DP Mathematics: Applications and Interpretation Questionbank

$d$.

Let $N$ be the number of cats weighing over $4.62 \text{kg}$.

Find the variance of $N$.

It is found that $12$ of the cats weigh more than $x \text{kg}$. Estimate the value of $x$.

Find the proportion of male Persian cats weighing between $5.5 \text{kg}$ and $6.5 \text{kg}$.

Find the probability that exactly one of them weighs over $4.62 \text{kg}$.

Find the probability that over a $12$-month period, there will be exactly $3$ months when Bill does not visit the garden.

Find $r_s$ for the seven employees working in the international department.

The tests are performed at the $5\%$ significance level.

Assuming that:

• there is no correlation between the marks in any of the sections and scores in any of the attributes,
• the outcome of each hypothesis test is independent of the outcome of the other hypothesis tests,

find the probability that at least one of the tests will be significant.

State the name of this type of test for reliability.

Describe one criticism that can be made about the validity of Juliet’s investigation.

State why the hypothesis test should be one-tailed.

Determine the value of $r$, Pearson’s product-moment correlation coefficient, for these remaining responses.

Perform the test, using a $5\%$ significance level, and state your conclusion in context.

Find the value of $c$, of $d$ and of $e$.

Calculate the value of $r$, the product moment correlation coefficient of the sample.

A packet is randomly selected. Given that the packet has a weight greater than $105 \text{g}$, find the probability that it has a weight greater than $110 \text{g}$.

Find the value of $a$.

Find the value of $b$.

The first $150$ athletes that completed the race won a prize.

Given that an athlete took between $22$ and $m$ minutes to complete the $5 \text{km}$ race, calculate the probability that they won a prize.

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

Find $\sigma$.

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

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

Write down ${\text{H}}_0$, the null hypothesis for this test.

Using this information, write down an equation in $p$ and $q$.

Using your answers to parts (b) and (c), find the height of Flower $\text{A}$.

Calculate $\overline{y}$, the mean time to fully charge the robot.

Describe the correlation between the wind speed and the time to fully charge the robot.

Find the value of $b$.

Find the number of potatoes in the sample with a weight of more than $200$ grams.

Find the probability that more than $7$ people arrive at the ride before Shunsuke.

Find the probability of a Type II error in the owner’s test.

the standard deviation.

Find the probability that a bag selected at random weighs more than $1 \text{kg}$.

Write down the null and alternative hypotheses.

Write down the degrees of freedom for this test.

State the null and alternative hypotheses for this test.

Find a $95\%$ confidence interval for $\mu$. You may assume that all conditions for a confidence interval have been met.

State the null and alternative hypotheses for the test.

Interpret your result.

Find the equation of the least squares regression quadratic curve for these four points.

Find the minimum score needed to obtain a grade $5$.

Find an expression for $a$ in terms of $b$.

Use the data in the second table to find the value of $m$ and the value of $b$ for the regression line, $\ln x=m(\ln d)+b$.

prefers a tablet or is $14\text{–}16$ years old.

maximum value of $h$.

a dart lands more than $15 \text{cm}$ from $\text{O}$.

Find Arianne’s expected score in the competition.

Using the data from Loreto’s sample, perform the hypothesis test at a $5\%$ significance level to determine if Loreto should employ extra staff.

Use Euler’s method with a step length of $0.5$ minutes to estimate the maximum value of $H$.

A bag that weighs more than $k$ grams is rejected by the factory for being overweight. The factory rejects $2\%$ of bags for being overweight.

Find the value of $k$.

A statistician in the company suggests it would be fairer if the company passes the inspection when the mean weight of five randomly chosen bags is greater than $950 \text{g}$.

Find the probability of passing the inspection if the statistician’s suggestion is followed.

Determine the probability that a Fuji apple selected at random will be a large apple.

$b$.

Calculate the $p$-value for this test.

Use your graphic display calculator to find an estimate of the mean reaction time.

State, giving a reason, whether Ms Calhoun should accept the null hypothesis.

Write down the value of $a$.

State the null hypothesis.

Write down a transition matrix T representing the movements between the two companies in a particular year.

Find the standard deviation of the changes.

Use all the coordinates in the table to find the equation of the least squares cubic regression curve.

Explain why for any transition state diagram the sum of the out degrees of the directed edges from a vertex (state) must add up to +1.

Write down the transition matrix M, for this Markov chain problem.

State the name for this type of sampling technique.

$c$.

It is not required to state units for this value.

Calculate Pearson’s product moment correlation coefficient for this data.

Write down the coefficient of determination, $R^2$.

Hence predict the temperature of the water after 3 minutes.

Show that $\text{ln}y=qx+\text{ln}p$.

Show that the expected frequency for 20 < $x$ ≤ 4 is 31.5 correct to 1 decimal place.

By finding the equation of a suitable regression line, show that $p=1.83$ and $q=0.986$.

$c$.

Find the value of $a$, of $b$, and of $c$. Give your answers to $3$ decimal places.

By calculating an appropriate ${\chi }^2$ statistic, test, at the 5% significance level, whether or not the binomial distribution gives a good fit to these data.

Test this claim at the 5% level of significance.

Calculate the mean number of brown eggs in a box.

Find the mean and standard deviation of the sample data in the table above. Show how you arrived at your answers.

Some time later, the actual value of $\mu$ is 503. Find the probability of a Type II error.

By drawing a Venn diagram, or otherwise, find $\text{P}\left(A\cup B\right)$.

Write down an expression, in set notation, for the shaded region represented by Diagram 3.

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

Find the value of y.

Find the probability that the 1st July was calm given that the 3rd July is windy.

Find $\text{P}(B)$.

Find the probability that Sara’s baggage arrives in London.

For these data, write down the upper quartile.

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

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.

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.

Using the equation of the regression line, estimate the concentration of dissolved oxygen in the river when the temperature is 18 °C.

Plot and label the point M$\left(\overline{m},\overline{p}\right)$ on the scatter diagram.

Write down the number of degrees of freedom.

Write down the associated p-value.

Write down the probability that this flight arrived on time.

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.

Use the cumulative frequency curve to find the lower quartile.

Find the value of a and of b.

Find the amount of money an employee earned for working 40 hours;

Find the value of a and the value of b.

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

Complete the cumulative frequency table.

A customer buys a large bouquet.

Find the probability that there are 12 roses in this bouquet.

Find the value of q.

If the teacher chooses a response at random, find the probability that it is a response to the Calculus question;

The teacher groups the responses by topic, and chooses two responses to the Logic question. Find the probability that both are not satisfactory.

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.

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

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

Jim’s model is $d=-2.24t+105$, for $0⩽t⩽20$. Use his model to predict the decrease in temperature for any 2 minute interval.

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

Write down the number of leaves with a length less than or equal to 8 cm.

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

Write down the value of  $r$.

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

Find $p$.

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

Find $\text{E}(Y)$.

A customer is selected at random. Find the probability that the customer is male and buys a sandwich.

Var$\left(4X-3Y\right)$.

$\text{E}\left(X^2-Y^2\right)$.

It is given that one in five cups of coffee contain more than 120 mg of caffeine.
It is also known that three in five cups contain more than 110 mg of caffeine.

Assume that the caffeine content of coffee is modelled by a normal distribution.
Find the mean and standard deviation of the caffeine content of coffee.

Given that $Y>9$, find .

Find the probability, in terms of $n$, that the game will end on her second draw.

Calculate the expected value of the score.

Find $\text{P}\left(A^{\prime }\cup B\right)$.

Find the value of k.

Find .

Using this distribution model, find .

Find the probability that Jorgé chooses a red disc.

Find the probability that this student studies Mathematics.

Write down the elements that belong to $A\cap B^{\prime }$.

Write down the standardized value for $x=1$.

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

Sketch the probability density function for X, and shade the region representing P(μ − 2σ < X < μ + σ).

Now suppose that Archie received exactly 20 emails in total in a consecutive two day period. Show that the probability that he received exactly 10 of them on the first day is independent of λ.

Find the value of p.

Find μ, the expected value of X.

Find the remainder when $p\left(x\right)$ is divided by $\left(x-2\right)$.

Another model of smartphone whose battery life may be assumed to be normally distributed with mean μ hours and standard deviation 1.2 hours is tested. A researcher measures the battery life of six of these smartphones and calculates a confidence interval of [10.2, 11.4] for μ.

Calculate the confidence level of this interval.

Find the probability that a student, chosen at random arrives between 45 minutes and 55 minutes after the school opens.

A second school, Mulberry Park, also opens at 08:00 every morning. The arrival times of the students at this school follows exactly the same distribution as Malthouse school.

Given that, on one morning, 15 students arrive at least 60 minutes after the school opens, estimate the number of students at Mulberry Park school.

Find the standard deviation of the times taken by female runners.

Calculate the new value of $\mu$ giving your answer correct to two decimal places.

Copy and complete the tree diagram.

Find the probability that the liquid turns blue.

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

Find the probability that a fish from this lake will have a weight of more than 560 grams.

Find an unbiased estimate for ${\sigma }^2$.

Assuming that $T$ is normally distributed, find

(i)  the 90% confidence interval for the mean time taken to travel to work by the workers of this company,

(ii)  the 95% confidence interval for the mean time taken to travel to work by the workers of this company.

