Further Normal Distribution (inc Central Limit Theorem) (DP IB Maths: AI HL)

Caleb owns a fishing lake. He catches 22 fish from the lake. He measures the fish that he catches and finds that they have mean length of 34.2 cm with standard deviation 7 cm. It is assumed that the lengths of the fish are normally distributed.

Find a 95% confidence interval for the population mean.

Caleb produces an advert for his fishing lake, stating that the average length of fish is 37 cm. Comment on Caleb’s claim.

Explain whether or not you needed to use the central limit theorem in your answer to part (a).

Dan enjoys a cup of coffee in his favourite mug every morning. He wants to check that the amount of coffee dispensed by his coffee machine stays consistent.

He measures the volume of coffee in his mug each morning and records the data in a spreadsheet over 31 days. The mean of Dan’s data is 219 ml and the standard deviation is 4.6 ml. 

Dan decides that the machine needs its settings adjusted if the amount of coffee it is dispensing on average is different to 220 ml.

Using a 95% confidence interval for the population mean, decide if Dan needs to adjust the settings on the coffee machine, justifying your answer.

SME Juices manufactures 250 ml cartons of apple juice. The quality control manager needs to make sure that the volume in each carton is suitably close to the advertised volume of 250 ml. She takes a random sample of 34 cartons and measures the volume of juice that they contain. It is assumed that the volume of apple juice in each carton follows a normal distribution. Her findings are summarised below.

Find unbiased estimates for the mean and variance of the population.

Find a 95% confidence interval for the population mean.

A customer complains that the mean volume in the apple juice cartons is 245 ml. State whether the customer’s complaint is justified, giving a reason for your answer.

A manager says that customers are only likely to complain if there is less than 246 ml of juice in a carton. He sets a target of less than 3% of customers making a complaint.

Assuming that the estimated mean and variance of the population are in fact the actual mean and variance

At an alligator sanctuary, the lengths of alligators, L metres, are assumed to be normally distributed with mean  and standard deviation .

A random sample of 12 alligators are safely captured so their health can be monitored. The sample can be summarised as follows

Find an unbiased estimate for the mean of L.

Use the formula  to find an unbiased estimate of the variance of L.

Find a 90% confidence interval for the mean length of alligators.

Explain how the sanctuary could obtain a smaller 90% confidence interval for the mean length of alligators.

A vet explains that in a population of alligators, it is likely that there are far more extremely long alligators than extremely short alligators. Explain how this affects the validity of the answer to part (c).

A veterinary nurse is investigating the weights of cats who attend her clinic. Over 1 week she weighs 34 cats. She records their weights and at the end of the week finds that the mean of her sample is 4.2 kg and the standard deviation is 0.5 kg.

Find a 90% confidence interval for the mean weight of cats visiting her clinic overall.

She decides to extend her study so that it lasts for a whole month. Her sample now includes 135 cats in total, with a mean of 4.18 kg.

Explain what is likely to happen to the width of the 90% confidence interval as a result of extending her study.

The veterinary nurse later finds a database containing the whole population of cats who have ever visited the veterinary practice. The database shows that the standard deviation of the weight of the cats is 0.4 kg.

Using this new information, and the sample for the whole month, find a new 90% confidence interval for the mean weight of cats visiting the practice.

A factory manufactures wheels for trains. The radii of the wheels follow a normal distribution. The mean of the radii of the wheels is 459.9 mm and the standard deviation is 0.2 mm. 

Once the wheels are made, the circumference of each wheel is coated in a uniform protective layer of thickness 3 mm, as shown in the diagram.

Find, including the protective layer:

An ecologist in Antarctica is weighing adult from a large colony. The mass of the adult penguins is normally distributed with mean 4.8 kg and standard deviation 1.2 kg.

The ecologist decides that any adult penguins that have a mass less than 3 kg are at risk of malnutrition.

Find the probability that a penguin selected at random from the colony will be considered at risk of malnutrition.

The ecologist is making a video to show the health of penguins in the colony, and she decides to select 8 of the penguins at random to feature.

Find the probability that the mean mass of the 8 selected penguins would be classed as being at risk of malnutrition.

Explain why the value for part (b) is lower than part (a).

The number of goals scored in a soccer match follows a Poisson distribution with an average of 2.5 goals per match.

Find the probability that there are no goals scored in a match.

Gary is analysing the number of goals scored per match at the World Cup, where there were 64 matches played.

Using the central limit theorem, estimate the probability that the mean number of goals per match at the World Cup is 3 or more.

