Mathematics 9709 · AS & A Level
Hypothesis testing
88 practice questions on Hypothesis testing, with worked solutions and instant marking.
The lengths of a particular lizard species are normally distributed with standard deviation $3.2\text{ cm}$. A naturalist records the lengths of a random sample of 100 lizards of this species and obtains an $\alpha\%$ confidence interval for the population mean. The full width of the interval is $1.25\text{ cm}$.
Feb/March 2020
Before the one-way system was brought in, Freda’s mean time for a certain daily trip was $39.2$ minutes. Now she wants to check whether the mean time for that journey has gone down. She records the times, $t$ minutes, for 40 journeys chosen at random and summarises them as follows: $n = 40$, $\sum t = 1504$, $\sum t^2 = 57760$.
Feb/March 2020
A nationwide survey indicates that $95\%$ of year 12 students use social media. Arvin thinks that the proportion of year 12 students at his college who use social media is below the national percentage. He takes a random sample of $20$ students from his college and records how many use social media. He then carries out a test at the $2\%$ significance level.
Feb/March 2020
An architect wants to find out whether, on average, the buildings in one city are taller than those in other cities. He draws a large random sample of buildings from the city and calculates the mean height of the sample. He works out the test statistic, $z$, and obtains $z = 2.41$.
Feb/March 2021
It is given that $8\%$ of adults in one town own a Chantor car. Following an advertising campaign, a car dealer wants to find out whether this proportion has gone up. He selects a random sample of $25$ adults from the town and records how many own a Chantor car.
Feb/March 2021
Harry has a spinner with five sectors, coloured blue, green, red, yellow and black. Harry suspects that the spinner could be biased. He intends to carry out a hypothesis test with the following hypotheses: $H_0:\;P(\text{the spinner lands on blue}) = \frac{1}{5}$ and $H_1:\;P(\text{the spinner lands on blue}) \neq \frac{1}{5}$. Harry spins the spinner 300 times, and blue comes up on 45 of the spins.
Feb/March 2022
Previously, the time in minutes that students needed to finish a particular challenge had a mean of $25.5$ and a standard deviation of $5.2$. A different challenge is now set, and it is anticipated that, on average, students will need less than $25.5$ minutes to finish it. A random sample of $40$ students is taken, and the sample mean time for the new challenge is $23.7$ minutes.
Feb/March 2022
For a certain road, the number of accidents in each 3-month period follows the distribution $\text{Po}(\lambda)$. Historically, $\lambda$ has had the value $5.7$. After some changes to the road, the council carries out a hypothesis test to decide whether $\lambda$ has fallen. If a randomly chosen 3-month period contains fewer than $3$ accidents, the council will decide that $\lambda$ has decreased.
Feb/March 2023
During the previous year, the average amount of time that students at a school needed to finish a particular test was $25$ minutes. Akash thinks that the average time needed by this year’s students is below $25$ minutes. To check this idea, he selects a large random sample of this year’s students and records the time for each student. He carries out a test, at the $2.5\%$ significance level, for the population mean time, $\mu$ minutes. Akash uses the null hypothesis $H_0: \mu = 25$.
Feb/March 2023
The heights, in centimetres, of adult females in Litania have mean $\mu$ and standard deviation $\sigma$. In 2004, the respective values of $\mu$ and $\sigma$ were $163.21$ and $6.95$. The government states that this year’s value of $\mu$ is higher than the 2004 figure. To check this statement, a researcher intends to perform a hypothesis test at the $1\%$ significance level. He measures the heights of a random sample of $300$ adult females in Litania this year and obtains the sample mean.
Feb/March 2024
A researcher has recorded the time, $T$ seconds, needed by adults to finish a questionnaire. The findings for a random sample of $60$ adults who completed the questionnaire this year are summarised as follows: $n = 60$, $\sum t = 3678$, $\sum t^2 = 226313.36$.
