Mathematics 9709 · AS & A Level
Sampling and estimation
100 practice questions on Sampling and estimation, with worked solutions and instant marking.
For 10 values of $x$, the mean is $86.2$ and $\sum (x - a) = 362$.
Feb/March 2016
A survey was carried out on the journey times of $63$ people who cycle to work in one town. The findings are set out in the cumulative frequency table below: Journey time (minutes) $\leq 10, \leq 25, \leq 45, \leq 60, \leq 80$ with cumulative frequencies $0, 18, 50, 59, 63$.
Feb/March 2016
Hannah selects $5$ singers from $15$ applicants to take part in a concert. She arranges the $5$ singers in the sequence they will perform. Out of the $15$ applicants, $10$ are female and $5$ are male.
Feb/March 2016
A plate of cakes contains $12$ different cakes. Determine the number of ways to divide these cakes between Alex and James if each person gets an odd number of cakes.
Feb/March 2017
There are 900 students in one year group. The same puzzle is issued to every student, and the time taken, $t$ minutes, to finish it is recorded. The results are arranged in the frequency table below: Time taken, $t$ minutes: $t \leq 3$, $3 < t \leq 4$, $4 < t \leq 5$, $5 < t \leq 6$, $6 < t \leq 8$, $8 < t \leq 10$, $10 < t \leq 14$ with matching frequencies $120$, $180$, $200$, $160$, $110$, $80$, $50$.
Feb/March 2018
A set of $3$ letters is chosen from the $8$ letters in COLLIDER.
Feb/March 2018
Last Saturday, Sarah noted the colour and style of $160$ cars in a car park. Every car that was neither red nor silver was placed in the ‘other’ group. Her findings are displayed in the table below: Red - Saloon $20$, Hatchback $40$, Estate $12$; Silver - Saloon $14$, Hatchback $26$, Estate $10$; Other - Saloon $6$, Hatchback $24$, Estate $8$.
Feb/March 2018
For a set of $n$ values of $x$, the information below was obtained: $\sum(x - 20) = 136$, $\sum(x - 20)^2 = 2888$. The mean of these $n$ values of $x$ is $24.25$.
Feb/March 2018
The digits $1, 3, 5, 6, 6, 6, 8$ may be rearranged to make numerous different $7$-digit numbers.
Feb/March 2018
A survey carried out at one large college found that the fraction of students who own a car is $\frac{1}{4}$.
Feb/March 2018
For $40$ readings of the variable $x$, the information given is that $\Sigma (x - c)^2 = 3099.2$, with $c$ fixed. The standard deviation of those $x$-values is $3.2$.
Feb/March 2019
Shown below are the weights, in kg, of the 11 members of the Dolphins swimming team and the 11 members of the Sharks swimming team.
Feb/March 2019
Find how many distinct arrangements of all $9$ letters in the word CAMERAMAN can be made in each of the following situations.
Feb/March 2019
A construction company records, in $t$ days, how long it takes to construct each house of one particular design. For a random sample of $60$ such houses, the summary data are given by: $\sum t = 4820$, $\sum t^2 = 392050$.
Feb/March 2021
A random sample of 12 rods from a particular machine has lengths, in millimetres, given by $200,\;201,\;198,\;202,\;200,\;199,\;199,\;201,\;197,\;202,\;200,\;199$.
Feb/March 2022
From a random sample of 500 households in a particular town, the confidence interval for the proportion, $p$, of all households in that town owning two or more cars was obtained as $0.355 \le p \le 0.445$.
Feb/March 2022
Anita conducted a survey of 140 students chosen at random from her college. She found that 49 of those students watched a TV programme called Bunch.
Feb/March 2023
The lengths, $X\text{ cm}$, for a sample of 100 insects of one particular type were summarised like this: $n = 100$, $\sum x = 36.8$, $\sum x^2 = 17.34$.
Feb/March 2024
A simple random sample of 250 residents of Barapet was selected. It was discovered that 78 of these residents owned a BETEC phone.
Feb/March 2024
Nikki is looking into what students at her school think about the school sports facilities. She intends to hand out a survey to a sample of students.