Gary wants to analyse the mean number of goals per match, just in the matches involving England. England played in 7 matches at the tournament.

Comment on whether the method used in part (b) would still be suitable.

In a role-playing game a 12-sided fair dice, numbered 1 to 12, is rolled to determine if a character’s magic spell is successful. If the rolled number is 8 or higher; their spell is successful. If the rolled number is less than 8 their spell is unsuccessful. In a particular magical battle, 6 spells are cast.

Define the probability distribution that could be used to model the number of successful spells in the magical battle. State any assumptions that are necessary.

Find

During the whole game, there are 48 magical bottles; each involving 6 spells, as modelled previously. Find the probability that there are on average, 2 or fewer successful spells per battle.

Explain why it is valid to use the central limit theorem for part (c).

It is suggested that the number of potholes (small holes in the road) in a 1 km stretch of road can be modelled by a Poisson distribution. There are estimated to be 15 potholes per kilometre in the UK on average.

Lucy works in the council and receives a complaint from a local resident, stating that they encountered over 100 potholes on their 5 km journey to work. Lucy decides to model the frequency of potholes using a Poisson distribution. 

Using a Poisson distribution, find the probability that there are over 100 potholes on a 5 km stretch of road.

Lucy claims that there are fewer potholes per 1 km in her local area than the national average. She decides to investigate this by taking a random sample of 50 separate 1 km sections of road.

Lucy calculates the mean number of potholes in a 1 km section of road is 13.5 (10% below the national average).

State a possible reason why a Poisson distribution may not be an accurate to model the occurrence of potholes.

At a cheese factory, cheese is made into ‘wheels’. Their weights are normally distributed with a mean of 6.98 kg and a standard deviation 0.1 kg.

The cheese wheels are transported on pallets. Each pallet holds 9 cheese wheels and is labelled with “Weight: 63 kg”. 

The cheese inspector takes a random sample of 2 pallets. 

Find the probability that the mean weight of the 2 pallets is less than labelled. 

A manufacturer producing bags of jellybeans claims that each bag contains an average of 60 jellybeans. Sebastian buys 35 bags of jellybeans check their claim. 

Sebastian found that from his 35 bags

 where  represents the number of jellybeans in each bag.

Use the formula  to help find a 95% confidence interval for the mean number of jellybeans per bag. State any assumptions you use.

Suggest, with justification,

The jellybean manufacturer would like to reward Sebastian for his research, and they decide to send him one million jellybeans. The manufacturer sends Sebastian 16820 bags. 

Using the data from Sebastian’s sample, estimate the probability that the 16820 bags contain less than one million jellybeans in total. State any assumptions you use.

It is known that the weights of male Border Terriers (a breed of dog) are normally distributed with a mean of 6.5 kg and a variance of 0.62 kg².

A group of 68 male Border Terriers are sampled from the population.

Find the probability that the mean weight of the 68 Border Terriers in the sample is less than 6.2 kg.

72 females are now added to the sample. The weights of female Border Terriers are normally distributed with mean 5.8 kg and variance 0.65² kg². A random dog from the combined group of males and females is selected.

Find the probability that the dog is a female, given that it weighs over 6 kg.

The table below shows the heights in cm of a class of 25 students, all aged 14.

Height, h cm	Frequency
	3
	5
	9
	7
	1

A student who is 160 cm tall says “I think I am average height compared to other students our age in the country”.

Investigate this claim using a 95% confidence interval for the mean. Clearly state what the population is in this case.

The tallest student in the class, at a height of 174.5 cm, says “I think my height is in the top 1% for students our age in the country”. 

Assuming the unbiased estimates for the population mean and variance are the actual population parameters, investigate this student’s claim. You may assume that the population is normally distributed.

Andrew and Bob are inspecting the quality of two types of bolts which are going to be used to replace current bolts in two different parts of an aircraft.

Andrew is inspecting the replacements for Bolt A, which are used to help secure a compartment in the cabin containing the pilot’s snacks. They should have a diameter of 1 cm. He takes a sample of 32 replacement bolts and produces a report about their suitability.

Bob is inspecting the replacements for Bolt B, which are used to help secure the engines to the wings. They should have a diameter of 8 cm. He takes a sample of 32 replacement bolts and produces a report about their suitability.

Bolt A (Andrew’ report)

Unbiased estimate of mean: 0.995 cm

Unbiased estimate of the variance: 0.01² cm²

Using a 99% confidence interval I found that the interval was entirely below 1 cm, so I have decided that Bolt A has FAILED.