Feb/March 2025
Amir thinks that 20% of the students at his college are left-handed. His friend thinks that the true proportion, $p$, is below 20%. Amir intends to use the binomial distribution to test the null hypothesis, $H_0: p = 0.2$, against the alternative hypothesis, $H_1: p < 0.2$. He plans to select 35 students at random. If 3 or fewer of these students are left-handed, Amir will reject his belief.
Feb/March 2025
Earlier, this crop’s yield, measured in tonnes per hectare, had a mean of $0.56$ and a standard deviation of $0.08$. After introducing a new fertilizer, the farmer wants to test at the $2.5\%$ significance level whether the mean yield has gone up. Over 10 years, he records a mean yield of $0.61$ tonnes per hectare.
May/June 2020
A fair spinner has five sectors labelled $1, 2, 3, 4, 5$. Let the score from one spin be denoted by $X$.
May/June 2020
The masses, measured in grams, of plums of this kind follow the distribution $N(40.4,\,5.2^2)$. These plums are put into bags, and each bag has 6 plums selected at random. A bag is rejected whenever the combined mass of the plums in it is below $220\text{ g}$.
May/June 2020
A shop sources apples from one particular farm. It has been established that $5\%$ of the apples from this farm are Grade A. After the growing conditions at the farm were altered, the shop management intend to run a hypothesis test to determine whether the proportion of Grade A apples has risen. They choose 25 apples at random. If the number of Grade A apples is greater than 3 they will decide that the proportion has risen.
May/June 2020
The counts of customers entering a particular shop from 9.00 am to 10.00 am have the distribution $\text{Po}(\lambda)$. In the past, the value of $\lambda$ was $5.2$. After some new advertising, the manager wants to check whether $\lambda$ has gone up. He takes a random sample of $20$ days and finds that the total number of customers who visited the shop between $9.00$ am and $10.00$ am on those days is $125$.
May/June 2020
A market researcher is examining how long customers stay at an information desk. He plans to take a sample of 50 customers on one specified day.
May/June 2020
At one large school, it was discovered that the fraction of students who were not wearing the correct uniform was $0.15$. A letter was then sent to parents, requesting that they make sure their children wear the correct uniform. The school now wants to check whether the fraction not wearing the correct uniform has gone down.
May/June 2021
A ball is tossed in a game and ends up in one of 4 slots, called $A$, $B$, $C$ and $D$. Raju wants to check whether the chance of the ball landing in slot $A$ is $\frac{1}{4}$.
May/June 2021
At a certain gym, the amount of time customers spend there, measured in minutes, is distributed as $N(\mu, 38.2)$. Previously, the value of $\mu$ was $42.4$. After some new equipment has been fitted, the management wants to check whether $\mu$ is now different.
May/June 2021
Previously, the time taken, in hours, for one particular train journey had mean $1.40$ and standard deviation $0.12$. After new signals were introduced, it is necessary to test whether the mean journey time has fallen.
May/June 2021
The local council says that the mean number of accidents each year on one specific road is $0.8$. Jane argues that the actual mean is above $0.8$. She examines the records from a random sample of $3$ recent years and discovers that there were $5$ accidents in total across those $3$ years.
May/June 2021
Arvind plays a game with a standard fair 6-sided die. He thinks he has a method for predicting the score that will appear each time the die is rolled. Before every roll, he records the score he expects. He says that he can record the right score more often than random guessing would allow. His friend Laxmi checks this by getting him to note his prediction before each of 15 rolls of the die. Arvind gives the correct score on exactly 5 of the 15 rolls.
May/June 2022
Previously, Jenny’s mean time for her morning run was 28.2 minutes. She has done some additional training and wants to check whether her mean time has decreased. After the training, Jenny takes a random sample of 40 morning runs. She decides that if the sample mean run time is below 27 minutes, she will conclude that the training was effective. You may assume that, after the training, Jenny’s run time has a standard deviation of 4.0 minutes.
May/June 2022
Earlier, the average height of plants in a certain species was $2.3\,\text{m}$. A random sample of $60$ plants from this species was treated with fertiliser, and the mean height of those $60$ plants was measured as $2.4\,\text{m}$. Assume that the standard deviation of the heights of plants treated with fertiliser is $0.4\,\text{m}$.