Feb/March 2025
The counts of people using a particular bus at different points in the day are: $17, 5, 2, 23, 16, 31, 8, 22, 14, 25, 35, 17, 27, 12, 6, 23, 19, 21, 23, 8, 26$.
May/June 2010
Find how many distinct ways there are to distribute a set of $10$ different mugs between Lucy and Monica, provided that each person receives an odd number of mugs.
May/June 2010
The times in minutes taken by seven students to become proficient at a new computer game were recorded. The results are given below: $15, 10, 48, 10, 19, 14, 16$.
May/June 2010
The cumulative frequency graphs show the birth weights of random samples of $900$ babies from country $A$ and $900$ babies from country $B$.
May/June 2010
Nine cards, all in different colours, are to be placed in a straight line. The $9$ cards include a pink card and a green card.
May/June 2010
The values of height, $x$ cm, for a set of 82 children are given by: $\Sigma(x - 130) = -287$, standard deviation of $x = 6.9$.
May/June 2010
Three identical cans of cola, 2 identical cans of green tea, and 2 identical cans of orange juice are placed in a line.
May/June 2010
Some insects of the same kind from two countries, $X$ and $Y$, had their lengths measured. The stem-and-leaf diagram gives the results. Key: $5\,|\,81\,|\,3$ means that an insect from country $X$ has length $0.815\text{ cm}$, while one from country $Y$ has length $0.813\text{ cm}$.
May/June 2010
A cricket side of $11$ players must be selected from $21$ players made up of $10$ batsmen, $9$ bowlers and $2$ wicketkeepers. The side has to contain at least $5$ batsmen, at least $4$ bowlers and at least $1$ wicketkeeper.
May/June 2011
In one country, there are $5000$ schools. The cumulative frequency table gives the number of pupils in a school together with the matching number of schools.
May/June 2011
For a sample of $36$ data values, $x$, the results were $\Sigma (x - 45) = -148$ and $\Sigma (x - 45)^2 = 3089$.
May/June 2011
Find how many distinct ways the $9$ letters in HAPPINESS can be arranged in a line.
May/June 2011
The hotel contains $90$ rooms. The table gives details of how many rooms were occupied on each day over a span of $200$ days.
May/June 2011
Red Street Garage is offering 9 used cars for sale. Fairwheel Garage is offering 15 used cars for sale. The mean age of the cars at Red Street Garage is $3.6$ years and the standard deviation is $1.925$ years. For Fairwheel Garage, $\Sigma x = 64$ and $\Sigma x^2 = 352$, where $x$ represents the age of a car in years.
May/June 2011
Fahad owns $4$ differently coloured pairs of shoes: white, red, blue and black; $3$ differently coloured pairs of jeans: blue, black and brown; and $7$ differently coloured tee shirts: red, orange, yellow, blue, green, white and purple.
May/June 2011
This cumulative frequency table gives the examination marks for $300$ candidates in country $A$ and $300$ candidates in country $B$.
May/June 2011
Let $X$ be normally distributed with mean $\mu$ and standard deviation $\tfrac{1}{4}\mu$. It is known that $\mathrm{P}(X > 20) = 0.04$.
May/June 2011
The diagonal lengths, measured in metres, for the $9$ most popular flat screen TVs and the $9$ most popular conventional TVs are listed here. Flat screen: $0.85$, $0.94$, $0.91$, $0.96$, $1.04$, $0.89$, $1.07$, $0.92$, $0.76$. Conventional: $0.69$, $0.65$, $0.85$, $0.77$, $0.74$, $0.67$, $0.71$, $0.86$, $0.75$.
May/June 2012
The body-feather lengths of a given bird species are represented by a normal distribution. A researcher takes a random sample of $600$ body feathers from birds of this species and records that $63$ are shorter than $6\text{ cm}$ and $155$ exceed $12\text{ cm}$ in length.
May/June 2012
Seven friends and their partners meet for a meal. To mark the occasion, a photograph is taken of all $14$ of them standing in a line. How many distinct line-ups are possible if every friend stands beside his or her partner?
