Mathematics 9709 · AS & A Level
The normal distribution
58 practice questions on The normal distribution, with worked solutions and instant marking.
For one variety of apples, the weights are normally distributed with mean 82 grams. 22% of the apples in this variety weigh more than 87 grams.
Feb/March 2020
In Greenton, $70\%$ of adults have a car, and a random sample of $8$ adults is selected from Greenton.
Feb/March 2020
The times spent by shoppers in a large shopping centre are normally distributed, with mean $96$ minutes and standard deviation $18$ minutes.
Feb/March 2021
A school in one country has $400$ students. Every student was asked if they preferred swimming, cycling or running, and the outcomes are shown in the table below. One student is selected at random.
Feb/March 2021
For male leopards in one particular region, the weights are normally distributed, with mean $55\text{ kg}$ and standard deviation $6\text{ kg}$.
Feb/March 2022
In a cycling event, the completion times for the course are represented using a normal distribution with mean $62.3$ minutes and standard deviation $8.4$ minutes.
Feb/March 2023
A firm sells rice in small and large bags. For the small bags, the masses follow a normal distribution with mean $1.20\text{ kg}$ and standard deviation $0.16\text{ kg}$.
Feb/March 2024
The daily mass of peaches sold in a supermarket follows a normal distribution with mean $65.8\text{ kg}$ and standard deviation $9.6\text{ kg}$.
Feb/March 2025
For female snakes of this species, the lengths follow a normal distribution with mean $54\text{ cm}$ and standard deviation $6.1\text{ cm}$.
May/June 2020
Trees in the Redian forest are grouped into tall, medium, or short categories according to their height. Their heights may be represented by a normal distribution with mean $40\text{ m}$ and standard deviation $12\text{ m}$. Any tree with height below $25\text{ m}$ is placed in the short category.
May/June 2020
For one town, the weekly television-watching time, $X$ hours, is normally distributed with mean $15.8$ hours and standard deviation $4.2$ hours.
May/June 2020
A factory makes a specific kind of metal rod. These rods have lengths that are normally distributed with mean $25.2\,\text{cm}$ and standard deviation $0.4\,\text{cm}$. One random sample of $500$ of these rods is selected.
May/June 2021
The masses of bags of sugar are normally distributed with mean $1.04\text{ kg}$ and standard deviation $\sigma\text{ kg}$. In a random sample of $2000$ bags of sugar, $72$ had a mass greater than $1.10\text{ kg}$.
May/June 2021
The lengths of the leaves on a certain kind of tree are represented by a normal distribution. A scientist takes a random sample of $500$ leaves from this tree species and records that $42$ are shorter than $4\text{ cm}$ and $100$ exceed $10\text{ cm}$ in length.
May/June 2021
For one type of leaf, lengths measured in cm follow the distribution $N(5.2,\,1.5^2)$. A second leaf type is also represented by a normal distribution. A scientist records the lengths of a random sample of 500 leaves of this type and discovers that 46 are shorter than $3\text{ cm}$ and 95 are longer than $8\text{ cm}$.
May/June 2022
The masses, in kg, of bags of rice made by Anders are modelled by the distribution $N(2.02, 0.03^2)$.
May/June 2022
In a large college, $28\%$ of the students play no musical instrument, $52\%$ play exactly one musical instrument, and the rest play two or more musical instruments. A random sample of $12$ students is selected from the college.
May/June 2022
A mathematical puzzle is presented to a large group of students. The completion times are normally distributed, with mean $14.6$ minutes and standard deviation $5.2$ minutes.
May/June 2023
Western bluebird lengths are normally distributed, with mean $16.5\text{ cm}$ and standard deviation $0.6\text{ cm}$. One random sample of $150$ of these birds is chosen.
May/June 2023
Anil is standing as a candidate in an election, and he gained $40\%$ of the votes. A random sample of 120 voters is selected.
May/June 2023
The amount of grapes sold each day by a large shop may be represented by a normal distribution with mean $28\text{ kg}$. On $10\%$ of days, the quantity sold is below $16\text{ kg}$.
May/June 2023
Adult raccoons of one particular species have tail lengths that are normally distributed, with mean $28\text{ cm}$ and standard deviation $3.3\text{ cm}$.
May/June 2024
For a particular region of the Arctic, the chance of snow on any one day is $0.7$, and this does not depend on any other day.
May/June 2024
A normal distribution with mean $131$ grams and standard deviation $54$ grams can be used to represent orange weights. Oranges are put into the categories small, medium or large. A large orange has a weight of at least $184$ grams, and $20\%$ of oranges are in the small category.
May/June 2024
In one country, adults’ heights follow a normal distribution with mean 1.64 m and standard deviation 0.25 m.
May/June 2024
The masses of bags of rice produced by a company are normally distributed with mean $\mu$ kg and standard deviation $0.14$ kg. The chance that a randomly selected bag of this rice has mass below $1.48$ kg is $0.22$.
May/June 2025
Each Saturday, one local community runs a ‘Puzzle’ event to collect funds for a new Leisure Centre. Those taking part try to finish a puzzle in the shortest possible time. Last Saturday, 600 competitors took part. The times to complete the puzzle were normally distributed with mean $32.4$ minutes and standard deviation $2.5$ minutes.
May/June 2025
In Millford, $70\%$ of residents have a bicycle. A random sample of $160$ residents is chosen.
May/June 2025
For kestrels, adult wingspans follow a normal distribution with mean $74.8\text{ cm}$ and standard deviation $3.2\text{ cm}$. A random sample of 120 adult kestrels is taken.