May/June 2022
The number of cars that reach a particular road junction on a weekday morning follows a Poisson distribution with mean $4.6$ per minute. Traffic lights are fitted at the junction, and a council officer wants to test at the $2\%$ significance level whether fewer cars are now arriving. He records the number of cars that arrive in one randomly selected $2$-minute interval.
May/June 2022
Anton thinks that $10\%$ of students at his college are left-handed. Aliya thinks this is an underestimate. She intends to carry out a hypothesis test of the null hypothesis $p = 0.1$ against the alternative hypothesis $p > 0.1$, where $p$ is the true proportion of students at the college who are left-handed. She takes a random sample of $20$ students from the college. She will reject the null hypothesis if at least $5$ of these students are left-handed.
May/June 2022
Batteries of type $A$ have a mean life of $150$ hours. The task is to determine whether a fresh battery type, type $B$, has a smaller mean life than type $A$ batteries.
May/June 2022
Earlier, the yearly quantity of wheat produced per farm by a large number of similarly sized farms in one region had a mean of $24.0$ tonnes and a standard deviation of $5.2$ tonnes. Last summer a new fertiliser was applied on all the farms, and it was expected that the mean quantity of wheat produced per farm would exceed $24.0$ tonnes. To check whether this was correct, a scientist recorded the amounts of wheat produced by a random sample of $50$ farms last summer. He found that the sample mean was $25.8$ tonnes.
May/June 2023
The weekly count of accidents at one factory follows a Poisson distribution. In previous years, the mean has been $1.9$ accidents per week. Last year, the manager issued every employee a new safety booklet. He wishes to test, at the $5\%$ significance level, whether the average number of accidents has gone down. He records the accident count for $4$ weeks selected at random this year.
May/June 2023
In country $A$, the newborn-baby masses in kilograms are modelled by random variable $X$, whose mean is $\mu$ and variance is $\sigma^2$. A random sample of 500 newborn babies was taken, and the summary of the observed masses is: $n = 500$, $\sum x = 1625$, $\sum x^2 = 5663.5$.
May/June 2023
If a child finishes an online exercise known as a Mathlit, a medal may be given. The publishers state that the chance that a child chosen at random who finishes a Mathlit is awarded a medal is $\frac{1}{3}$. Asha wants to test this statement. She says that if she receives no medals after completing $10$ Mathlits, she will decide that the true probability is below $\frac{1}{3}$. Let the true probability of being awarded a medal be $p$.
May/June 2023
In the previous year, the average pizza-delivery time from Pete’s Pizza Pit was $32.4$ minutes. This year, the delivery time, $t$ minutes, from Pete’s Pizza Pit was noted for a random sample of $50$ deliveries. The results were: $n = 50 \qquad \sum t = 1700 \qquad \sum t^2 = 59\,050$
May/June 2023
A new light was fitted along a footpath. A town councillor chose to use a hypothesis test to check whether the number of people using the path in the evening had risen. Before the light was fitted, the mean number of people using the path in any 20-minute period in the evening was $1.01$. Once the light had been fitted, the total number, $n$, of people using the path in 3 randomly selected 20-minute periods in the evening was recorded.
May/June 2023
A random sample of 8 cereal boxes from one supplier was selected. After weighing each box, the recorded masses in grams were: $261, 249, 259, 252, 255, 256, 258, 254$. Determine unbiased estimates for the population mean and variance.
May/June 2024
Each July, as part of a research project, Rita gathers information on sightings of one particular species of bird. On every day in July she records whether or not she sees this bird, and she counts the number $X$ of days on which it is observed. She represents the distribution of $X$ by $\mathrm{B}(31, p)$, where $p$ is the chance of seeing this bird on a randomly selected day in July. Results from earlier years indicate that $p = 0.3$, but in 2022 Rita thought that the value of $p$ might have fallen. She chose to perform a hypothesis test. During July 2022, she observed this bird on $4$ days.
