Write down a formula for $A$, the surface area to be coated.

Express this volume in $\text{c}{\text{m}}^3$.

Write down, in terms of $r$ and $h$, an equation for the volume of this water container.

Show that $A=\pi r^2+\frac{1000000}{r}$.

Find $\frac{\text{d}A}{\text{d}r}$.

Using your answer to part (e), find the value of $r$ which minimizes $A$.

Find the value of this minimum area.

Find the least number of cans of water-resistant material that will coat the area in part (g).

Find $q$.

Find $p$.

Write down the probability of drawing three blue marbles.

Explain why the probability of drawing three white marbles is $\frac{1}{6}$.

The bag contains a total of ten marbles of which $w$ are white. Find $w$.

Jill plays the game nine times. Find the probability that she wins exactly two prizes.

Grant plays the game until he wins two prizes. Find the probability that he wins his second prize on his eighth attempt.

Show that $b=0.3-a$.

Find the difference between the greatest possible expected value and the least possible expected value.

Find the value of $\sigma$.

On a randomly selected day, find the probability that Suzi’s drive to work will take longer than $45$ minutes.

Find the probability that she will be late to work at least one day next week.

Given that Suzi will be late to work at least one day next week, find the probability that she will be late less than three times.

Find the probability that Suzi will receive a bonus.

Find the value of a and of b.

Use the regression equation to estimate the value of y when x = 3.57.

The relationship between x and y can be modelled using the formula y = kxⁿ, where k ≠ 0 , n ≠ 0 , n ≠ 1.

By expressing ln y in terms of ln x, find the value of n and of k.

Write down the number of students in the group who take art class.

the student does not take art class.

the student takes either art class or music class, but not both.

Use your graphic display calculator to write down $\overline{x}$, the mean project mark.

Use your graphic display calculator to write down $\overline{y}$, the mean examination score.

Use your graphic display calculator to write down r , Pearson’s product–moment correlation coefficient.

Find the exact value of m and of c for these data.

Show that the point M ($\overline{x}$, $\overline{y}$) lies on the regression line y on x.

Use the regression line y on x to estimate Jerome’s examination score.

Justify whether it is valid to use the regression line y on x to estimate Jerome’s examination score.

In his final IB examination Jerome scored 65.

Calculate the percentage error in Jerome’s estimated examination score.

Sketch a diagram to represent this information.

Show that $m=15.7$.

Find the probability that Emlyn plays between $13 \text{minutes}$ and $18 \text{minutes}$ in a game.

Find the probability that Emlyn plays more than $20 \text{minutes}$ in a game.

Find the value of $x$.

Find the probability he plays between $13 \text{minutes}$ and $18 \text{minutes}$ in one game and more than $20 \text{minutes}$ in the other game.

Find the expected number of successful shots Emlyn will make on Monday, based on the results from Saturday and Sunday.

Emlyn claims the results from Saturday and Sunday show that his expected number of successful shots will be more than Johan’s.

Determine if Emlyn’s claim is correct. Justify your reasoning.

Find the value of $a$.

Find the value of $b$.

Find the number of students who visited at least two types of main attraction.

Write down the value of $n( R\cap W)$.

Find the probability that a randomly selected student visited the rollercoasters.

Find the probability that a randomly selected student visited the virtual reality rides.

Hence determine whether the events in parts (d)(i) and (d)(ii) are independent. Justify your reasoning. 

Find the probability that the student plays a sport and is involved in theatre.

Find the probability that the student is involved in theatre, but does not play a sport.

Find $\text{P}\left(G\cap T\right)$.

Determine if the events $G$ and $T$ are independent. Justify your answer.

The heights of adult males in a country are normally distributed with a mean of 180 cm and a standard deviation of $\sigma \text{ cm}$. 17% of these men are shorter than 168 cm. 80% of them have heights between $(192-h)\text{ cm}$ and 192 cm.

Find the value of $h$.

Find the range of the average body weights for these seven species of mammal.

For the data from these seven species calculate $r$, the Pearson’s product–moment correlation coefficient;

For the data from these seven species describe the correlation between the average body weight and the average weight of the brain.

Write down the equation of the regression line $y$ on $x$, in the form $y=mx+c$.

Use your regression line to estimate the average weight of the brain of grey wolves.

