quality of its healthcare. The data, showing the number of patients who became infected during their stay in hospital, was recorded in the following table.

A ${\mathrm{\chi <!--\; \chi \; -->}}^2$-test was performed at the 5% significance level.

The critical value for this test is 7.815.

The null hypothesis for the test is

${\text{H}}_0$: Becoming infected during a stay in the hospital is independent of the ward.

In a class of $30$ students, $19$ play tennis, $3$ play both tennis and volleyball, and $6$ do not play either sport.

The following Venn diagram shows the events “plays tennis” and “plays volleyball”. The values $t$ and $v$ represent numbers of students.

A group of $130$ applicants applied for admission into either the Arts programme or the Sciences programme at a university. The outcomes of their applications are shown in the following table.

An applicant is chosen at random from this group. It is found that they were accepted into the programme of their choice.

Let $\mathrm{\theta <!--\; \theta \; -->}$ be an obtuse angle such that $\text{sin}\mathrm{\theta <!--\; \theta \; -->}=\frac{3}{5}$.

Let $f\left(x\right)={\text{e}}^x\text{sin}x-<!--\; -\; -->\frac{3x}{4}$.

Jae Hee plays a game involving a biased six-sided die.

The faces of the die are labelled −3, −1, 0, 1, 2 and 5.

The score for the game, X, is the number which lands face up after the die is rolled.

The following table shows the probability distribution for X.

Jae Hee plays the game once.

A group of 60 sports enthusiasts visited the PyeongChang 2018 Winter Olympic games to watch a variety of sporting events.

The most popular sports were snowboarding (S), figure skating (F) and ice hockey (H).

For this group of 60 people:

4 did not watch any of the most popular sports,
x watched all three of the most popular sports,
9 watched snowboarding only,
11 watched figure skating only,
15 watched ice hockey only,
7 watched snowboarding and figure skating,
13 watched figure skating and ice hockey,
11 watched snowboarding and ice hockey.

All the children in a summer camp play at least one sport, from a choice of football ($F$) or basketball ($B$). 15 children play both sports.

The number of children who play only football is double the number of children who play only basketball.

Let $x$ be the number of children who play only football.

There are 120 children in the summer camp.

Abhinav carries out a χ² test at the 1 % significance level to determine whether a person’s gender impacts their chosen professional field: engineering, medicine or law.

He surveyed 220 people and the results are shown in the table.

A survey was carried out to investigate the relationship between a person’s age in years ( $a$) and the number of hours they watch television per week ($h$). The scatter diagram represents the results of the survey.

The mean age of the people surveyed was 50.

For these results, the equation of the regression line $h$ on $a$ is $h=0.22a+15$.

The number of sick days taken by each employee in a company during a year was recorded. The data was organized in a box and whisker diagram as shown below:

For this data, write down

Pablo drives to work. The probability that he leaves home before 07:00 is $\frac{3}{4}$.

If he leaves home before 07:00 the probability he will be late for work is $\frac{1}{8}$.

If he leaves home at 07:00 or later the probability he will be late for work is $\frac{5}{8}$.

Anita is concerned that the construction of a new factory will have an adverse affect on the fish in a nearby lake. Before construction begins she catches fish at random, records their weight and returns them to the lake. After the construction is finished she collects a second, random sample of weights of fish from the lake. Her data is shown in the table.

Anita decides to use a t-test, at the 5% significance level, to determine if the mean weight of the fish changed after construction of the factory.

A bag contains 5 green balls and 3 white balls. Two balls are selected at random without replacement.

A game is played where two unbiased dice are rolled and the score in the game is the greater of the two numbers shown. If the two numbers are the same, then the score in the game is the number shown on one of the dice. A diagram showing the possible outcomes is given below.

Let $T$ be the random variable “the score in a game”.

Find the probability that

Let $f\left(x\right)=x^3-<!--\; -\; -->2x^2+ax+6$. Part of the graph of $f$ is shown in the following diagram.

The graph of $f$ crosses the $y$-axis at the point P. The line L is tangent to the graph of $f$ at P.

Dune Canyon High School organizes its school year into three trimesters: fall/autumn ($F$), winter ($W$) and spring ($S$). The school offers a variety of sporting activities during and outside the school year.

The activities offered by the school are summarized in the following Venn diagram.

The Malthouse Charity Run is a $5$ kilometre race. The time taken for each runner to complete the race was recorded. The data was found to be normally distributed with a mean time of $28$ minutes and a standard deviation of $5$ minutes.