May/June 2024
The masses of cereal boxes packed by one particular machine have mean $510$ grams. The machine is then adjusted, and an inspector wants to check whether the mean mass of cereal boxes filled by the machine has fallen. Once the adjustment has been made, he takes a random sample of $120$ cereal boxes. The sample mean mass is $508$ grams. Assume that the standard deviation of the masses is $10$ grams.
May/June 2024
Do not use an approximating distribution in this question. In last year’s election in Menham, 24% of voters backed the Today Party. A student wants to check whether support for the Today Party has fallen since last year. He takes a random sample of 25 voters in Menham and discovers that exactly 2 of them say they support the Today Party.
May/June 2024
The numbers of green sweets in 200 randomly selected packets of Frutos are shown in the table.
May/June 2024
The random variable $T$, measured in minutes, for a particular daily bus trip is normally distributed. The bus company says that the mean value of $T$ is $45$. One passenger thinks that the true mean of $T$ exceeds $45$. She records the journey times on a random sample of $60$ days. The information is summarised below: $n = 60$, $\sum t = 2750$, $\sum t^2 = 127000$.
May/June 2025
A cell-phone manufacturer says that $25\%$ of students own a Pumpkin phone. Jeyeraj believes that the proportion of students at his large college who own a Pumpkin phone is below $25\%$. He intends to test the claim made by the manufacturer. He picks a random sample of $30$ students at his college. If fewer than $5$ students own a Pumpkin phone, Jeyeraj will reject the manufacturer’s claim.
May/June 2025
Birgitte has a six-sided dice. She thinks that the dice may be biased, with the probability, $p$, of getting a six on a single throw being less than $\frac{1}{6}$. She rolls the dice 30 times and gets a six on exactly 2 throws.
May/June 2025
The pencil lengths produced in a factory are normally distributed. Their standard deviation is $\sigma$ cm, and the intended mean length is $10$ cm. An inspector suspects that the true mean exceeds $10$ cm. He selects a random sample of $50$ pencils from the factory and obtains a sample mean of $10.03$ cm. He then carries out a hypothesis test and finds that the test statistic $z$ has value $1.995$ correct to $3$ decimal places.
May/June 2025
Earlier, 1/4 of people applying for jobs at a particular firm possessed first-class degrees. The job description is then altered, and a director of the firm thinks that, on average, the proportion of applicants with first class degrees has fallen. In the month after the alteration, there were 35 job applicants, and $r$ of these had first-class degrees. The firm carried out a hypothesis test at the $4\%$ significance level to check the director’s view.
May/June 2025
It is stated that $28\%$ of the voters in a particular town back the Forward Now political party. A researcher thinks the actual proportion is below $28\%$. She takes a random sample of $30$ voters from the town and discovers that $4$ people in the sample said they support the Forward Now party. She intends to perform a hypothesis test at the $10\%$ significance level.
May/June 2025
A firm manufactures a particular kind of battery-powered toy. Let the battery life be denoted by $X$ hours, and suppose the population mean of $X$ is claimed to be $12$. The Quality Control department wanted to investigate whether the true population mean of $X$ is actually below $12$. They took a random sample of $50$ of these toys and obtained the sample mean $\bar{X}=11.4$.
May/June 2025
Anjan’s Monday commute to work, measured in minutes, has a mean of $38.4$ and a standard deviation of $6.9$.
Oct/Nov 2020
The areas, $X\;\text{cm}^2$, of petals from one species of flower have mean $\mu\;\text{cm}^2$. It was previously established that $\mu = 8.9$. After a change in climate, a botanist states that the mean is not still $8.9$. The areas of a random sample of $200$ petals from this species are recorded, and the data are summarized by $\Sigma x = 1850$, $\Sigma x^2 = 17\,850$.
Oct/Nov 2020
A biscuit maker says that, on average, $1$ out of every $3$ packets of biscuits includes a prize offer. Gerry thinks the proportion of packets with the prize offer is below $\frac{1}{3}$. To check the maker’s statement, he buys $20$ packets chosen at random. He discovers that exactly $2$ of those packets include the prize offer.