Find the probability that a randomly chosen male bird weighs between 4.75 kg and 4.85 kg.

State suitable hypotheses to investigate whether or not $U$, $V$ are independent.

Find unbiased estimates of $\mu$ and ${\sigma }^2$.

Find the values of the constants $a$ and $b$.

During the interval $p$ < $t$ < $q$, he volume of water in the container increases by $k$ m³. Find the value of $k$.

When $t$ = 0, the volume of water in the container is 2.3 m³. It is known that the container is never completely full of water during the 4 hour period.

Find the minimum volume of empty space in the container during the 4 hour period.

State the central limit theorem as applied to a random sample of size $n$, taken from a distribution with mean $\mu$ and variance ${\sigma }^2$.

Given that the probability that Josie makes a Type II error is 0.25, find the value of $\mu$, giving your answer correct to three significant figures.

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

Peter uses the regression line of $y$ on $x$ as $y=0.248x+83.0$ and calculates that a student with a Mathematics test score of 73 will have a running time of 101 seconds. Comment on the validity of his calculation.

Calculate the probability that Audrey will run at least two marathons in a particular year.

Write down an element that belongs to ${\left(A\cup B\right)}^{\mathrm{\prime }}\cap C$.

Determine the mean of X.

Show that the probability that Chloe wins the game is $\frac{3}{8}$.

will test negative.

$a$.

has the disease given that they tested negative.

The medical centre finds the actual number of positive results in their sample is different than predicted by the tree diagram. Explain why this might be the case.

Sketch a diagram showing the above information.

Sketch a diagram showing the above information.

during the first and third month only.

After the first year, a number of baby magpies start to visit Hank’s garden. It may be assumed that each of these baby magpies visits the garden randomly and independently, and that the number of times each baby magpie visits the garden per month is modelled by a Poisson distribution with mean $2.1$.

Determine the least number of magpies required, including Bill, in order that the probability of Hank’s garden having at least $30$ magpie visits per month is greater than $0.2$.

Find a prediction for the ratio of the number of patients Doctor Black will have, compared to Doctor Green, after two years.

Without calculation, explain why it might not be appropriate to calculate a correlation coefficient for the whole sample of $18$ employees.

Use Juliet’s data to find the value of $a$ and of $b$.

Find the probability that at least three people from the sample took a holiday in the Lake District in $2019$.

Packets are delivered to supermarkets in batches of $80$. Determine the probability that at least $20$ packets from a randomly selected batch have a weight less than $95 \text{g}$.

Given that Andre did not become the champion, find the probability that he lost in the semi-final.

Find the probability that the first ball chosen is labelled $\text{A}$.

Using your answers to parts (b) and (c), find the height of Flower $\text{D}$.

On graph paper, draw a scatter diagram to show the results of Don’s investigation. Use a scale of $1 \text{cm}$ to represent $2$ units on the $x$-axis, and $1 \text{cm}$ to represent $5$ units on the $y$-axis.

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

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.

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

Find the median weight.

The Principal selects the students for the sample by asking those who took part in a previous survey if they would like to take part in another. She takes the first of those who reply positively, up to the maximum needed for the sample.

State which two of the sampling methods listed below best describe the method used.

Stratified          Quota          Convenience          Systematic          Simple random

State a more appropriate model for the water temperature in the lake over an extended period of time. You are not expected to calculate any parameters.

Find the value of $u_1$.

A game is played in which the arrow attached to the centre of the disc is spun and the sector in which the arrow stops is noted. If the arrow stops in sector $1$ the player wins $10$ points, otherwise they lose $2$ points.

Let $X$ be the number of points won

Find $\text{E}\left(X\right)$.

Determine the conclusion of the test, clearly justifying your answer.

Use the given value of $s_n$ to find the value of $s_{n-1}$.

Find the critical region for this test.

Write down the null and alternative hypotheses for the test.

Calculate the $p$-value for this test.

Find an estimate for how many copies the vendor expects to sell each day.

Write down the conclusion to the test. Give a reason for your answer.

State the null and alternative hypotheses.

It is sunny today. Find the probability that it will be sunny in three days’ time.

For this data, find the value of the Pearson’s product-moment correlation coefficient, $r$.

A 57-year-old male also ran in the $5000 \text{m}$ race.

Use the equation of the regression line to estimate the time he took to complete the $5000 \text{m}$ race.

$c$.

Assuming that the model found in part (a) remains valid, estimate the percentage of trees in stock when $d=25$.

Find the time, in seconds, it takes for the blade $\text{[BC]}$ to make one complete rotation under these conditions.

Find the probability that Arianne scores at least $5$ points in the competition.

a dart lands less than $13 \text{cm}$ from $\text{O}$.

Find the probability that Arianne scores at least $5$ points and less than $8$ points.

Find the probability that Arianne does not score a point on a turn of three darts.

Use the general solution from part (d) and the initial condition $h\left(0\right)=3.2$ to predict the value of $T$.

Find this new height.

Find the $p$-value for the test.

Given that Karl has two socks of the same colour find the probability that he has two brown socks.

Find the probability of making a Type I error when weighing a male cuttlefish.

the upper quartile.

Find the value of the Spearman’s rank correlation coefficient $r_s$.

Find the maximum number of tickets that could be sold if the expected number of passengers who arrive to board the flight must be less than or equal to $72$.

their mean standardized score.

the standard deviation of their standardized score.

State the alternative hypothesis.

median reaction time.

Write down the modal class from the table.

Find the probability he will choose a male student at most 9 times.

Product research leads a company to believe that the revenue ($R$) made by selling its goods at a price ($p$) can be modelled by the equation.

$R\left(p\right)=cp{\text{e}}^{dp}$,  $c$, $d\in \mathbb{R}$

There are two competing models, A and B with different values for the parameters $c$ and $d$. 

Model A has $c$ = 3, $d$ = −0.5 and model B has $c$ = 2.5, $d$ = −0.6.

The company experiments by selling the goods at three different prices in three similar areas and the results are shown in the following table.

The company will choose the model with the smallest value for the sum of square residuals.

Determine which model the company chose.

State a suitable null and alternative hypotheses for Tom’s test.

Find the probability he will choose a female student 8 times.

The Head of Year, Mrs Smith, decides to select a student at random from the year group to read the notices in assembly. There are 80 students in total in the year group. Mrs Smith calculates the probability of picking a male student 8 times in the first 20 assemblies is 0.153357 correct to 6 decimal places.

Find the number of male students in the year group.

Find the eigenvalues and corresponding eigenvectors of T.

Calculate Spearman’s rank correlation coefficient for this data.

Interpret what the value of $R^2$ tells you about the model.

Calculate the number of volunteers in the sample under the age of 30.

Give an example of a set of data with 7 numbers in it that does have an outlier, justify this fact by stating the Interquartile Range.

State the name of this test for reliability.

Find the value of this area.

Find unbiased estimates for the population mean.

Hence explain how a straight line graph could be drawn using the coordinates in the table.

Hence predict the total number of people infected by this disease after several months.

Use the logistic model to find the day when the rate of increase of people infected is greatest.

By considering the claims of both Aimmika and Nichakarn, explain whether the advertising was beneficial to the store.

The number of cars passing a certain point in a road was recorded during 80 equal time intervals and summarized in the table below.

Carry out a ${\chi }^2$ goodness of fit test at the 5% significance level to decide if the above data can be modelled by a Poisson distribution.

Write down appropriate hypotheses.

State suitable hypotheses.

In order to test for the goodness of fit, the test statistic was calculated to be 1.0847. Show how this was done.

State your hypotheses, critical number, decision rule and conclusion (using a 5% level of significance).

Find the significance level of this procedure.

State the number of boys who answered questions in Portuguese.

Complete the tree diagram below.

Complete the tree diagram.

Write down the modal number of pets.

State the alternative hypothesis.

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

Find the correlation coefficient.

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

the standard deviation.

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

Show that $x=3$.

Find the value of x.

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

Write down the χ² statistic.

Use the cumulative frequency curve to find the upper quartile.

$q$.

Write down the number of employees who worked 50 hours or less.

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

Draw a box-and-whisker diagram on the grid below to represent the Vitamin C content, in milligrams, for this sample.

Find the standardized value for 289 g.

By considering the graph of f write down the median of $X$;

Find the value of the interquartile range.

Write down the number of degrees of freedom.

Complete the following table.

Find the median test grade of the students.

Find $\text{E}(X)$.

A female customer is selected at random. Find the probability that she buys a sandwich.

$\text{E}\left(2X+7Y\right)$.

Find the value of $\mu$.

Find the probability, in terms of $n$, that the game will end on her first draw.

third draw.

Write down the probability that the second spin is yellow, given that the first spin is blue.

Find the value of $k$.

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 $h$.

Calculate the value of $w$.

It is also known that P($x$ > 2) = $b$.

Find $s$.

an estimate for the mean number of emails received per working day.

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).

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

Find the probability that a runner selected at random will complete the marathon in less than 3 hours.

Complete the Venn diagram for these students.

Determine whether the events $S$ and $M$ are independent.

Find .

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

State suitable hypotheses for the test.