Find the percentage error in your estimate in part (d).

State whether it is valid to use the regression line to estimate the average weight of the brain of mice. Give a reason for your answer.

Show that event A and event D are not independent.

Find $\text{P}\left(A\cap D^{\prime }\right)$.

 Given that all passengers for a flight arrive on time, find the probability that the flight does not depart on time.

Find the value of $\sigma$.

All flights have two pilots. Find the percentage of flights where both pilots flew more than 30 hours last week.

Calculate the mean test grade of the students;

Calculate the standard deviation.

Find the median test grade of the students.

Find the interquartile range.

Find the probability that this student scored a grade 5 or higher.

Given that the first student chosen at random scored a grade 5 or higher, find the probability that both students scored a grade 6.

Calculate the probability that a student chosen at random spent at least 90 minutes preparing for the test.

Calculate the expected number of students that spent at least 90 minutes preparing for the test.

On graph paper, draw a scatter diagram for these data. Use a scale of 2 cm to represent 5 hours on the $x$-axis and 2 cm to represent 10 points on the $y$-axis.

(i)     $\overline{x}$, the mean number of hours spent on social media;

(ii)     $\overline{y}$, the mean number of IB Diploma points.

Plot the point $(\overline{x},\overline{y})$ on your scatter diagram and label this point M.

Write down the value of $r$, the Pearson’s product–moment correlation coefficient, for these data.

Write down the equation of the regression line $y$ on $x$ for these eight male students.

Draw the regression line, from part (e), on your scatter diagram.

Use the given equation of the regression line to estimate the number of IB Diploma points that this girl obtained.

Write down a reason why this estimate is not reliable.

Draw a diagram that shows this information.

(i)     Find the probability that a box of cereal, chosen at random, is sold.

(ii)     Calculate the manufacturer’s expected daily income from these sales.

Calculate the manufacturer’s expected daily recycling cost.

Calculate the value of $w$.

Find the median of the examination results.

Find the interquartile range.

Find the final examination result required to obtain the highest possible grade.

Write down the modal class.

Write down the mid-interval value of the modal class.

Calculate an estimate of the mean examination result.

Calculate an estimate of the standard deviation, giving your answer correct to three decimal places.

The teacher sets a grade boundary that is one standard deviation below the mean.

Use the cumulative frequency graph to estimate the number of students whose final examination result was below this grade boundary.

Find the probability that both people chosen are not allergic to nuts.

Copy and complete the tree diagram.

Find the probability that this adult is allergic to nuts and the liquid turns blue.

Find the probability that the liquid turns blue.

Find the probability that the tested adult is allergic to nuts given that the liquid turned blue.

Estimate the number of employees, from this 38, who are allergic to nuts.

Write down the value of a.

Write down the value of b.

Use the tree diagram to find the probability that an employee encountered traffic and was late for work.

Use the tree diagram to find the probability that an employee was late for work.

Use the tree diagram to find the probability that an employee encountered traffic given that they were late for work.

Find the value of x.

Find the value of y.

Find the number of employees who, in the last year, did not travel to work by car, bicycle or public transportation.

Find $n\left(\left(C\cup B\right)\cap P^{\prime }\right)$.

Write down the value of $a$ and of $b$.

Use this regression line to estimate the monthly honey production from a hive that has 270 bees.

Write down the number of low production hives.

Find the value of $k$;

Find the number of hives that have a high production.

Adam decides to increase the number of bees in each low production hive. Research suggests that there is a probability of 0.75 that a low production hive becomes a regular production hive. Calculate the probability that 30 low production hives become regular production hives.

Write down the mode.

Find the value of the range.

Find the mean.

Find the variance.

Find the probability that an orange weighs between 289 g and 310 g.

Find the standardized value for 289 g.

Hence, find the value of σ.

To the nearest gram, find the minimum weight of an orange that the grocer will buy.

Find the probability that the grocer buys more than half the oranges in a box selected at random.

The grocer selects two boxes at random.

Find the probability that the grocer buys more than half the oranges in each box.

Find the probability that there is at least one defective lamp in the sample.

Given that there is at least one defective lamp in the sample, find the probability that there are at most two defective lamps.

Find the probability that a basketball player has a weight that is less than 61 kg.

In a training session there are 40 basketball players.