May/June 2012
The ages of 150 cars, measured in $x$ years, are summarised by $\sum x = 645$ and $\sum x^2 = 8287.5$. Find $\sum (x - \bar{x})^2$, where $\bar{x}$ represents the mean of $x$.
May/June 2012
The back-to-back stem-and-leaf diagram displays the values for two variables $A$ and $B$. Key: $4 | 16 | 7$ indicates that $A = 0.164$ and $B = 0.167$.
May/June 2012
An English examination has $8$ questions in Part A and $3$ in Part B. Candidates must select $6$ questions. The order of selection is irrelevant.
May/June 2012
The box holds $30$ biscuits altogether. Some biscuits are wrapped in gold foil and the rest are unwrapped, and some biscuits are chocolate-covered. Of the $12$ biscuits wrapped in gold foil, $7$ are chocolate-covered. In total, there are $17$ chocolate-covered biscuits.
May/June 2012
Ashfaq and Kuljit carried out a school statistics project about the prices of one specific model of headphones for MP3 players. Ashfaq gathered prices from 21 shops. Kuljit used the internet to collect prices from 163 websites.
May/June 2012
The heights, $x$ cm, of a set of young children are summarised by $\Sigma(x - 100) = 72$, $\Sigma(x - 100)^2 = 499.2$. The mean height is $104.8$ cm.
May/June 2012
In how many different ways may the 9 letters of the word TELEPHONE be placed in a straight line if P and L are required to be at the ends?
May/June 2012
Suzanne owns 20 pairs of shoes, and some of them have designer labels. She has 6 pairs of high-heeled shoes, and 2 of those pairs have designer labels. She has 4 pairs of low-heeled shoes, and 1 pair of these has a designer label. The remaining pairs are sports shoes. In total, Suzanne has 8 pairs of shoes with designer labels.
May/June 2012
For 30 values of $x$, the following totals were obtained: $\sum (x-c) = 234$, $\sum (x-c)^2 = 1957.5$, where $c$ is a constant.
May/June 2013
The back-to-back stem-and-leaf diagram below gives the yearly salaries of 39 females and 39 males. Key: $2|20|3$ represents $ a320\,200$ for females and $ a320\,300$ for males.
May/June 2013
The random variable $Y$ follows a normal distribution with positive mean $\mu$ and standard deviation $\frac{1}{2}\mu$. Determine the probability that a value of $Y$ chosen at random is negative.
May/June 2013
Fiona uses her calculator to generate $12$ random integers from $7$ to $21$, inclusive. The random variable $X$ counts how many of the $12$ integers are multiples of $5$.
May/June 2013
Four families visit a theme park together. Mr and Mrs Lin bring their $2$ children. Mr O’Connor brings his $2$ children. Mr and Mrs Ahmed bring their $3$ children. Mrs Burton brings her son. The $14$ people must all pass through a turnstile one at a time to enter the theme park.
May/June 2013
Box $A$ has $8$ white balls and $2$ yellow balls, whereas box $B$ has $5$ white balls and $x$ yellow balls. One ball is selected at random from box $A$ and transferred into box $B$. A ball is then selected at random from box $B$. The tree diagram shows the possible colours of the balls selected.
May/June 2013
For the 22 cars that passed a particular point, the speeds, $x$ kilometres per hour, were summarised by the information $\sum (x - 50) = 81.4$ and $\sum (x - 50)^2 = 671.0$.
May/June 2013
Robert’s calculator is used to produce 5 random integers from 1 to 9 inclusive.
May/June 2013
The annual amounts of money spent on clothes, rounded to the nearest $\$10$, for 27 people are listed below: $10, 40, 60, 80, 100, 130, 140, 140, 140, 150, 150, 150, 160, 160, 160, 160, 170, 180, 180, 200, 210, 250, 270, 280, 310, 450, 570$.
May/June 2013
A town council is planning to plant 12 trees down the middle of a main road. The council purchases the trees from a garden centre that has 4 different hibiscus trees, 9 different jacaranda trees and 2 different oleander trees on sale.
May/June 2013
On one side of a street, the $12$ houses are labelled with even numbers from $2$ up to $24$. On Monday, a free newspaper is delivered to $3$ distinct houses chosen at random from those $12$.