Interpret your $p$-value at the 5% level of significance, justifying your conclusion.

Jack takes a random sample of size 100 and calculates that $\overline{x}=60.2$. Find an approximate 90 % confidence interval for $\mu$.

Write down the modal grade of the eggs.

Sungwon will play the game for $11$ turns.

Find the expected number of times the score on a turn is greater than $4$.

Find the total number of patients who visited the centre during this day.

Find a matrix $\mathit{P}$, with integer elements, such that $\mathit{T}=\mathit{P}\mathit{D}\mathbf{ }{\mathit{P}}^{-1}$, where $\mathit{D}$ is a diagonal matrix.

Use an appropriate test, at the $5\%$ significance level, to determine whether a new employee staying with the firm is independent of their interview rating. State the null and alternative hypotheses, the $p$-value and the conclusion of the test.

Hence comment on the validity of the written assessment as a measure of the level of performance of employees in this department. Justify your answer.

Calculate the mean annual income for these remaining responses.

The critical value for this test, at the $5\%$ significance level, is $0.549$. Juliet assumes that the population is bivariate normal.

Determine whether there is significant evidence of a positive correlation between annual income and happiness. Justify your answer.

Find the coefficient of determination for each of the two models she considers.

Find the least value of $n$ required to ensure that the width of the confidence interval is less than $2 \text{grams}$.

Use your graphic display calculator to find an estimate of the standard deviation of the weights of mangoes from this harvest.

Find the probability that the second ball chosen is labelled $\text{A}$, given that the first ball chosen was labelled $\text{N}$.

Find the value of $a$.

Show that $m=15.7$.

Find the value of $x$.

Two girls are selected at random.

Calculate the probability that one girl answered questions in Mandarin and the other answered questions in Hindi.

Find the $p$-value for the owner’s test.

Calculate the number of grade $12$ students who should be in the sample.

Assuming the data follows a linear model for this period, find the regression line of $T$ on $m$ for the remaining data.

Explain why your line should not be used to estimate the value of $m$ at which the temperature is $10.0$ $°\mathrm{C}$.

Use your line to find an estimate for the water temperature on the first day of May.

Explain in context why your line should not be used to predict the value for December (month $12$).

Find the probability it will rain at least once while Paula is outside.

Find the probability it will rain in each of the three hours Paula is working outside.

Use your answer to part (b) to write down the value of $n$ to the nearest integer.

Find the equation of the least squares regression line of ${\log }_{10} P$ against ${\log }_{10} d$.

Explain why this graph indicates that $P$ is inversely proportional to $d^n$.

Which of the correlation coefficients would you recommend is used to assess whether or not there is an association between total number of minutes late and distance from school? Fully justify your answer.

Write down the null and alternative hypotheses.

Hence find upper and lower bounds for the number of fish in the lake when $t=8$.

Find the mean and standard deviation of the mass of the melons for this year.

State whether the result of the test supports Arriane’s claim. Justify your reasoning.

State, in words, the null hypothesis.

Suggest, with justification, a valid conclusion that Talha could make.

Find an unbiased estimate for the mean number $\left(\mu \right)$ of chocolates per packet.

Comment on your answer to part (b), referring to at least one limitation of the model.

State which of the two sampling methods, systematic or quota, Jason has used.

State whether it is valid to use the regression line $y$ on $x$ for Jason’s estimate. Give a reason for your answer.

Calculate the value of $r_s$.

in less than $6.5 \text{m}$.

in more than $7 \text{m}$.

Find the eigenvalues and eigenvectors of $\mathit{T}$.

By considering the gradient of this curve when $x=4$, explain why it may not be a good model.

$b$.

Find the number of students who obtained a grade $5$.

The sacks of potatoes are transported in crates. There are $10$ sacks in each crate and the weights of the sacks of potatoes are independent of each other.

Find the probability that the total weight of the sacks of potatoes in a crate exceeds $500 \text{kg}$.

Calculate the angle, in degrees, that the blade $\text{[BC]}$ turns through in one second.

Given that Arianne scores at least $5$ points, find the probability that Arianne scores less than $8$ points.

Find the equation of the least squares quadratic regression curve.

Using the given information, complete the following Venn diagram.

Write down the expected frequency of rolling a $1$.

Find the probability that a randomly chosen bag is rejected for being underweight.

Find the value of $k$.

State what your conclusion means in context.

The airline sells $74$ tickets for this flight. Find the probability that more than $72$ passengers arrive to board the flight.

Find $s_{n-1}$ for this sample.

Find the value of the Spearman’s rank correlation coefficient, $r_s$.

Hence write down the number of customers that company X can expect to have in the long term.

By finding a critical value, perform this test at a $5 \%$ significance level.

Find the value of this area.

State what conclusion Charles can make from the answer in part (c).

Write down an expression for the area enclosed by the cubic regression curve, the $x$-axis, the $y$-axis and the line $x=10$.

Explain how your answer to part (f) fits with your answer to part (c).

Explain which part of the transition state diagram confirms this.

Calculate Pearson’s product moment correlation coefficient for this data.

Find the p-value for this t-test.

Find the minimum number of tosses of the coin that Abi will have to make to be at least 95% certain of having finished the game by reaching state C.

State the conclusion of this test, in context, giving a reason.

Hence, show that a data set with only 5 numbers in it cannot have any outliers.

The current drug.

Use the normal distribution model to find the score required to pass.

the number of new people infected on day 6.

Hence find the area enclosed by the exponential function, the $x$-axis, the $y$-axis and the line $x=4.4$.

Find unbiased estimates for the population Variance.

in the long term.

Write down the value of $R^2$ for this model.

Explain why it would not be appropriate to conduct a hypothesis test on the value of $r$ found in (a)(ii).

Use linear regression to estimate the value of $k$ and of $L$.

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

Showing all steps clearly, test whether the die is fair

(i)   at the 5% level of significance;

(ii)  at the 1% level of significance.

Calculate the value of the ${\chi }^2$ statistic and state your conclusion using a 10% level of significance.

The data from the two samples above are combined to form a single set of data. The following frequency table gives the observed frequencies for the combined sample. The data has been divided into five intervals.

Test, at the 5% level, whether the combined data can be considered to be a sample from a normal population with a mean of 380.

Find the value of x.

Find P(A ∩ B′ ).

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.

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, $\overline{y}$.

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

Calculate the expected number of teenagers that prefer cats.

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

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

Write down the mid-interval value for 10 ≤ t < 15.

Write down the correlation coefficient.

Use the cumulative frequency curve to find an estimate for the number of students who worked at most 35 hours per month.

(i)     Find the number of students who spent between 25 and 30 hours browsing the Internet.

(ii)     Given that 10% of the students spent more than k hours browsing the Internet, find the maximum value of $k$.

Any suitcase that weighs more than $k$ kg is identified as excess baggage.
19.6 % of the suitcases at this airport are identified as excess baggage.

Find the value of $k$.

For this test find the p-value.

State the null hypothesis for this test.

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

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

For this test, write down the ${\chi }^2$ statistic.

Show that event A and event D are not independent.

It is further given that P(X ≤ 1) = 0.09478 correct to 4 significant figures.

Determine the value of n and the value of p.

Find the probability that a randomly selected Infiglow battery will have a life of at least 15 hours.

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

Use your graphic display calculator to estimate the mean 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 ${\chi }^2$ statistic.

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

Write down an equation, in terms of $x$ and $y$, for the total number of times the die was rolled.

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

Find the value of $k$.

Find $q$.

Show that $a=32$ and $b=\frac{1}{12}$.

Find the gradient of $L$.

Hence or otherwise, find the obtuse angle formed by the tangent line to $f$ at $x=8$ and the tangent line to $f$ at $x=2$.

only a sandwich.

Complete the tree diagram.

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

In the table indicate which two of the given statements are true by placing a tick (✔) in the right hand column.

Calculate the manufacturer’s expected daily recycling cost.

Find P(−1.6 < $z$ < 2.4). Write your answer in terms of $a$ and $b$.

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

Prove that $p\left(x\right)$ has only one real zero.

Write down the transformation that will transform the graph of $y=p\left(x\right)$ onto the graph of $y=8x^3-12x^2+16x-24$.

Given that $\text{P}(X=2)=0.241667$ and $\text{P}(X=3)=0.112777$, use part (a) to find the value of $\mu$.

Calculate the probability that Iqbal passes at least two of the papers he attempts.

Given that $\mu =253$ and $\sigma =1.5$ find the probability that a randomly chosen packet of biscuits is underweight.

Write down $n(W\cap S)$.

Find the probability that at least one of Francesco’s light bulbs is defective.

State the distribution of your test statistic, including the parameter.

Find the p-value for the test.

At the start of the day, one employee, Amanda, has a queue of four customers. A second employee, Brian, has a queue of three customers. You may assume they work independently.

Find the probability that Amanda’s queue will be dealt with before Brian’s queue.

Write down the conclusion reached.

Find the probability that the weight of a randomly chosen male bird is more than twice the weight of a randomly chosen female bird.

Use a two-tailed test to determine the $p$-value for the above results.

Find unbiased estimates of the mean of $T$.

Calculate the probability that he makes a Type II error.

Carry out an appropriate test and state the $p$-value obtained.