A runner who completed the race is chosen at random.

In an effort to study the level of intelligence of students entering college, a psychologist collected data from 4000 students who were given a standard test. The predictive norms for this particular test were computed from a very large population of scores having a normal distribution with mean 100 and standard deviation of 10. The psychologist wishes to determine whether the 4000 test scores he obtained also came from a normal distribution with mean 100 and standard deviation 10. He prepared the following table (expected frequencies are rounded to the nearest integer):

The mass of a certain type of Chilean corncob follows a normal distribution with a mean of 400 grams and a standard deviation of 50 grams.

A farmer labels one of these corncobs as premium if its mass is greater than $a$ grams. 25% of these corncobs are labelled as premium.

A polygraph test is used to determine whether people are telling the truth or not, but it is not completely accurate. When a person tells the truth, they have a $20\%$ chance of failing the test. Each test outcome is independent of any previous test outcome.

$10$ people take a polygraph test and all $10$ tell the truth.

The price per kilogram of tomatoes, in euro, sold in various markets in a city is found to be normally distributed with a mean of 3.22 and a standard deviation of 0.84.

At Springfield University, the weights, in $\text{kg}$, of $10$ chinchilla rabbits and $10$ sable rabbits were recorded. The aim was to find out whether chinchilla rabbits are generally heavier than sable rabbits. The results obtained are summarized in the following table.

A $t$-test is to be performed at the $5\%$ significance level.

In a group of 20 girls, 13 take history and 8 take economics. Three girls take both history and economics, as shown in the following Venn diagram. The values $p$ and $q$ represent numbers of girls.

The diagram shows a circular horizontal board divided into six equal sectors. The sectors are labelled white (W), yellow (Y) and blue (B).

A pointer is pinned to the centre of the board. The pointer is to be spun and when it stops the colour of the sector on which the pointer stops is recorded. The pointer is equally likely to stop on any of the six sectors.

Eva will spin the pointer twice. The following tree diagram shows all the possible outcomes.

The following Venn diagram shows the sets $A$, $B$, $C$ and $U$.

$x$ is an element of $U$.

Let $f(x)=1+{\text{e}}^{-<!--\; -\; -->x}$ and $g(x)=2x+b$, for $x\in <!--\; \in \; -->\mathbb{R}$, where $b$ is a constant.

A college runs a mathematics course in the morning. Scores for a test from this class are shown below.

$25 33 51 62 63 63 70 74 79 79 81 88 90 90 98$

For these data, the lower quartile is $62$ and the upper quartile is $88$.

The box and whisker diagram showing these scores is given below.

Test scores

Another mathematics class is run by the college during the evening. A box and whisker diagram showing the scores from this class for the same test is given below.

Test scores

A researcher reviews the box and whisker diagrams and believes that the evening class performed better than the morning class.

As part of a study into healthy lifestyles, Jing visited Surrey Hills University. Jing recorded a person’s position in the university and how frequently they ate a salad. Results are shown in the table.

Jing conducted a $\mathrm{\chi <!--\; \chi \; -->}$² test for independence at a 5 % level of significance.

At the end of a school day, the Headmaster conducted a survey asking students in how many classes they had used the internet.

The data is shown in the following table.

The mean number of classes in which a student used the internet is 2.

Kayla wants to measure the extent to which two judges in a gymnastics competition are in agreement. Each judge has ranked the seven competitors, as shown in the table, where 1 is the highest ranking and 7 is the lowest.

A group of 20 students travelled to a gymnastics tournament together. Their ages, in years, are given in the following table.

The lower quartile of the ages is 16 and the upper quartile is 18.5.

In a high school, 160 students completed a questionnaire which asked for the number of people they are following on a social media website. The results were recorded in the following box-and-whisker diagram.

The following incomplete table shows the distribution of the responses from these 160 students.

A health inspector analysed the amount of sugar in 500 different snacks prepared in various school cafeterias. The collected data are shown in the following box-and-whisker diagram.

Amount of sugar per snack in grams

Stephen was invited to perform a piano recital. In preparation for the event, Stephen recorded the amount of time, in minutes, that he rehearsed each day for the piano recital.

Stephen rehearsed for $32$ days and data for all these days is displayed in the following box-and-whisker diagram.

Stephen states that he rehearsed on each of the $32$ days.