May/June 2013
John plays two games of squash. The chance that he wins his first game is $0.3$. If he wins his first game, the chance that he wins his second game is $0.6$. If he loses his first game, the chance that he wins his second game is $0.15$. Given that he wins his second game, find the probability that he won his first game.
May/June 2013
The weights, $x$ kilograms, of $144$ people were collected. The findings are shown in the cumulative frequency table below.
May/June 2013
At a dog show, there are $10$ spaniels, $14$ retrievers and $6$ poodles. $7$ dogs are chosen to progress to the final.
May/June 2013
Mrs Hunt is given $6$ paperback books and $2$ hardback books by a book club. She randomly selects $4$ of them to take away on holiday. The random variable $X$ stands for the number of paperback books she picks.
May/June 2014
The playground equipment includes swings ($S$), roundabouts ($R$), climbing frames ($C$) and play-houses ($P$). The quantities of equipment in each of $3$ playgrounds are listed below: Playground $X$: $3S, 2R, 4P$; Playground $Y$: $6S, 3R, 1C, 2P$; Playground $Z$: $8S, 3R, 4C, 1P$. Every day Nur takes her child to one of these playgrounds. The chance that she goes to playground $X$ is $\frac{1}{4}$. The chance that she goes to playground $Y$ is $\frac{1}{4}$. The chance that she goes to playground $Z$ is $\frac{1}{2}$. On arrival, she selects one item of equipment at random.
May/June 2014
Determine how many distinct arrangements of all $8$ letters in the word TANZANIA are possible so that
May/June 2014
A typing test is completed by $111$ people. Their numbers of typing errors are shown in the table below, with classes $1$--$5$, $6$--$20$, $21$--$35$, $36$--$60$, $61$--$80$ and matching frequencies $24$, $9$, $21$, $15$, $42$.
May/June 2014
A school club contains members from $3$ year-groups: Year $1$, Year $2$ and Year $3$. There are $7$ members in Year $1$, $2$ members in Year $2$ and $2$ members in Year $3$. Five members of the club are chosen.
May/June 2014
The running times for $57$ athletes over $100$ metres are summarised in the following cumulative frequency table: Time (seconds) $<10.0$, $<10.5$, $<11.0$, $<12.0$, $<12.5$, $<13.5$ with cumulative frequencies $0$, $4$, $10$, $40$, $49$, $57$ respectively.
May/June 2014
The amount of time Rafa does homework each day during term-time is normally distributed with mean $1.9$ hours and standard deviation $\sigma$ hours. On $80\%$ of these days, he works on his homework for more than $1.35$ hours.
May/June 2014
Some adults and some children were each asked to estimate, without using a watch, how many seconds had passed in a fixed time-interval. Their estimates are shown below. Adults: 55, 58, 67, 74, 63, 61, 63, 71, 56, 53, 54, 78, 73, 64, 62. Children: 86, 95, 89, 72, 61, 84, 77, 92, 81, 54, 43, 68, 62, 67, 83.
May/June 2014
A pet shop keeps 6 rabbits and 3 hamsters. 5 of these animals are picked at random. The random variable $X$ is the number of hamsters picked.
May/June 2014
The heights, $x$ cm, of a set of 28 people were recorded. The mean height was 172.6 cm and the standard deviation was 4.58 cm. One person with height 161.8 cm then left the group.
May/June 2014
The nine cards carry the numbers 1, 2, 2, 3, 3, 4, 6, 6, 6.
May/June 2014
The table gives a summary of the lengths, in centimetres, of $104$ dragonflies.
May/June 2015
In one country, $68\%$ of households own a printer. Determine the probability that, from a random sample of $8$ households, exactly $5$, $6$ or $7$ of them have a printer.
May/June 2015
Determine how many distinct numbers may be formed by rearranging all nine digits of the number $223\,677\,888$ with no extra conditions.
May/June 2015
A newspaper article was read by 120 people. The times they took, correct to the nearest second, are shown in the table below: Time (seconds) $1$-$25$, $26$-$35$, $36$-$45$, $46$-$55$, $56$-$90$; Number of people $4$, $24$, $38$, $34$, $20$.