Oct/Nov 2020
The weekly count of worker absences at a factory is distributed as $\mathrm{Po}(2.1)$.
Oct/Nov 2020
On Mondays, Anjan’s journey-to-work time, measured in minutes, has mean $38.4$ and standard deviation $6.9$.
Oct/Nov 2020
A random sample of $75$ students from a large college was used for a survey. $15$ of the students said that they owned a car. Using this information, an approximate $\alpha\%$ confidence interval for the proportion of all students at the college who own a car was found. The interval width was $0.162$.
Oct/Nov 2021
The fraction of people who have a certain medical condition is $1$ in $100\,000$. A random sample of $2500$ people is taken. Let $X$ represent the number of people in the sample who have the condition.
Oct/Nov 2021
For apples taken from a particular farm, the masses in grams have mean $\mu$ and standard deviation $5.2$. The farmer states that $\mu$ is $64.6$. A quality control inspector argues that $\mu$ is actually below $64.6$. To check this claim, a random sample of $100$ apples is selected from the farm.
Oct/Nov 2021
A particular type of firework is expected to burn for an average of 30 seconds after being lit. An inspector thinks these fireworks may in fact burn for a shorter average time. He selects a random sample of 100 fireworks of this type. Each firework in the sample is ignited and its duration is recorded. It is known that the population standard deviation of the times that fireworks of this type last is 5 seconds.
Oct/Nov 2021
A machine is meant to generate random digits. Bob believes that the machine is biased and that the chance of producing the digit $0$ is smaller than $\frac{1}{10}$. To check this, he counts how many times the digit $0$ appears in $30$ digits produced by the machine. He carries out a test at the $10\%$ significance level.
Oct/Nov 2021
From a random sample of $75$ students chosen at a large college, a survey showed that $15$ said they owned a car. Using this outcome, an approximate $\alpha\%$ confidence interval for the proportion of all students at the college who own a car was found. The width of the interval was $0.162$.
Oct/Nov 2021
For a given population, the chance that one person has this specific medical condition is $1$ in $100\,000$. A random selection of $2500$ people is then taken. Let $X$ denote the number of people in the sample who have the condition.
Oct/Nov 2021
The masses, measured in grams, of apples from one farm have mean $\mu$ and standard deviation $5.2$. The farmer states that $\mu = 64.6$. A quality control inspector asserts instead that $\mu$ is below $64.6$. To check this claim, a random sample of 100 apples is selected from the farm.
Oct/Nov 2021
A spinner is split into five sections, and each section is marked with a different colour. Susma and Sanjay both want to check whether the spinner is biased so that it lands on red less often than it would if it were fair. Susma turns the spinner 40 times. She records that it lands on red exactly 4 times.
Oct/Nov 2022
Earlier, Laxmi’s journey to college, measured in minutes, had a mean of $32.5$ and a standard deviation of $3.1$. After altering her route, she wants to check whether the mean travel time has fallen. She records her journey times for a random sample of $50$ journeys and obtains a sample mean of $31.8$ minutes. Assume that the standard deviation stays the same.
Oct/Nov 2022
Earlier, the mean length of a specific variety of worm was $10.3\,$cm, and the standard deviation was $2.6\,$cm. After a change in climate, it is believed that the mean length of this variety of worm has fallen. A random sample of $100$ worms of this variety is taken, and the sample mean is $9.8\,$cm.
Oct/Nov 2022
The faults in cloth produced on a particular machine follow a Poisson distribution with mean $2.4$ per $10\text{ m}^2$. The machine is then adjusted. At the $5\%$ significance level, the aim is to test whether the mean number of faults has gone down. A random $30\text{ m}^2$ piece of cloth is inspected and the number of faults is recorded.
Oct/Nov 2022
A spinner is split into five sectors, and each sector carries a different colour. Susma and Sanjay both want to check whether the spinner is biased so that it lands on red less often than it would if it were fair. Susma spins the spinner $40$ times, and it lands on red exactly $4$ times.