Bolt B (Bob’s report)

Unbiased estimate of mean: 7.99 cm

Unbiased estimate of the variance: 0.05² cm²

Using an 80% confidence interval I found that the interval contained the value 8 cm, so I have decided that Bolt B has PASSED. 

Find the confidence intervals, correct to 4 decimal places, that Andrew and Bob used.

In the coastal town of Burnham-on-Sea, historical data from 1911 to 2001 shows that the sea flows over the flood defences and floods the road once every 5 years on average. It can be assumed that the number of times this happens in any length of time can be modelled by a Poisson distribution.

Find the probability that the road is flooded by the sea at least 3 times in a 10-year period.

Clarissa, a climate scientist, collects data from other locations around the world with the same historical rate of flooding as Burnham-on-Sea. There are 32 different locations in her sample, and her data covers the 5-year period 2016 to 2020 in each location. Clarissa records the number of times that the roads are flooded in those 32 locations.

The mean number of times the roads were flooded during the 5-year period was in the top 5% according to the model. Find the range of values for the mean.

Using the sample of the 32 locations, find the probability that the mean number of times that the roads were flooded during the 5-year period is more than 10% higher than the historical mean.

A coffee shop; Küste Kaffee, has a loyalty card scheme which allows them to track customers’ spending habits. They have found that adults have a mean spend of €6.80 and a standard deviation of €2. Teenagers have a mean spend of €4.75 with a standard deviation of €3.10. The spending of both adults and teenagers are modelled as normal distributions.

On Wednesday between 2pm and 3pm, 17 adults and 8 teenagers visit Küste Kaffee and they each make a single purchase.

Find the probability that the coffee shop receives over €160 in the hour. State any assumptions you have made.

On Thursday between 2pm and 3pm, the coffee shop experiments with a “half price hour” where all items are 50% off. They predict that 30 adults and 20 teenagers will each make a single purchase during this hour.

Find the probability that Küste Kaffee receives over €160 on Thursday between 2pm and 3pm. State any assumptions you have made.

A game at a funfair involves throwing a ball at some coconuts and knocking them off their stand. There are 5 coconuts and players get three balls to throw in turn. To win, players must knock over 3 coconuts in a row. Players win a large stuffed toy unicorn as a prize if they win.

When there are  coconuts still standing, the probability of hitting one is  

On Friday night, 35 players play the game. Find the probability that at least 5 players win.

A statistician, Stacey, visits the funfair and takes a random sample of 35 players each evening who play the game to see how many of them win. Stacey is really keen on gathering lots of statistics, so she does this over the course of 38 days.

Find the probability that over 38 days, the mean number of winners out of the 35 players in a sample is less than 2.84.

Explain why it is valid to use the central limit theorem for part (b).

A class of primary school children are painting a wall with pictures of themselves. They are going to measure their heights, and then make the paintings an enlargement of scale factor 1.5; so that they cover more of the wall.

Explain how the 1.5 scale factor will affect

A child in the class thinks that this still won’t cover enough of the wall. They suggest that once the enlarged pictures are painted on the wall, they should draw a 30 cm hat on every picture’s head. 

Explain how adding the hats to all the paintings will affect

In the class there are 20 children. Their heights can be modelled as a normal distribution with mean 128 cm and standard deviation 6 cm. They all paint their pictures onto the wall, which is 2.25 m high, using a scale factor of 1.5. Once the pictures are painted, 12 students are randomly selected to have a 30 cm hat drawn on top of their paintings’ heads.

Find the probability that at least 1 student’s painting is too tall to fit on the wall.

Caleb owns a fishing lake. He creates an advert for social media 

Come to Caleb’s Fishing Lake!

Average catch size: a huge 37.8 cm!

(10 fish sampled with standard deviation of 18 cm)

£5 per day entry

Refund if you catch 10 fish and their mean is less than 35 cm! 

Caleb models the length of fish in the lake using a normal distribution.

It costs Caleb £80 per day to run the fishing lake. On a given day, 28 people pay the entry fee to come fishing.

Assuming that all 28 people each catch exactly 10 fish, find the probability that Caleb does not make a profit on this day. You may assume that the unbiased estimates for the mean and variance can be used as the actual mean and variance.

Caleb decides he wants to put a different statistic in his next advert. 

Find a 90% confidence interval for the mean using Caleb’s sample of 10 fish.

Stacey the statistician visits the fishing lake and determines that the lengths of fish in the lake are not normally distributed.

Decide whether Caleb should use the confidence interval, that was calculated in part (b), in his next advert. Justify your answer.