Use a chi-squared goodness of fit test to investigate whether or not, at the 5 % level of significance, the N(0, 1) distribution can be used to model these results.

Write down the gradient of the curve of $f$ at P.

Find the value of $a$.

State suitable hypotheses $H_0$ and $H_1$ to test Peter’s claim, using a two-tailed test.

It is known that $20\%$ of the runners took more than $28$ minutes and less than $k$ minutes to complete the race.

Find the value of $k$.

Explain why it is valid to use the regression equation to estimate the airfare between Hong Kong and Tokyo.

Write down the number of degrees of freedom.

Calculate the probability that the customer is an adult.

The random variable $X$ has the Poisson distribution $\text{Po}(m)$. Given that $\text{P}(X>0)=\frac{3}{4}$, find the value of $m$ in the form $\ln a$ where $a$ is an integer.

Draw a Venn diagram to illustrate this information, placing all relevant information on the diagram.

will not have the disease and will test positive.

A cat is selected at random from all $160$ cats.

Find the probability that the cat was female, given that its weight was over $4.7 \text{kg}$.

Ten of the cats are chosen at random. Find the probability that exactly one of them weighs over $6.25 \text{kg}$.

Show that $11$ employees are selected for the sample from the national department.

Describe one way in which Juliet could improve the reliability of her investigation.

Juliet classifies response $\text{K}$ as an outlier and removes it from the data. Suggest one possible justification for her decision to remove it.

State the null and alternative hypotheses for this test.

Jenna is a keen book reader. The number of books she reads during one week can be modelled by a Poisson distribution with mean $2.6$.

Determine the expected number of weeks in one year, of $52$ weeks, during which Jenna reads at least four books.

Find the probability that a randomly selected packet has a weight less than $100 \text{g}$.

Find the value of $v$.

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

Write down the modal group for these data.

Find the probability that the first ball chosen is labelled $\text{A}$ or labelled $\text{N}$.

Find the probability that both balls chosen are labelled $\text{N}$.

Write down a second equation in $p$ and $q$.

Hence or otherwise estimate the charging time when the wind speed is $27 {\text{km\hspace{0.05em}h}}^{-1}$.

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

Sketch a diagram to represent this information.

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

Find the probability it will not rain while Paula is outside.

Find the transition matrix for the maze.

A scientist sets up the robot and then leaves it moving around the maze for a long period of time.

Find the probability that the robot is in room $\text{B}$ when the scientist returns.

Find an expression for $P$ in terms of $d$.

Find the probability there will be space for him on the $9.01$ car.

Find the sum of the square residuals for Jorge’s model using the values $t=1, 2, 3, 4$.

have a total mass greater than $6.0 \text{kg}$.

State an assumption that is being made for $X$ to be considered as following a binomial distribution.

Show that an estimate for $\text{Var}\left(X\right)$ is $38.25$.

the mean.

Find the expected score of a game.

the minimum number of sick days taken during the year.

the lower quartile.

Determine the transition matrix for this graph.

Write down the median time to read the book.

Complete the following transition diagram to represent this information.

Katie works for $180$ days in a year.

Find the probability that Katie cycles to work on her final working day of the year.

State the null and alternative hypotheses.

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

Find the time, in seconds, that point $\text{C}$ is above a height of $100 \text{m}$, during each complete rotation.

Write down the amplitude of the function.

Perform the test, clearly stating the conclusion in context.

Show that $\frac{dH}{dt}\approx 0.2514-0.009873t-0.1405\sqrt{H}$, where $0\le t\le T$.

Find the number of surveyed students who did not like any of the three flavours.

Write down the percentage of bags that weigh more than $500 \text{g}$.

Find the probability of a Type I error.

Two different applicants are chosen at random from the original group.

Find the probability that both applicants applied to the Arts programme.

Calculate the $p$-value for this test.

Find an unbiased estimate of the population mean of $d$.

Given that all assumptions for this test are satisfied, carry out an appropriate hypothesis test. State and justify your conclusion. Use a $5\%$ significance level.

Find the probability of making an error using the zoologist’s method.

Find the probability of making a Type II error when weighing a female cuttlefish.

Write down the expected number of passengers who will arrive to board the flight if $72$ tickets are sold.

Name the type of sampling that best describes the method used by the Principal.

Calculate the expected score.

Find a 95 % confidence interval for the population mean, giving your answer to 4 significant figures.

In a coffee shop, the time it takes to serve a customer can be modelled by a normal distribution with a mean of 1.5 minutes and a standard deviation of 0.4 minutes.

Two customers enter the shop together. They are served one at a time.

Find the probability that the total time taken to serve both customers will be less than 4 minutes.

Clearly state any assumptions you have made.

The average number of fish caught in an hour is actually 2.5.

Find the probability of a Type II error.

Use an appropriate test to determine whether there is evidence, at the 5 % significance level, that the students in school B have improved more than those in school A.

State whether this estimate is reliable. Justify your answer.

Use an appropriate test to determine whether showing an improvement is independent of gender.

Write down the coefficient of determination.

Find the equation of the regression line of $\text{ln}\left(T-25\right)$ on $t$.

Write down the transition matrix for this Markov chain.

Find ${\mathbf{v}}_1\text{,}{\mathbf{v}}_2\text{,}{\mathbf{v}}_3\text{,}{\mathbf{v}}_4$.

Find the steady state probability vector for M in terms of $a$ and $b$.

State the hypotheses for this t-test.

State the name of this test for validity.

Find the probability of making a Type I error.

Hence state the probability of a Type I error for this test.

Calculate unbiased estimates of the population mean and the population variance.

Six coins are tossed simultaneously 320 times, with the following results.

At the 5% level of significance, test the hypothesis that all the coins are fair.

Carry out a test at the 1% significance level, and state your conclusion.

A horse breeder records the number of births for each of 100 horses during the past eight years. The results are summarized in the following table:

Stating null and alternative hypotheses carry out an appropriate test at the 5% significance level to decide whether the results can be modelled by B (6, 0.5).

State suitable hypotheses for testing this belief.

Without doing any further calculations, explain briefly how you would carry out a test, at the 5% significance level, to decide if the data can be modelled by B(6, $p$), where $p$ is unspecified.

Write down an expression, in set notation, for the shaded region represented by Diagram 2.

Shade, on the Venn diagram, the region represented by the set $J\cap K$.

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

Find the value of x.

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

Find the probability that exactly one of the selected balls is green.

Write down the value of $n(F)$.

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, $r$.

For these data, write down the lower quartile.

Calculate the expected number of male engineers.

Abhinav rejects H₀.

State a reason why Abhinav is incorrect in doing so.

For these data, find the equation of the regression line y on x.

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

Find the value of a and of b.

On the same grid, complete the cumulative frequency curve for these data.

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.

Write down a reason why this estimate is not reliable.

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

Sketch a normal curve to represent this probability.

Draw the regression line on the scatter diagram.

State the conclusion of this ${\chi }^2$ test. Give a reason for your answer.

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

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 of rolling two or more red faces.

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

Find the value of $k$.

Hence state the interquartile range of $X$.

Write down the number of low production hives.

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

Using the mean score, write down a second equation in terms of $x$ and $y$.

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

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

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

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

Find .

Find $\text{P}(Y⩾3)$.

Find $\lambda$.

Hence find the expected number of times per day that Steffi is fed at Will’s house.

Show that the expected number of occasions per year on which Steffi visits Will’s house and is not fed is at least 30.

Write down the value of k.

(i)     show that $\text{P}(A)=\frac{1}{3}$;

(ii)     hence find $\text{P}(B)$.

Place the numbers $2\pi \text{,}-5\text{,}{3}^{-1}$ and $2^{\frac{3}{2}}$ in the correct position on the Venn diagram.

Find the coordinates of P.

Hence, find the equation of L in terms of $a$.

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.

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

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

Find the value of P(μ − 2σ < X < μ + σ).

Find $\text{P}(A)$.

Find P(X > μ).

Find the expected number of weeks in the year in which Lucca eats no bananas.

State suitable hypotheses for a two-tailed test.

Calculate the probability that, on a randomly selected day, Timmy makes a profit.

Find the probability that Pablo leaves home before 07:00 and is late for work.

Find the probability that Pablo is late for work.

The shop is open for 24 days every month.

Calculate the probability that, in a randomly selected month, Timmy makes a profit on between 5 and 10 days (inclusive).

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

Determine whether rock-climbing is offered by the school in the fall/autumn trimester.

Each of the 25 students in a class are asked how many pets they own. Two students own three pets and no students own more than three pets. The mean and standard deviation of the number of pets owned by students in the class are $\frac{18}{25}$ and $\frac{24}{25}$ respectively.

Find the number of students in the class who do not own a pet.

Find the probability that the time taken by an employee to deal with a queue of three customers is less than nine minutes.

Find unbiased estimates of the variance of $T$.

Two randomly chosen male birds and three randomly chosen female birds are placed on a weighing machine that has a weight limit of 18 kg. Find the probability that the total weight of these five birds is greater than the weight limit.

Line $L$ passes through the origin and has a gradient of $\text{tan}\theta$. Find the equation of $L$.

Find the derivative of $f$.

Calculate the mean score.