Find the expected number of players with a weight less than 61 kg in this training session.

Sketch a normal curve to represent this probability.

Find the value of q.

Given that P(W > k) = 0.225 , find the value of k.

For this test state the null hypothesis.

For this test find the p-value.

State a conclusion for this test. Justify your answer.

Given that 5.3% of newborn babies have a low birth weight, find $w$.

Find the number of students in the school that are taught in Spanish.

Find the number of students in the school that study Mathematics in English.

Find the number of students in the school that study both Biology and Mathematics.

Write down $n\left(S\cap \left(M\cup B\right)\right)$.

Write down $n\left(B\cap M\cap S^{\prime }\right)$.

Find the probability that this student studies Mathematics.

Find the probability that this student studies neither Biology nor Mathematics.

Find the probability that this student is taught in Spanish, given that the student studies Biology.

Find the value of $a$ and of $b$.

Write down the correlation coefficient.

Using the regression equation, estimate the number of hot chocolates that Lucy will sell on a day when the maximum temperature is $12°\text{C}$.

Given that $\text{P}(A\cap B)=x$, find the value of $x$.

Find $\text{P}\left(A\text{'} \overline{) B\text{'}}\right)$.

By stating and solving an appropriate equation, show, correct to two decimal places, that $\mu =32.29$.

Find the $86$th percentile time to complete the jigsaw puzzle.

Find the probability that a randomly chosen person will take more than $30$ minutes to complete the jigsaw puzzle.

Find the probability that at least five of them will take more than $30$ minutes to complete the jigsaw puzzle.

Having spent $25$ minutes attempting the jigsaw puzzle, a randomly chosen person had not yet completed the puzzle.

Find the probability that this person will take more than $30$ minutes to complete the jigsaw puzzle.

Find the value of $c$.

Determine whether this lottery is a fair game in the first week. Justify your answer.

Given that the grand prize is not won and the grand prize continues to double, write an expression in terms of $n$ for the value of the grand prize in the $n\text{th}$ week of the lottery.

The $w\text{th}$ week is the first week in which the player is expected to make a profit. Ryan knows that if he buys a lottery ticket in the $w\text{th}$ week, his expected profit is $\$p$.

Find the value of $p$.

State the alternative hypothesis.

Calculate the expected frequency of flights travelling at most 500 km and arriving slightly delayed.

Write down the number of degrees of freedom.

Write down the χ² statistic.

Write down the associated p-value.

State, with a reason, whether you would reject the null hypothesis.

Write down the probability that this flight arrived on time.

Given that this flight was not heavily delayed, find the probability that it travelled between 500 km and 5000 km.

Two flights are chosen at random from those which were slightly delayed.

Find the probability that each of these flights travelled at least 5000 km.

Given that $2\%$ of the flight times are longer than $82$ minutes, find the value of $\sigma$.

Find the probability that a randomly selected flight will have a flight time of more than $80$ minutes.

Given that a flight between the two cities takes longer than $80$ minutes, find the probability that it takes less than $82$ minutes.

Find the expected number of flights that will have a flight time of more than $80$ minutes.

Find the probability that more than $6$ of the flights on this particular day will have a flight time of more than $80$ minutes.

Write down the value of $a$ and the value of $b$.

Use this model to predict the value of $y$ when $x=18$.

Write down the value of $\overline{x}$ and the value of $\overline{y}$.

Draw the line of best fit on the scatter diagram.

Find the value of $a$ and the value of $b$.

Write down the value of Pearson’s product-moment correlation coefficient, $r$.

Interpret, in context, the value of $a$ found in part (a)(i).

On another day, Sarah visits the café to order a coffee. Seven customers have already ordered their coffee and are waiting to receive it.

Use the result from part (a)(i) to estimate Sarah’s waiting time to receive her coffee.

Find the value of $x$.

Find the standard deviation.

Write down the null hypothesis, H₀ , for this test.

State the number of degrees of freedom.

the expected frequency of female students who chose to take the Chinese class.

the ${\chi }^2$ statistic.

State whether or not H₀ should be rejected. Justify your statement.

Find the probability that the student does not take the Spanish class.

Find the probability that neither of the two students take the Spanish class.

Find the probability that at least one of the two students is female.