May/June 2015
In an open-plan office there are $88$ computers. The lengths of time taken by these $88$ computers to reach a particular web page are shown on the cumulative frequency diagram.
May/June 2015
A box has $5$ discs, labelled $1$, $2$, $4$, $6$, $7$. William chooses $3$ discs at random, without replacement, and records the numbers written on them.
May/June 2015
Find how many distinct arrangements of the $7$ letters in the word BANANAS are possible when the first letter is $N$ and the last letter is $B$.
May/June 2015
For Joanna’s cooking, the chance that the meal is served on time is $\frac{1}{5}$. The chance that the kitchen ends up in a mess is $\frac{3}{5}$. The chance that the meal is not served on time and the kitchen is not left in a mess is $\frac{3}{10}$. Some of this information is displayed in the table below.
May/June 2015
A pet shop is offering $9$ rabbits, and $6$ of them are white. Two rabbits are picked at random without replacement.
May/June 2015
Seventy fertiliser samples were taken, and the nitrogen content of each one was recorded. The cumulative frequency table is shown below.
May/June 2015
Rachel owns 3 types of ornament. She has 6 distinct wooden animals, 4 distinct sea-shells and 3 distinct pottery ducks.
May/June 2015
On any given day, the chance that the school bus arrives on time is $0.6$. If it is on time, the chance that Sam gets a cup of coffee is $0.9$. If it is late, the chance that Sam gets a cup of coffee is $0.3$.
May/June 2016
A box holds $2$ green sweets and $5$ blue sweets. Two sweets are chosen at random from the box, without replacement. The random variable $X$ represents the number of green sweets chosen.
May/June 2016
Plastic drinking straws are made so that they will slot into drinks cartons with a hole in the top. A straw passes through the hole when its diameter is below $3\,\text{mm}$. The straw diameters follow a normal distribution with mean $2.6\,\text{mm}$ and standard deviation $0.25\,\text{mm}$.
May/June 2016
Find the number of integers from $100$ to $999$ with all three digits different.
May/June 2016
The table below summarises the amounts spent by $160$ shoppers at a supermarket: amount spent $(\$x)$ in the classes $0<x\leq30$, $30<x\leq50$, $50<x\leq70$, $70<x\leq90$, $90<x\leq140$ and the matching numbers of shoppers $16, 40, 48, 26, 30$.
May/June 2016
Ayman’s breakfast drink is tea, coffee or hot chocolate with probabilities 0.65, 0.28, 0.07 respectively. If he has tea, the probability that it includes milk is 0.8. If he has coffee, the probability that it includes milk is 0.5. If he has hot chocolate, it always contains milk.
May/June 2016
For a dental appointment, the chance of waiting for under 5 minutes is 0.16, and the chance of waiting for under 10 minutes is 0.88.
May/June 2016
These figures show the maximum daily wind speeds, in kilometres per hour, for the first two weeks of April in the two towns, Bronlea and Rogate.
May/June 2016
Determine how many distinct arrangements can be formed using all 10 letters of WALLFLOWER when no restrictions are imposed.
May/June 2016
Out of a group of 30 adults, 25 are right-handed and 8 wear spectacles. The number who are right-handed but do not wear spectacles is 19.
May/June 2016
A group of children took part in a computer game that recorded how many seconds they needed to complete a task. A summary of the times for the girls and the boys in the group is shown.
May/June 2016
For 9 apartments in one city, the monthly rental prices, $x$, are given and can be summarised by the following: $\sum (x - c) = 1845$ and $\sum (x - c)^2 = 477\,450$. The mean monthly rental price is $\$2205$.
May/June 2016
Determine how many different arrangements of all 9 letters in the word EVERGREEN are possible if:
May/June 2016
The table summarises the times, $t$ seconds, taken by 1140 people to solve a puzzle.
May/June 2017
Eight children of different ages are arranged at random in a line. Find how many different arrangements are possible if none of the three youngest children are beside one another.
May/June 2017
Rani and Diksha are shopping for clothes.
May/June 2017