State suitable hypotheses to test the inspector’s claim.

Write down the probability that Josie makes a Type I error.

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

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

Use your regression equation to estimate the cost of a flight from Hong Kong to Tokyo with Galois Airways.

Find the probability that Sungwon scores greater than $4$ on both of her first two turns.

Show that the expected number of children who chose shrimp is $31$, correct to two significant figures.

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

Determine the variance of X.

Find the probability that exactly seven rooms will have fewer than three faults in the carpet.

Find the probability that the carpet laid in the first room has fewer than three faults.

Find the proportion of male Persian cats weighing between $5.5 \text{kg}$ and $6.5 \text{kg}$.

$b$.

State the sampling method being used.

$c$.

Determine the expected number of cats in this group that have a weight of less than $5.3 \text{kg}$.

Hence, or otherwise, determine the expected ratio of the number of patients Doctor Black would have compared to Doctor Green in the long term.

Hence comment on the reliability of the written assessment.

State whether your $p$-value suggests that $X$ and $Y$ are independent.

Given a further value $x=5.2$ from the distribution of $X$, $Y$, predict the corresponding value of $y$. Give your answer to one decimal place.

From a random sample of $n$ holidaymakers, the probability that at least one of them took a holiday in the Lake District in $2019$ is greater than $0.999$.

Determine the least possible value of $n$.

From a random sample of $500$ packets, determine the number of packets that would be expected to have a weight lying within $1.5$ standard deviations of the mean.

Given that the weights of the broccoli actually follow a normal distribution with mean $392 \text{grams}$ and variance $80 {\text{grams}}^2$, find the probability of Anjali making a Type II error.

Write down the correlation coefficient.

Don concluded from his investigation: “There is no causation between wind speed and the time to fully charge the robot”.

In the context of the question, briefly explain the meaning of “no causation”.

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

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

Find the probability it will not rain while Paula is outside.

Write down the probability it rains during Paula’s lunch break.

Given this result, comment on the validity of the linear model used in part (a).

the mode.

Paul claims that this box and whisker diagram can be used to infer that the percentage of employees who took fewer than six sick days is smaller than the percentage of employees who took more than eleven sick days.

State whether Paul is correct. Justify your answer.

Calculate the interquartile range.

Hence find the long-term percentage of sunny days in Vokram.

Find the lower quartile of the weights of the sacks of potatoes.

State the conclusion for the test in context. Give a reason for your answer.

Write down the number of degrees of freedom.

Find the height of $\text{C}$ above the ground when $t=2$.

At any given instant, find the probability that point $\text{C}$ is visible from Tim’s window.

a dart lands less than $13 \text{cm}$ from $\text{O}$.

Find the probability that Arianne does not score a point on a turn of three darts.

Find the probability that Arianne throws two consecutive darts that land more than $15 \text{cm}$ from $\text{O}$.

Write down null and alternative hypotheses for Loreto’s test.

Find the probability that Arianne throws two consecutive darts that land more than $15 \text{cm}$ from $\text{O}$.

Suggest why Eva’s use of the linear regression equation in this way could be unreliable.

Hence, write down a suitable domain for Eva’s function $h\left(t\right)=p{t}^2+qt+r$.

Interpret the meaning of parameter $a$ in the context of the model.

By solving the differential equation $\frac{dh}{dt}=-R^2\sqrt{70 560h}$, show that the general solution is given by $h=17 640{\left(c-R^{2}t\right)}^2$, where $c\in \mathrm{ℝ}$.

Show that $\frac{dh}{dt}=-R^2\sqrt{70 560h}$.

Write down the degrees of freedom.

Find the probability that Karl takes two socks of the same colour.

Find the probability that the company fails the inspection.

Using your answer to (a), or otherwise, find the long-term probability of the switch being in state $\text{A}$. Give your answer in the form $\frac{c}{d}$, where $c, d\in {\mathrm{\mathbb{Z}}}^{+}$.

Determine the probability that fewer than $7$ people will pass this polygraph test.

State the conclusion of the test. Justify your answer.

Calculate the expected number of people who will pass this polygraph test.

Calculate the probability that exactly $4$ people will fail this polygraph test.

Find an unbiased estimate of the population variance of $d$.

When calculating the ranks, Chester incorrectly read the Netherlands’ score as $478$. Explain why the value of the Spearman’s rank correlation $r_s$ does not change despite this error.

the mean mark.

Perform an appropriate test at the $5\%$ significance level to see if the mean marks achieved by the students in the school are higher than the national standard. It can be assumed that the marks come from a normal population.

Jae Hee plays the game twice and adds the two scores together.

Find the probability Jae Hee has a total score of −3.

Find the estimated number of teenagers who have a reaction time greater than $0.4$ seconds.

Using the value of $r$, interpret the relationship between Stan’s score and Minsun’s score.

State why comparing only the final IB points of the students from the two schools would not be a valid test for the effectiveness of the two different teaching methods.

Use a paired $t$-test to determine whether there is significant evidence that the students in school A have improved their IB points since the start of the course.

Find the mean and variance for the sample data given in the table.

Hence, deduce the form of ${\mathbf{v}}_n$.

State the conclusion of this test, in context, giving a reason.

$A$.

It is not required to state units for this value.

Find the steady state probability vector for this Markov chain.

Draw a transition state diagram for this Markov chain problem.

Find the equation of the least squares exponential regression curve for $28-T$.

State an assumption that Anita is making, in order to use a t-test.

Write down the coefficient of determination.

Hence comment on the suitability of the model from (b)(ii) in comparison with the linear model found in part (a).

Show that this data leads to an estimated value of $p=0.4$.

Copy and complete the table, showing how you arrived at your answers.

Hence estimate $p$, the probability that a randomly chosen egg is brown.

Find the exact value of the mean of this distribution.

Most of the expected frequencies have been calculated in the third column. (Frequencies have been rounded to the nearest integer, and frequencies in the first and last classes have been extended to include the rest of the data beyond 15 and 225. Find the values of $a$, $b$ and $c$ and show how you arrived at your answers.

the level of significance of a hypothesis test.

Calculate the mean of these data and hence estimate the value of $p$.

Write down an expression, in set notation, for the shaded region represented by Diagram 1.

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

In the table indicate whether the given statements are True or False.

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

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

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

For these data, find Pearson’s product-moment correlation coefficient, r.

Draw the line of best fit, by eye, on the scatter diagram.

Using your line of best fit, estimate the physics test score for a student with a score of 20 in their mathematics test.

State the alternative hypothesis.

Use the cumulative frequency curve to find the median distance.

Find the value of $\text{Var}(X)$.

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

State a conclusion for this test. Justify your answer.

Use your graphic display calculator to find the ${\chi }^2$ statistic for this data.

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 production cost, $\overline{C}$.

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

State, giving a reason, whether the null hypothesis should be rejected.

$\text{P}\left(X>\mu +5\right)$.

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

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

Find the least possible value of n.

Write down the value of $x$.

By considering the graph of f write down the mean of $X$;

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

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.

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

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

Jim describes the correlation as very strong. Circle the value below which best represents the correlation coefficient.

$$0.9920.2510-0.251-0.992$$

Write down the value of $x$.

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

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

Find the interquartile range.

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

Given that , and that 0.25 < s < 0.4 , find the value of s.

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

Write down the probability of drawing three blue marbles.

Find $\text{Var}(X)$.

Find $\left(f\circ f\right)\left(x\right)$.

In any given year of 365 days, the probability that Steffi does not visit Will for at most $n$ days in total is 0.5 (to one decimal place). Find the value of $n$.

fourth draw.

Find the value of k.

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

Find the value of $p$.

Find the value of $q$.

Find the probability that a randomly chosen student will be accepted by Marron College.

Given that Naomi attends Marron College, find the probability that she achieved a mark of at least 500 on the test.

Find .

Find the value of $a$.

Estimate the interquartile range of the distribution.

Find the price that is two standard deviations above the mean price.

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

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

Find the value of $q$.

Show that $\text{P}(A\cup B)=\text{P}(A)+\text{P}(A^{\prime }\cap B)$.

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

(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.

Given that $z$ > −1.6, find the probability that z < 2.4 . Write your answer in terms of $a$ and $b$.

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

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

Show that $\text{P}(X=x+1)=\frac{\mu }{x+1}\times \text{P}(X=x),x\in \mathbb{N}$.

Find the probability that a student, chosen at random arrives at least 60 minutes after the school opens.

Two days next week Pablo will drive to work. Find the probability that he will be late at least once.

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

Write down the probability that a light bulb produced by Home Shine is not defective.

Find the probability that both light bulbs are not defective.

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

Find the product-moment correlation coefficient $r$.

Use an appropriate regression line to estimate the weight of a fish with length 360 mm.

Find an unbiased estimate for $\mu$.

Find the $p$-value for the test.

(i)     Hence show that $X$ has two modes $m_1$ and $m_2$.

(ii)     State the values of $m_1$ and $m_2$.

Determine the minimum value of $x$ such that the probability Kati receives at least one free gift is greater than 0.5.

Find the critical region for Josie’s test, giving your answer correct to two decimal places.

The line $y=kx-5$ is a tangent to the curve of $f$. Find the values of $k$.

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.