Draw a scatter diagram for this data. Use a scale of 2 cm for 5000 folders on the horizontal axis and 2 cm for 10 000 Euros on the vertical axis.

Write down, for this set of data the mean number of folders produced, $\overline{x}$;

Write down, for this set of data the mean production cost, $\overline{C}$.

Label the point $\text{M}(\overline{x},\overline{C})$ on the scatter diagram.

Use your graphic display calculator to find the Pearson’s product–moment correlation coefficient, $r$.

State a reason why the regression line $C$ on $x$ is appropriate to model the relationship between these variables.

Use your graphic display calculator to find the equation of the regression line $C$ on $x$.

Draw the regression line $C$ on $x$ on the scatter diagram.

Use the equation of the regression line to estimate the least number of folders that the factory needs to sell in a month to exceed its production cost for that month.

Find the value of k.

Calculate the expected value of the score.

The die is rolled 80 times. On how many rolls would you expect to obtain a three?

Find the value of a and of b.

Write down the correlation coefficient.

Another athlete on this sports team has a hand length of 21.5 cm. Use the regression equation to estimate the height of this athlete.

Write down the total number of people, from this group, who are pet owners.

Write down the modal number of pets.

For these data, write down the median number of pets.

For these data, write down the lower quartile.

For these data, write down the upper quartile.

Write down the ratio of teenagers to non-teenagers in its simplest form.

State the null hypothesis.

State the alternative hypothesis.

Write down the number of degrees of freedom for this test.

Calculate the expected number of teenagers that prefer cats.

Use your graphic display calculator to find the $p$-value for this test.

State the conclusion for this test. Give a reason for your answer.

Find the number of buses that travelled a distance between 15000 and 20000 kilometres.

Use the cumulative frequency curve to find the median distance.

Use the cumulative frequency curve to find the lower quartile.

Use the cumulative frequency curve to find the upper quartile.

Hence write down the interquartile range.

Write down the percentage of buses that travelled a distance greater than the upper quartile.

Find the number of buses that travelled a distance less than or equal to 12 000 km.

Find the value of m.

The smallest distance travelled by one of the buses was 2500 km.
The longest distance travelled by one of the buses was 23 000 km.

On graph paper, draw a box-and-whisker diagram for these data. Use a scale of 2 cm to represent 5000 km.

Find $\text{P}\left(\mu -1.5\sigma <X<\mu +1.5\sigma \right)$.

Find the value of $\mu$ and of $\sigma$.

medium.

large.

premium.

The selling prices of the different categories of avocado at this supermarket are shown in the following table:

The supermarket pays the farm $\$ 200$ for the avocados and assumes it will then sell them in exactly the same proportion as purchased from the farm.

According to this model, find the minimum number of avocados that must be sold so that the net profit for the supermarket is at least $\$ 438$.

Find the probability that it will take Fiona between $15$ minutes and $30$ minutes to walk to the bus stop.

Find $\sigma$.

Find the probability that the bus journey takes less than $45$ minutes.

Find the probability that Fiona will arrive on time.

This year, Fiona will go to school on $183$ days.

Calculate the number of days Fiona is expected to arrive on time.

Find the value of $p$.

Write down the coordinates of A.

Write down the rate of change of $f$ at A.

Find the coordinates of B.

Find the the rate of change of $f$ at B.

Let $R$ be the region enclosed by the graph of $f$ , the $x$-axis, the line $x=b$ and the line $x=a$. The region $R$ is rotated 360° about the $x$-axis. Find the volume of the solid formed.

Write down the value of $a$ and of $b$.

Find the correlation coefficient.

Use the regression equation to estimate the BMI of an adult man whose waist size is 95 cm.

Find the probability that a bag selected at random is rejected.

Estimate the number of bags which will be rejected from a random sample of $100$ bags.

Given that a bag is not rejected, find the probability that it has a mass greater than $1005$ grams.

Find the probability of rolling exactly one red face.

Find the probability of rolling two or more red faces.

Show that, after a turn, the probability that Ted adds exactly $10 to his winnings is $\frac{1}{3}$.

Write down the value of $x$.

Hence, find the value of $y$.

Ted will always have another turn if he expects an increase to his winnings.

Find the least value of $w$ for which Ted should end the game instead of having another turn.

Find P(24.15 < X < 25).

Find $\sigma$, the standard deviation of X.

