4.9 Further Normal Distribution (inc Central Limit Theorem)

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.

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. 

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.

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.

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”.

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”. 

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. 

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.

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.

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.

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.

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  

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.

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.

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. 

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.

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.

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

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