Consider f(x), g(x) and h(x), for x∈$\mathbb{R}$ where h(x) = $\left(f\circ g\right)$(x).

Given that g(3) = 7 , g′ (3) = 4 and f ′ (7) = −5 , find the gradient of the normal to the curve of h at x = 3.

Write down the probability that the runner completed the race in more than $28$ minutes.

Write down $n\left(B\right)$.

Find the probability that Sungwon’s score on her first turn is greater than $4$.

The random variable $Y$ has the Poisson distribution $\text{Po}(2m)$. Find $\text{P}(Y>1)$ in the form $\frac{b-\ln c}{c}$ where $b$ and $c$ are integers.

Assuming that the distribution of $X, Y$ is bivariate normal with product moment correlation coefficient $\rho$, calculate the $p$-value of your result when testing the hypotheses $H_0 : \rho =0 ; H_1 : \rho >0$.

The probability that a randomly selected packet has a weight greater than $w$ grams is $0.444$. Find the value of $w$.

Assuming that the shopkeeper’s claim is correct, find the probability that the weight of six randomly chosen carrots is more than two times the weight of one randomly chosen broccoli.

Find the value of $t$.

Write down the value of the median distance in kilometres (km).

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

Find the probability that Fiona will arrive on time.

Find the expected number of short trips when it rained.

Plot and label the point $\text{M}$ on your scatter diagram.

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

Find the probability that an apple from the tree has a weight greater than $90$ grams.

The weight of the smallest potato in the sample is $20$ grams and the weight of the largest is $400$ grams.

Use the scale shown below to draw a box and whisker diagram showing the distribution of the weights of the potatoes. You may assume there are no outliers.

Write down the probability it rains during Paula’s lunch break.

Find the probability of scoring a six when rolling the novelty die.

Find the value of $a$ and the value of $b$, giving your answers correct to one decimal place.

both have a mass greater than $3.0 \text{kg}$.

a player scores $6$, given that they scored at least $3$.

George goes fishing. From experience he knows that the mean number of fish he catches per hour is $1.1$. It is assumed that the number of fish he catches can be modelled by a Poisson distribution.

On a day in which George spends $8$ hours fishing, find the probability that he will catch more than $9$ fish.

Find the standard deviation of the weights of these bags of apples.

Find the $p$-value for this test.

Write down the conclusion to the test. Give a reason for your answer.

Write down the degrees of freedom for this test.

Find an estimate for how many copies the vendor expects to sell each day.

Write down the conclusion to the test. Give a reason for your answer.

If the mouse was left to wander indefinitely, use your graphic display calculator to estimate the percentage of time that the mouse would spend at point $\text{F}$.

Sheila finds $126$ coffees were sold during the $5$-hour period.

State Sheila’s conclusion to the test. Justify your answer.

Describe the correlation.

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

Write down the equation of the regression line of $t$ on $x$, in the form $t=ax+b$.

Write down the $p$-value.

Find the probability that Arianne scores at least $5$ points and less than $8$ points.

Given that Arianne scores at least $5$ points, find the probability that Arianne scores less than $8$ points.

a dart lands more than $15 \text{cm}$ from $\text{O}$.

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

Find the value of $k$.

A student is chosen at random from the surveyed students.

Find the probability that this student likes kiwi fruit smoothies given that they like mango smoothies.

Find $\text{P}(Y>100)$.

Find the probability that the applicant applied for the Arts programme.

Find the probability that a randomly chosen applicant from this group was accepted by the university.

Determine if the Netherlands’ score is an outlier for this data. Justify your answer.

$a$.

State whether the data is discrete or continuous.

the standard deviation of the marks.

State, giving a reason, whether the null hypothesis should be accepted.

Determine the $90\text{th}$ percentile of the reaction times from the cumulative frequency graph.

Write down the value of $b$.

Suggest how, if at all, the estimated mean and estimated median reaction times will change if the errors are corrected. Justify your response.

Calculate the $p$-value for this test.

If you were to repeat the test performed in part (e) intending to compare the quality of the teaching between the two schools, suggest two ways in which you might choose your sample to improve the validity of the test.

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

Hence write down matrices P and D such that T = PDP⁻¹.

Find an expression for the number of customers company X has after $n$ years, where $n\in \mathbb{N}$.

Hence, find the expected improvement between predicted and final points for an increase of one unit in effort grades, giving your answer to one decimal place.

find the value of $a$ and of $b$.

Explain why it is appropriate to use Spearman’s rank correlation coefficient to measure the strength of the relationship between $P$ and $d$.

Show that $\text{ln}\left(T-25\right)=bt+\text{ln}a$.

Calculate Spearman’s rank correlation coefficient for this data.

predict the temperature of the metal rod after 3 minutes.

Hence determine, with a reason, if the new exam is a valid indicator of future performance.

Find the steady state probability vector for this Markov chain problem.

Explain why having a steady state probability vector means that the matrix M must have an eigenvalue of $\lambda =1$.

Recalling definitions, such as the Lower Quartile is the $\frac{n+1}{4}th$ piece of data with the data placed in order, find an expression for the Interquartile Range.

Explain why $28-T$ can be modeled by an exponential function.

Use the trapezoidal rule to find an estimate for the area.

With reference to the shape of the graph, explain whether your answer to part (a)(i) will be an over-estimate or an underestimate of the area.

Give two reasons why the prediction in part (b)(ii) might be lower than 14.

Given that $p=0.2$ find the probability of making a Type II error.

Use all the coordinates in the table to find the equation of the least squares cubic regression curve.

Write down an expression for the area enclosed by the cubic function, the $x$-axis, the $y$-axis and the line $x=4.4$.

Perform a suitable test, at the 5% significance level, to determine if there is a difference between the mean scores of males and females. You should clearly state your hypotheses, the p-value and your conclusion.

Use an exponential regression to find the value of $a$ and of $b$, correct to 4 decimal places.

$L$.

Show that Jonas’s network satisfies the requirement of there being less than a $2\%$ probability of the network failing after a power surge.

State one assumption that Aimmika needs to make about the sales of bags of rice to support her belief that it follows a Poisson distribution.

Stating null and alternative hypotheses, carry out an appropriate test at the 5 % level to decide whether the farmer’s claim can be justified.

Calculate the mean number of eggs laid by these birds.

Explain what is meant by “level of significance” in part (a).

Test, at the 5% level of significance, whether or not the data can be modelled by a Poisson distribution.

The scientists wish to investigate the claim that Group B gain weight faster than Group A. Test this claim at the 5% level of significance, noting which hypothesis test you are using. You may assume that the weight gain for each group is normally distributed, with the same variance, and independent from each other.

Calculate an appropriate value of ${\chi }^2$ and state your conclusion, using a 1% significance level.

Shade, on the Venn diagram, the region represented by the set ${\left(H\cup I\right)}^{\prime }$.

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

Find $\text{P}(A^{\prime }\cap B)$.

Draw a tree diagram to represent this information for the first three days of July.

Find the probability that the 3rd July is calm.

Find $\text{P}(A\cup B)$.

On the Venn diagram, shade the region $A\cap (B\cup C{)}^{\prime }$.

Complete the Venn diagram using the above information.

The health inspector visits two school cafeterias. She inspects the same number of meals at each cafeteria. The data is shown in the following box-and-whisker diagrams.

Meals prepared in the school cafeterias are required to have less than 10 grams of sugar.

State, giving a reason, which school cafeteria has more meals that do not meet the requirement.

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

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

Find the p-value for this test.

Write down the mid-interval value for the 100 < x ≤ 150 group.

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.

Write down the total number of customers in terms of k.

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

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.

Find the mean number of hours spent browsing the Internet.

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.

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

For the students in this group write down the median age.

Find the mean number of hours that the people surveyed watch television per week.

If the teacher chooses a response at random, find the probability that it is a satisfactory response, given that it is a response to the Calculus question.

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

Hence, find the value of σ.

By considering the graph of f write down the mode of $X$.

Calculate $P(X⩽4|X⩾3)$.

Find the value of $k$;

Use your graphic display calculator to find the ${\chi }^2$ statistic for this test.

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

Calculate the mean test grade of the students;

Find the value of m.

Show that $a=\frac{2}{3}$.

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

Find u.

Hence, write down $f^{-1}\left(x\right)$.

(i)     Find the value of $c$.

(ii)     Show that $b=\frac{\pi }{6}$.

(iii)     Find the value of $a$.

Find the probability that more than 100 cakes will be sold on a typical day.

Find the probability that on a randomly selected day, Steffi does not visit Will’s house.

Find the probability that both spins are yellow.

Find the probability that this applicant took at least 40 minutes to complete the test.

Write down the probability that the mass of one of these corncobs is greater than 400 grams.

Find the probability that the price of a kilogram of tomatoes, chosen at random, will be between 2.00 and 3.00 euro.

To stimulate reasonable pricing, the city offers a free permit to the sellers whose price of a kilogram of tomatoes is in the lowest 20 %.

Find the highest price that a seller can charge and still receive a free permit.

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

Find $f^{\prime }\left(x\right)$.

Write down the elements that belong to $A\cap B$.

A newborn baby has a low birth weight.

Find the probability that the baby weighs at least 2.15 kg.