Hence, find the probability that a seashell selected at random has a length greater than 26 mm.

Find E(Y).

Find the probability that exactly three of these seashells have a length greater than 26 mm.

A seashell selected at random has a length less than 26 mm.

Find the probability that its length is between 24.15 mm and 25 mm.

Events $A$ and $B$ are independent and $\text{P}\left(A\right)=3\text{P}\left(B\right)$.

Given that $\text{P}(A\cup B)=0.68$, find $\text{P}\left(B\right)$.

Find the value of a and the value of b.

Let L₂ be the regression line of y on x. The lines L₁ and L₂ pass through the same point with coordinates (p , q).

Find the value of p and the value of q.

Write down the probability that Ayako avoids the trap in this wall.

Find the probability that only one of Ayako and Natsuko falls into a trap while attempting to pass through a door in the first wall.

Copy the probability tree diagram and write down the relevant probabilities along the branches.

A contestant is chosen at random. Find the probability that this contestant fell into a trap while attempting to pass through a door in the second wall.

A contestant is chosen at random. Find the probability that this contestant fell into a trap.

120 contestants attempted this game.

Find the expected number of contestants who fell into a trap while attempting to pass through a door in the third wall.

State the name for this type of sampling technique.

Show that 3 students will be selected from grade 12.

Calculate the number of students in each grade in the sample.

State the name for this type of sampling technique.

Calculate Pearson’s product moment correlation coefficient, $r$.

Interpret the meaning of the value of $r$ in the context of the principal’s concerns.

Explain why the value of $r$ makes it appropriate to find the equation of a regression line.

Another student at the school, Jasmine, has a self-esteem value of 29.            

By finding the equation of an appropriate regression line, estimate the time Jasmine spent on social media the previous day.

Draw a scatter diagram for temperature difference from 37 °C ($x$) against heartbeat ($y$). Use a scale of 2 cm for 0.1 °C on the horizontal axis, starting with −0.3 °C. Use a scale of 1 cm for 2 heartbeats per minute on the vertical axis, starting with 60 beats per minute.

Write down, for this set of data the mean temperature difference from 37 °C, $\overline{x}$.

Write down, for this set of data the mean number of heartbeats per minute, $\overline{y}$.

Plot and label the point M($\overline{x}$, $\overline{y}$) on the scatter diagram.

Use your graphic display calculator to find the Pearson’s product–moment correlation coefficient, $r$.

Hence describe the correlation between temperature difference from 37 °C and heartbeat.

Use your graphic display calculator to find the equation of the regression line $y$ on $x$.

Draw the regression line $y$ on $x$ on the scatter diagram.

Draw a Venn diagram to represent the given information, using sets labelled $B$, $C$ and $H$.

Show that $x=3$.

Write down the value of $n(B\cap C)$.

Find the probability that this person

(i)     went on at most one trip;

(ii)     went on the coach trip, given that this person also went on both the helicopter trip and the boat trip.

Find the value of a and of b.

Write down the correlation coefficient.

Use your equation to estimate the mean weight of a child that is 1.95 years old.

Find the mean number of hours spent browsing the Internet.

During week 2, the students worked on a major project and they each spent an additional five hours browsing the Internet. For week 2, write down

(i)     the mean;

(ii)     the standard deviation.

During week 3 each student spent 5% less time browsing the Internet than during week 1. For week 3, find

(i)     the median;

(ii)     the variance.

Determine the value of Pearson’s product‐moment correlation coefficient, $r$, for these data.

Interpret, in context, the value of $r$ found in part (a) (i).

Find the equation of the regression line of $p$ on $t$.

Use the regression equation from part (b) to predict this patient’s systolic blood pressure.

A $16$‐year‐old male patient enters the medical clinic for his appointment.

Explain why the regression equation from part (b) should not be used to predict this patient’s systolic blood pressure.

A person is selected at random from Group X. Find the probability that their reaction time is greater than 0.65 seconds.

The probability that the reaction time of a person in Group Y is greater than 0.65 seconds is 0.396. Find the value of σ.

A randomly selected participant has a reaction time greater than 0.65 seconds. Find the probability that the participant is in Group X.

Ten of the participants with reaction times greater than 0.65 are selected at random. Find the probability that at least two of them are in Group X.

Calculate the probability a flight is not late.