Oct/Nov 2022
Previously, for Laxmi’s journey to college, the time taken in minutes had a mean of $32.5$ and a standard deviation of $3.1$. After she changes her route, Laxmi wants to check whether the mean time has gone down. She records the journey times for a random sample of $50$ trips and obtains a sample mean of $31.8$ minutes. Assume that the standard deviation stays the same.
Oct/Nov 2022
From a survey of 300 randomly selected adults in Rickton, 134 reported that they exercised regularly. This was then used to form an $\alpha\%$ confidence interval for the proportion of Rickton adults who exercise regularly. The upper limit of the interval was $0.487$, accurate to 3 significant figures.
Oct/Nov 2023
Previously, the number of enquiries per minute at a customer service desk was represented by a random variable with distribution $\text{Po}(0.31)$. After the desk was moved, the mean number of enquiries per minute is expected to rise. To check whether this has happened, the total number of enquiries in one randomly selected $5$-minute interval is recorded. Assume that a Poisson model remains suitable.
Oct/Nov 2023
A biologist wants to investigate whether the mean concentration $\mu$, in suitable units, of a particular pollutant in a river is below the allowed level of $0.5$. She records the concentration, $x$, of the pollutant at $50$ randomly selected places in the river. The data are summarised below. $n = 50 \quad \sum x = 23.0 \quad \sum x^2 = 13.02$
Oct/Nov 2023
A researcher examined a magazine piece that said boys aged $1$ to $3$ prefer green rather than orange. It claimed that, if a green cube and an orange cube are offered to play with, a boy is more likely to select the green cube. The researcher does not agree with this claim. She thinks boys in this age range are equally likely to choose either colour. To test this belief, the researcher carried out a hypothesis test at the $5\%$ significance level. She gave a green cube and an orange cube to each of $10$ randomly selected boys aged $1$ to $3$, and noted the number, $X$, of boys who chose the green cube. Of the $10$ boys, $8$ boys chose the green cube.
Oct/Nov 2023
The tree height $H$, measured in metres, for mature trees of a particular variety follows a normal distribution with standard deviation $0.67$. To test whether the population mean of $H$ is greater than $4.23$, the heights of a random sample of 200 trees are measured.
Oct/Nov 2023
In a survey of 300 adults chosen at random in Rickton, 134 reported that they exercised regularly. This data was then used to work out an $\alpha\%$ confidence interval for the proportion of adults in Rickton who exercise regularly. The upper endpoint of the interval was obtained as $0.487$, correct to 3 significant figures.
Oct/Nov 2023
Earlier, the number of enquiries per minute at a customer service desk was represented by a random variable with distribution $\mathrm{Po}(0.31)$. After the desk was repositioned, the mean number of enquiries per minute is expected to rise. To check whether this has happened, the total number of enquiries in a randomly selected 5-minute interval is recorded. Assume that a Poisson model is still appropriate.
Oct/Nov 2023
A biologist wants to investigate whether the mean concentration $\mu$, in suitable units, of a particular pollutant in a river is less than the permitted level of $0.5$. She records the concentration, $x$, of the pollutant at $50$ randomly selected sites in the river. The summary statistics are given below. $n = 50 \qquad \Sigma x = 23.0 \qquad \Sigma x^2 = 13.02$
Oct/Nov 2023
The lengths, in centimetres, of worms of a particular species are normally distributed with mean $\mu$ and standard deviation $2.3$. A magazine article gives the value of $\mu$ as $12.7$. A scientist wants to check whether this value is correct. He measures the lengths, $x$ cm, of a random sample of 50 worms of this species and obtains $\Sigma x = 597.1$. He intends to carry out a test, at the $1\%$ significance level, to find whether the true value of $\mu$ differs from $12.7$.
Oct/Nov 2024
The accident count per year on one particular road follows the distribution $\text{Po}(\lambda)$. In the past, the value of $\lambda$ was $3.3$. A new speed limit has now been introduced, and the council wants to check whether $\lambda$ has gone down. It records the total number, $X$, of accidents in two randomly selected years after the speed limit came into force, and performs a test at the $5\%$ significance level.