Using this distribution model, find the standard deviation of $X$.

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

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

Hence show that events $A^{\prime }$ and $B$ are independent.

Find the probability that a wolf selected at random is at least 5 years old.

Eight wolves are independently selected at random and their ages recorded.

Find the probability that more than six of these wolves are at least 5 years old.

Find the remainder when $p\left(x\right)$ is divided by $\left(x-3\right)$.

Given that more than 5 taxis arrive during T, find the probability that exactly 7 taxis arrive during T.

Complete the given probability tree diagram for Iqbal’s three attempts, labelling each branch with the correct probability.

Find the probability of making a Type II error.

Find P(M < 95) .

Calculate $T_1$.

Calculate the new value of $\sigma$.

Write down, in terms of $F$, $W$ and $S$, an expression for the set which contains only archery, baseball, kayaking and surfing.

Find the probability that this person is not allergic to nuts.

The maximum weight a hand net can hold is 6 kg. Find the probability that a catch of 11 fish can be carried in the hand net.

State the conclusion of the test, justifying your answer.

State the p-value and interpret it in this context.

Find the coordinates of the point of intersection of the planes defined by the equations $x+y+z=3,x-y+z=5$ and $x+y+2z=6$.

Find

(i)     $\text{E}(X)$;

(ii)     $\text{Var}(X)$.

Show that $\text{P}(X=3)=0.001$ and $\text{P}(X=4)=0.0027$.

a Type I error.

Find the value of $p$.

Calculate the probability that the runner completed the race in less than $26$ minutes.

the ${\chi }_2$ statistic.

the $p$-value.

Hence, show that the long-term transition matrix ${\mathit{T}}^{\infty }$ is given by ${\mathit{T}}^{\infty }=\left(\begin{array}{cc}\frac{10}{17}& \frac{10}{17}\\ \frac{7}{17}& \frac{7}{17}\end{array}\right)$.

on exactly $12$ occasions.

State the null and alternative hypotheses for this test.

Interpret, referring to income and happiness, what the value of $a$ represents.

Find the significance level for this test.

Find $m$.

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

Complete the values in the tree diagram.

Calculate $\overline{x}$, the mean wind speed.

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

Draw this regression line on your scatter diagram.

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

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 lower quartile.

Find the upper quartile.

Write down the value of $\underset{i=1}{\overset{9}{\text{Σ}}}{u}_i$.

Given it rains at least once while Paula is outside find the probability that it rains during her lunch hour.

Find the probability of scoring more than $2$ sixes when this die is rolled $5$ times.

State the degrees of freedom for the test.

Find the value of the ${\chi }^2$ test statistic $\left({\chi }_{calc}^2\right)$ for this test.

It is given that the critical value for this test is $9.49$.

State the conclusion of the test in context. Use your answer to part (c) to justify your conclusion.

Hence show that the variance of the proportion of marked fish in the sample, $\text{Var}\left(\frac{X}{300}\right)$, is $0.000425$.

Taking the value for the variance given in (d) (ii) as a good approximation for the true variance, find the upper and lower bounds for the proportion of marked fish in the lake.

Find the probability that a melon selected at random will have a mass greater than $3.0 \text{kg}$.

Write down the value of $n$ and the value of $p$.

Complete the table to show the probability distribution of $T$.

Identify the type of sampling used by the restaurant manager.

Find the $p$-value for the test.

the median.

Find the probability that Sheila will make a type I error in her test conclusion.

Find the percentage score calculated by Jason.

$6.75 \text{m}$ and $7 \text{m}$.

Find the equation of the new model.

Find the median of the scores obtained.

prefers a laptop given that they are $17\text{–}18$ years old.

prefers a tablet.

Find the period of the function.

minimum value of $h$.

The wind speed increases. The blades rotate at twice the speed, but still at a constant rate.

At any given instant, find the probability that Tim can see point $\text{C}$ from his window. Justify your answer.

Sketch the function $h\left(t\right)$ for $0\le t\le 5$, clearly labelling the coordinates of the maximum and minimum points.

Write down null and alternative hypotheses for this test.

Write down the null and alternative hypotheses.

Complete the tree diagram.

Assuming the null hypothesis to be true, state the distribution of $X$.

With reference to the box and whisker diagrams, state one aspect that may support the researcher’s opinion and one aspect that may counter it.

State the alternative hypothesis.

State whether it would be appropriate for Chester to use the equation of a regression line for $y$ on $x$ to predict a country’s Eurovision score. Justify your answer.

Find the value of $k$.

Find the probability that on any given day Mr Burke chooses a female student to answer a question.

The bags are labelled as being 1.5 kg mass. Comment on this claim with reference to your answer in part (b).

The Commissioner believes Minsun’s score for competitor G is too high and so decreases the score from 9.5 to 9.1.

Explain why the value of the Spearman’s rank correlation coefficient $r_s$ does not change.

Comment on the result obtained for $r_s$.

Identify a test that might have been used to verify the null hypothesis that the predictions from the standardized test can be modelled by a normal distribution.

Write down the equation of the regression line $y$ on $x$.

Find the mean change.

State what conclusion Kayla can make from the answer in part (a).

Explain why it is not appropriate to use Pearson’s product moment correlation coefficient to measure the strength of the relationship between $P$ and $d$.

Show that $\lambda =1$ is always an eigenvalue for M and find the other eigenvalue in terms of $a$ and $b$.

Write down the value of $b$.

when $n=5$.

Hence determine, with a reason, if the survey is reliable.

Find the p-value for this t-test.

State an assumption that the company is making, in order to use a t-test.

Perform a suitable test, at the 5% significance level, to determine if it is easier to achieve a distinction on the new exam. You should clearly state your hypotheses, the critical region and your conclusion.

the day when the total number of people infected will be greater than 1000.

$k$.

Perform a ${\chi }^2$ goodness of fit test at the 5% significance level. You should clearly state your hypotheses, the p-value, and your conclusion.

Use your answer to part (a) to show that the model predicts 16.7 people will be infected on the first day.

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

Hence state why Aimmika believes her data follows a Poisson distribution.

Perform the ${\chi }^2$ goodness of fit test and state, with reason, a conclusion.

Test the hypothesis at the 5% level of significance.

a goodness of fit test (a complete explanation required);

Write down the value of $n\left(\left(F\cup H\right)\cap S^{\prime }\right)$.

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.

Calculate $k$;

Find the value of $p$.

Find the number of children who play only football.

State the null hypothesis.

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

State the number of degrees of freedom.

State the null hypothesis, H₀, for this test.

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

$p$.

The mean number of squirrels in a certain area is known to be 3.2 squirrels per hectare of woodland. Within this area, there is a 56 hectare woodland nature reserve. It is known that there are currently at least 168 squirrels in this reserve.

Assuming the population of squirrels follow a Poisson distribution, calculate the probability that there are more than 190 squirrels in the reserve.

Find the expected value of T.

Determine the value of $\text{E}(X^2)$.

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.

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.

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

Write down the probability that a bouquet of roses sold is not small.

If the teacher chooses a response at random, find the probability that it is a satisfactory response to the Calculus question;

Show that the expected frequency of satisfactory Calculus responses is 12.

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

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

Find the expected frequency of the patients who became infected whilst in Nightingale ward.

 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 $\mu$ and the value of $\sigma$.

Find the probability of rolling exactly one red face.

Hence, find the value of $y$.

Show that $P(0⩽X⩽2)=\frac{1}{4}$.

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

Write down the independent variable.

Write down the boiling temperature of the liquid.

Write down the median length of these leaves.

Find the value of $x$ and of $y$.

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

Hence write down the interquartile range.

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

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

Find the acute angle between $y=x$ and $L$.

(i)     Find $w$.

(ii)     Hence or otherwise, find the maximum positive rate of change of $g$.

both a sandwich and a cake.

Find the expected number of cakes sold on a typical day.

Find the expected value of $X$.

Find .

Copy and complete the probability distribution table for Y.

Find the probability that at least one of the spins is yellow.

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.

Estimate the number of applicants who completed the test in less than 25 minutes.

On the following diagram, shade the region representing the probability that the price of a kilogram of tomatoes, chosen at random, will be higher than 3.22 euro.

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

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

Find the value of $p$;

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

Find $f^{\prime }\left(x\right)$.

The graph of $f$ has a local minimum at the point Q. The line L passes through Q.

Find the value of $a$.

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

Find the value of k for which P(μ − kσ < X < μ + kσ) = 0.5.

Find $\text{P}(B)$.

The random variable $X$ follows a Poisson distribution with a mean of $\lambda$ and $6\text{P}\left(X=3\right)=3\text{P}\left(X=2\right)-2\text{P}\left(X=1\right)+3\text{P}\left(X=0\right)$.

Find the value of $\lambda$.

Find the most likely number of taxis that would arrive during T.

Find the probability that Iqbal passes his third paper, given that he passed only one previous paper.

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 .

Determine unbiased estimates for $\mu$ and ${\sigma }^2$.

Find the critical region for testing $\overline{b}$ at the 5 % significance level.

One of the students who joined the sports club is chosen at random. Find the probability that this student joined both clubs.

Write down the number of sporting activities offered by the school during its school year.

Write down an expression, in terms of $a$, for the probability that at least one of Deborah’s three light bulbs is defective.