Find the value of $m$.

Calculate the probability that at least $7$ of these flights are on time.

Given that at least $7$ of these flights are on time, find the probability that exactly $10$ flights are on time.

SpeedWay increases the number of flights from city $\text{A}$ to city $\text{B}$ to $20$ flights each week, and improves their efficiency so that more flights are on time. The probability that at least $19$ flights are on time is $0.788$.

A flight is chosen at random. Calculate the probability that it is on time.

Write down the value of k.

Show that μ = 106.

Find P(M < 95) .

Find the probability that a bag of apples selected at random contains at most one small apple.

Find the expected number of bags in this crate that contain at most one small apple.

Find the probability that at least 48 bags in this crate contain at most one small apple.

Find P(A ∩ B′ ).

Given that P((A ∪ B)′ ) = 0.19, find P(A | B′ ).

Find the probability that a randomly selected chocolate muffin weighs less than $61 \text{g}$.

In a random selection of $12$ chocolate muffins, find the probability that exactly $5$ weigh less than $61 \text{g}$.

Find the probability that the randomly selected muffin weighs less than $61 \text{g}$.

Given that a randomly selected muffin weighs less than $61 \text{g}$, find the probability that it is chocolate.

Find the value of $\sigma$.

Write down the value of $a$ and of $b$.

Use your regression equation to predict the number of messages sent by a teenager that spent $154$ minutes talking on their phone in October.

State the median reaction time after sleeping well.

Verify that the measurement of $0.46$ seconds is not an outlier.

State why it appears that the mean reaction time is greater than the median reaction time.

Now consider the two box and whisker diagrams.

Comment on whether these box and whisker diagrams provide any evidence that might suggest that not sleeping well causes an increase in reaction time.

The die is tossed five times. Find the probability of obtaining at most three “sixes”.

The die is tossed five times. Find the probability of obtaining the third “six” on the fifth toss.

State whether $N$ is a discrete or a continuous variable.

Write down, for $N$, the modal class;

Write down, for $N$, the mid-interval value of the modal class.

Use your graphic display calculator to estimate the mean of $N$;

Use your graphic display calculator to estimate the standard deviation of $N$.

Find the expected frequency of students choosing the Science category and obtaining 31 to 40 correct answers.

Write down the null hypothesis for this test;

Write down the number of degrees of freedom.

Write down the $p$-value for the test;

Write down the ${\chi }^2$ statistic.

State the result of the test. Give a reason for your answer.

Find .

Find the value of $\mu$.

Find $\lambda$.

Given that $Y>9$, find .

Write down the value of $a$ and of $b$.

Use the model to predict how many edge pieces she had found when she had sorted a total of 750 pieces.

For these data, find the mean distance from a student’s home to school.

Find the value of $p$.

Find the interquartile range.

Write down the probability that the first disc selected is red.

Let $X$ be the number of red discs selected. Find the smallest value of $m$ for which .

Find Pearson’s product-moment correlation coefficient, $r$, for these data.

The relationship between the variables can be modelled by the regression equation $D=ah+b$. Write down the value of $a$ and the value of $b$.

One of these eight students was disappointed with her result and wished she had practised more. Based on the given data, determine how her score could have been expected to alter had she practised an extra five hours per week.

Find the value of $k$.

Write down $\text{P}(X=2)$.

Find $\text{P}(X=2|X>0)$.

Find the value of $a$ and of $b$.

Write down the value of  $r$.

Use the regression equation to estimate the number of visitors on a day when the maximum temperature is 15 °C.

A data set consisting of $16$ test scores has mean $14.5$ . One test score of $9$ requires a second marking and is removed from the data set.

Find the mean of the remaining $15$ test scores.

Find $k$.

Hence, find the probability that exactly $k$ students are left handed;

Hence, find the probability that fewer than $k$ students are left handed.

Show that $2k^2-k+0.12=0$.

Find the value of $k$, giving a reason for your answer.

Hence, find $\text{E}\left(X\right)$.

Find an expression for $q$ in terms of $p$.

Find the value of $p$ which gives the largest value of $\text{E}\left(X\right)$.

Hence, find the largest value of $\text{E}\left(X\right)$.

The graph of $f$ has a horizontal tangent line at $x=0$ and at $x=a$. Find $a$.