Oct/Nov 2024
A factory owner represents the daily number of workers who eat in the factory canteen using the distribution $B(25, p)$. Earlier, the value of $p$ was $0.8$. A new menu has been added in the canteen, and the owner wishes to test whether $p$ has risen. On one randomly selected day, he records that $23$ employees use the canteen.
Oct/Nov 2024
One-year-old trees of a particular variety are understood to have a mean height of $2.3\,\text{m}$. A scientist thinks that, on average, trees of this age and variety in her area are a little taller than those elsewhere. She intends to perform a hypothesis test at the $2\%$ significance level to check her idea. She chooses a random sample of 100 such trees in her area and records their heights, $h\,\text{m}$. Her data are shown below: $n=100, \quad \sum h=238, \quad \sum h^2=580$.
Oct/Nov 2024
The lengths, in centimetres, of worms from a particular species are normally distributed with mean $\mu$ and standard deviation $2.3$. A magazine article says that $\mu=12.7$. A scientist wants to check whether this claim is correct. He measures the lengths, $x$ cm, of a random sample of $50$ worms of this species and obtains $\Sigma x = 597.1$. He intends to perform a test, at the $1\%$ significance level, of whether the actual value of $\mu$ differs from $12.7$.
Oct/Nov 2024
The annual count of accidents on a particular road follows the distribution $\text{Po}(\lambda)$. In the past, the value of $\lambda$ was $3.3$. A new speed limit has now been introduced, and the council wants to check whether $\lambda$ has fallen. It records the total number, $X$, of accidents over two randomly selected years after the speed limit was introduced and performs a test at the $5\%$ significance level.
Oct/Nov 2024
The stated mean mass of Trueleaf tea packets is 500 grams. An inspector wants to check whether this figure is accurate. He measures 60 packets chosen at random and records the mass, $x$ grams, of each packet. The data are summarised by: $n = 60$, $\sum x = 29970$, $\sum x^2 = 14970300$.
Oct/Nov 2025
The given information is that 20% of households in a particular country have more than 4 people. Laxmi thinks that the proportion in her town is below 20%. She takes a random sample of 40 households in her town and records how many of them contain more than 4 people. She then carries out a test at the 2.5% significance level using a binomial distribution.
Oct/Nov 2025
An inspector thinks that 18% of the cups produced at a particular factory are defective. The factory owner says that the actual proportion is below 18%. A random sample of 40 cups is checked by the inspector, and 3 of those cups are found to be defective.
Oct/Nov 2025
The weekly profit, measured in dollars, for a particular firm follows a normal distribution. Previously, the weekly profit had distribution $N(736, 26^2)$. After a change in management, the mean weekly profit over 35 randomly selected weeks is $725$.
Oct/Nov 2025
The expected mean mass of packets of Trueleaf tea is 500 grams. An inspector wants to check whether this figure is accurate. He takes 60 packets at random and records the mass, $x$ grams, of each packet. The results are summarised as follows: $n = 60$, $\sum x = 29970$, $\sum x^2 = 14970300$.
Oct/Nov 2025
It is given that 20% of households in one country have more than 4 people. Laxmi thinks that the proportion in her town is below 20%. She takes a random sample of 40 households in her town and records how many have more than 4 people. She then performs a test at the 2.5% significance level with a binomial distribution.
Oct/Nov 2025
A researcher is checking whether the proportion of families in his town who do not own a car is different from the proportion for the country as a whole, which is 10.1%. He selects a large random sample of families from his town and records the proportion who do not own a car.
Oct/Nov 2025
A website records an average of $\mu$ hits each hour. In the past, the value of $\mu$ was 14.4. After some improvements, the owner wants to check whether $\mu$ has gone up. He randomly selects a 10-minute interval and records 6 hits in that interval. Assume that the number of hits the website gets in any fixed time period follows a Poisson distribution.
Oct/Nov 2025