May/June 2025
A firm sells pasta in bags. The masses of the large bags are normally distributed with mean $2.50\,\text{kg}$ and standard deviation $0.12\,\text{kg}$.
May/June 2025
In a particular forest, tree heights are grouped as tall, medium or small. Their heights may be represented by a normal distribution with mean $20\,\text{m}$ and standard deviation $5\,\text{m}$. Any tree with height below $14\,\text{m}$ is put into the small group.
May/June 2025
In one large school, the average number of pupils who take music lessons is two in five. A random selection of $80$ pupils is taken from this school.
May/June 2025
Davin’s daily time spent on his games machine is normally distributed, with mean $3.5$ and standard deviation $0.9$.
Oct/Nov 2020
Pia jogs 2 km each day, and her recorded times, in minutes, are normally distributed with mean $10.1$ and standard deviation $1.3$.
Oct/Nov 2020
The times that members of a large swimming club need to swim 100 metres are normally distributed, with mean $62$ seconds and standard deviation $5$ seconds.
Oct/Nov 2020
The 13 00 service from Jahor to Keman operates every day. The chance that it reaches Keman late is $0.35$.
Oct/Nov 2020
Karli's daily time on social media, measured in minutes, follows a normal distribution with mean 125 and standard deviation 24.
Oct/Nov 2021
The completion times, measured in minutes, for this task among workers at a large company are normally distributed with mean $32.2$ and standard deviation $9.6$.
Oct/Nov 2021
Raj wants to get fitter, so he runs each day. His run times, measured in minutes, are normally distributed with mean $41.2$ and standard deviation $3.6$.
Oct/Nov 2021
People living in Persham were asked how dependable they thought their internet service was. $12\%$ described it as ‘poor’, $36\%$ described it as ‘satisfactory’ and $52\%$ described it as ‘good’. A random sample of $8$ residents of Persham is chosen.
Oct/Nov 2022
For a large population, adults’ systolic blood pressure (SBP) follows a normal distribution with mean $125.4$ and standard deviation $18.6$.
Oct/Nov 2022
The rod lengths manufactured by a company are normally distributed with mean $55.6\text{ mm}$ and standard deviation $1.2\text{ mm}$.
Oct/Nov 2022
At a company call centre, $90\%$ of callers are put through straight away to a representative. A random sample of $12$ callers is selected.
Oct/Nov 2022
In a large college, $32\%$ of the students have blue eyes. A random sample of $80$ students is drawn.
Oct/Nov 2022
Company A makes bags of sugar. An inspector discovers that, on average, $10\%$ of the bags are underweight. $10$ of the bags are picked at random.
Oct/Nov 2022
A farmer trades eggs. Their weights, measured in grams, are modelled by a normal distribution with mean $80.5$ and standard deviation $6.6$. The eggs are sorted into small, medium or large by weight. An egg is small if it weighs below $76$ grams, and $40\%$ of the eggs fall into the medium category.
Oct/Nov 2023
A factory makes one particular kind of electrical component. It is known that $15\%$ of the components made are faulty. A random sample of $200$ components is selected.
Oct/Nov 2023
The heights of club members follow a normal distribution with mean $166\text{ cm}$ and standard deviation $10\text{ cm}$. Find the probability that a member chosen at random from the club has a height below $170\text{ cm}$.
Oct/Nov 2023
A company makes large bags of pasta whose weights are normally distributed with mean $1.5\,\text{kg}$ and standard deviation $0.05\,\text{kg}$.
Oct/Nov 2023
The weights of the green apples sold by a shop follow a normal distribution with mean $90$ grams and standard deviation $8$ grams.
Oct/Nov 2024
At Breven college, the heights of the female students follow a normal distribution: • $90\%$ of the female students are shorter than $182.7$ cm. • $40\%$ of the female students are shorter than $162.5$ cm.
Oct/Nov 2024
The heights, in metres, of white pine trees follow a normal distribution with mean $19.8$ and standard deviation $2.4$. In one forest, there are $450$ white pine trees.
Oct/Nov 2024
In Wimfield, 30% of residents own an electric car, and 3 residents are selected at random.
Oct/Nov 2024
In Molimba, adult males' heights, in cm, are normally distributed with mean $176\text{ cm}$ and standard deviation $4.8\text{ cm}$.
Oct/Nov 2024
A chocolate-bar factory makes chocolate bars. Their weights are normally distributed with mean 155 g and standard deviation 6 g. One random sample of 350 bars is selected. A second chocolate-bar factory also makes chocolate bars. Their weights are normally distributed with mean $\mu$ g and standard deviation $\sigma$ g. Tests indicate that 8% of the bars weigh more than 114.0 g and 20% weigh less than 99.5 g.
Oct/Nov 2025
The 124 Senior members of the Giraffes basketball club have heights that are normally distributed with mean 187.4 cm and standard deviation 6.4 cm.
Oct/Nov 2025
Obi’s times for walking to work each morning are modelled by a normal distribution with mean 14.8 minutes and standard deviation 1.5 minutes. He does this journey 5 days a week for 45 weeks in a year.
Oct/Nov 2025
Many runners entered two charity runs to collect money for a new community centre. In the first run, the finishing times were normally distributed with mean 46.3 seconds and standard deviation 6.4 seconds. In the second run, the finishing times were normally distributed with mean 39.8 seconds and standard deviation of $\sigma$ seconds. 10% of the runners took more than 48.6 seconds.
Oct/Nov 2025