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

Find the probability that the time taken for a randomly chosen customer to be dealt with by an employee is greater than 180 seconds.

Find $(g\circ f)(x)$.

Find the value of $\text{tan}\theta$.

The following diagram shows the graph of $f$  for 0 ≤ $x$ ≤ 3. Line $M$ is a tangent to the graph of $f$ at point P.

Given that $M$ is parallel to $L$, find the $x$-coordinate of P.

Find the least value of $|r|$ for which the test concludes that $\rho \ne 0$.

One of the players is chosen at random. Find the probability that this player’s score was 5 or more.

Explain why a normal distribution can be used to give an approximate model for $X$.

Using a 10% significance level and justifying your answer, state your conclusion in context.

Deduce that $\frac{\text{P}(X=n)}{\text{P}(X=n-1)}=\frac{0.9(n-1)}{n-3}$ for $n>3$.

a Type II error.

Find the volume of the container.

Find the value of $p$ and of $q$.

Find the equation of the normal to the curve of $f$ at P.

Determine the concavity of the graph of $f$ when and justify your answer.

Find the coordinates of B.

Carry out a suitable test at the 5 % significance level. With reference to the $p$-value, state your conclusion in the context of Peter’s claim.

State whether the weight of the eggs is a continuous or discrete variable.

The mean weight of these eggs is 64.9 grams, correct to three significant figures.

Use the table and your answer to part (c) to find the smallest possible number of eggs that could be within one standard deviation of the mean.

Write down the median rehearsal time.

State whether Stephen is correct. Give a reason for your answer.

On $k$ days, Stephen practiced exactly $24$ minutes.

Find the possible values of $k$.

Given that the customer is a child, calculate the probability that they chose pasta or fish.

Determine the expected number of cats in this group that have a weight of less than $5.3 \text{kg}$.

Write down a transition matrix $\mathit{T}$ indicating the annual population movement between clinics.

For the data in this table, test the null hypothesis, ${\text{H}}_0:\rho =0$, against the alternative hypothesis, ${\text{H}}_1:\rho >0$, at the $5\%$ significance level. You may assume that all the requirements for carrying out the test have been met.

Write down the number of tests they carry out.

The firm obtains a significant result when comparing section $2$ of the written assessment and attribute $\text{X}$. Interpret this result.

Juliet decides to use the coefficient of determination to choose between these two models.

Comment on the validity of her decision.

State the name of the test which Juliet should use.

Hence compare the two models.

State two assumptions made in order for this model to be valid.

Find the probability that a randomly selected student from the class plays tennis or volleyball, but not both.

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

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

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

The $p$-value for this test is $0.0206$.

State the conclusion to Isaac’s test. Justify your reasoning.

Find the value of $p$.

On the grid below, draw a histogram for the data in the table.

Find the height of Flower $\text{B}$.

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

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

The test is performed at the $5\%$ significance level.

State the conclusion of the test, giving a reason for your answer.

Find the probability it will rain at least once while Paula is outside.

Find the expected frequency for each of the numbers if the manufacturer’s claim is true.

Determine the degrees of freedom for Dana’s test.

a player scores at least $3$ in a game.

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

Find the mean weight of a bag of apples.

Write down the null and alternative hypotheses.

Use the formula $s_{n-1}^2=\frac{\text{Σ}{x}^2-{\displaystyle \frac{{\left(\text{Σ}x\right)}^2}{n}}}{n-1}$ to determine an unbiased estimate for the variance of the number of chocolates per packet.

Determine whether Jason is correct. Support your reasoning.

Copy and complete the information in the following table.

$6.5 \text{m}$ and $6.75 \text{m}$.

Find the $p$-value for the test.

Write down the matrix $\mathit{P}$.

Write down the matrix $\mathit{D}$.

Comment on your answer to part (a), using the information that Eduardo found.

Write down the value of $a$.

Find an expression, in terms of $b$, for the probability of a person not having blue eyes and having fair hair.

Find the probability that a sack is under its labelled weight.

is $11\text{–}13$ years old and prefers a mobile phone.

Find the probability that Arianne scores at least $5$ points in the competition.

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

Find the probability of a Type II error, if the number of cars now follows a Poisson distribution with a mean of $4.5$ cars per minute.

Show that the test score of $25$ would not be considered an outlier.

State the alternative hypothesis.

Interpret the value obtained for $r_s$.

Find, to the nearest integer, the expected increase or decrease in the money made by the airline if they decide to sell $74$ tickets rather than $72$.

the interquartile range.

$c$.

State one reason why the test might not be valid.

It was not possible to ask every person in the school, so the Headmaster arranged the student names in alphabetical order and then asked every 10th person on the list.

Identify the sampling technique used in the survey.

Find the probability he will choose a female student 8 times.

interquartile range of the reaction times.

State the null hypothesis.

Find the exact value of $p$.

Find the probability of a Type I error.

Copy and complete the information in the following table.

Use your regression equation from part (b) to estimate Minsun’s score when Stan awards a perfect 10.

State why it was important to test that both sets of points were normally distributed.

Perform a test on the data from school A to show it is reasonable to assume a linear relationship between effort scores and improvements in IB points. You may assume effort scores follow a normal distribution.

What does $b$ represent in this context?

Use the trapezoidal rule to find an estimate for the area.

We know that she went out for lunch on a particular Sunday, find the probability that she went out for lunch on the following Tuesday.

State the hypotheses for this t-test.

State a possible disadvantage of using this test for reliability.

The new drug.

Perform a suitable test, at the 5% significance level, to determine if the scores follow a normal distribution, with the mean and variance found in part (a). You should clearly state your hypotheses, the degrees of freedom, the p-value and your conclusion.

Explain why the number of degrees of freedom is 2.

Find the equation of the regression line of $Q\left(t\right)$ on $t$.

Explain what is meant by the 5% level of significance.

A different horse breeder collected data on the time and outcome of births. The data are summarized in the following table:

Carry out an appropriate test at the 5% significance level to decide whether there is an association between time and outcome.

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

Show that the events $A$ and $B$ are not independent.

Complete the Venn diagram using the given information.

Find the probability that the boy answered questions in Hindi.

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.

Write down the value of a.

Complete the following tree diagram.

Write down an expression, in terms of $x$, for the number of children who play only basketball.

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

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

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

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

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

Find the standard deviation

Find the interquartile range.

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

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

In the following table, write down the letter of the corresponding graph next to the given mean and standard deviation.

Find the probability that a suitcase weighs less than 15 kg.

For this test state the null hypothesis.

Draw a box-and-whisker diagram, for these students’ ages, on the following grid.

By placing a tick (✔) in the correct box, determine which of the following statements is true:

Diogo is 18 years old. Give a reason why the regression line should not be used to estimate the number of hours Diogo watches television per week.

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

.

One student sent k text messages, where k > 11 . Given that k is an outlier, find the least value of k.

Find the number of hives that have a high production.

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

Calculate the standard deviation.

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

Find the median of $X$.

(i)     Write down the value of $k$.

(ii)     Find $g(x)$.

Complete the probability distribution table for $X$.

Hayley plays the game when $n$ = 5. She pays $20 to play and can earn money back depending on the number of draws it takes to obtain a blue marble. She earns no money back if she obtains a blue marble on her first draw. Let M be the amount of money that she earns back playing the game. This information is shown in the following table.

Find the value of $k$ so that this is a fair game.

Write down $\text{P}(X>107)$.

Find the probability that exactly 4 taxis arrive during T.

Find $k$.

A girl is selected at random. Find the probability that she takes economics but not history.

Draw a diagram that shows this information.

an estimate for the standard deviation of the number of emails received per working day.

Give one piece of evidence that suggests Willow’s Poisson distribution model is not a good fit.

Suppose that the probability of Archie receiving more than 10 emails in total on any one day is 0.99. Find the value of λ.

During quiet periods of the day, taxis arrive at a mean rate of 1.3 taxis every 10 minutes.

Find the probability that during a period of 15 minutes, of which the first 10 minutes is busy and the next 5 minutes is quiet, that exactly 2 taxis arrive.

Find the probability that Lucca eats at least one banana in a particular day.

State the distribution of $\overline{X}$ giving its mean and variance.

Find $\mu$ and $\sigma$.

Given that Pablo is late for work, find the probability that he left home before 07:00.

Show that μ = 106.

Write down the elements of the set $F\cap W^{\prime }$;

Find a 90 % confidence interval for $\mu$.

Before seeing these results the managing director believed that the mean time was 26 minutes.

Explain whether your answers to part (b) support her belief.

Given that $\underset{x\to +\mathrm{\infty }}{lim}(g\circ f)(x)=-3$, find the value of $b$.

Use this model to find the values of $A$ and $B$ such that , where $A$ and $B$ are symmetrical about the mean of $X$.

Find the value of $p$.

Write down the coordinates of A.

Find the probability that she will run at least two marathons in exactly four out of the following five years.

Use your graphic display calculator to find an estimate for the standard deviation of the weight of the eggs.

Complete the following Venn diagram using all elements of $U$.

State ${\text{H}}_0$, the null hypothesis for this test.

Calculate the probability that the customer is an adult or that the customer chose shrimp.