4.9 Further Normal Distribution (inc Central Limit Theorem)

Let be a normally distributed random variable that represents the distance travelled in metres by a slug in one day. The distance covered by a random sample of 21 slugs on a randomly selected day can be summarized as follows

Find an unbiased estimate of the mean, , of .

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

Find a 95% confidence interval for μ.

Justin believes that the average slug travels 15 m per day.

State whether or not Justin’s statement is valid. Give a reason for your answer.

A farm grows pumpkins and transports them in crates of 24. The mass of the pumpkins follows a normal distribution with mean 7.9 kg and standard deviation 0.4 kg.

Find the mean mass of a crate of pumpkins.

Find the standard deviation of the mass of a crate of pumpkins.

Find the probability that a crate selected at random has a mass of between 170 kg and 190 kg.

The time taken for a customer services advisor to complete a phone call follows a normal distribution with mean 4.2 minutes and standard deviation 1.3 minutes.

A customer service advisor deals with 5 phone calls one after the other. It is assumed that the phone calls are independent events.

Find the expected total time to complete the 5 phone calls.

Find the variance of the total time to complete the 5 phone calls.

Find the probability that the total time taken to complete the 5 phone calls will be more than 25 minutes.

A fisherman catches 16 fish from a local population of mackerel. He measures the fish that he catches and finds that they have mean length of 30.5 cm with standard deviation 5 cm.

Find .

Find a 95 % confidence interval for the population mean.

The fisherman advertises the population from which he fishes as having an average length of 34 cm.

Comment on the fisherman’s claim using your answer from part (b).

A gardener is laying a pathway of pebbles from a large sack of pebbles. The mass of the pebbles is normally distributed with mean 564 g and standard deviation 57g.

Find the probability that a pebble that the gardener picks at random from the sack has a mass of less than 500 g.

The gardener decides that any pebbles that have a mass greater than 620 g are “oversized” and should not be used to create the pathway.

Find the probability that a pebble selected at random from the sack will be considered “oversized”.

The gardener decides to select 8 of the pebbles at random.

Find the probability that the mean mass of the 8 pebbles selected would fall in the “oversized” range.

In a busy office all workers are able to send jobs to the printer to be printed. It is assumed that each print job is an independent event, and that more than one print job does not arrive in the print queue at the same time. The number of jobs arriving in the print queue in 1 hour follows a Poisson distribution given by .

Find the probability that the number of print jobs sent to the printer in 1 hour is less than 15.

Helen wants to investigate the mean number of print jobs sent to the printer in an hour over the course of a working week of 35 hours.

Using the central limit theorem, define a probability distribution that may be used to approximate the distribution of the random variable .

Using the answer to part (b), find the probability that in a working week of 35 hours the mean number of print jobs in a single hour is less than 15.

A population of leatherback turtles has a mean swimming speed of 31 km/h with standard deviation 2.3 km/h.

Using the central limit theorem, find an estimate for the probability that a sample of 32 leatherback turtles have a mean swimming speed greater than 31.8 km/h.

Also using the central limit theorem, find an estimate for the probability that a sample of 50 leatherback turtles have a mean swimming speed greater than 31.8 km/h.

Explain why there is a difference between your answers to part (a) and part (b).

On a flower farm the height of a tulip, in centimetres, is normally distributed with mean μ and standard deviation σ. A random sample of 60 flowers is taken from the farm and can be summarised as follows

Find an unbiased estimate for μ.

Given that , find an unbiased estimate for the variance of the height of the tulips.

It is subsequently discovered that the actual standard deviation, , of the tulip population is 6.47 cm.

A second sample of 40 tulips is picked.

 denotes the mean height of the new sample.

State a distribution that may reasonably be used to model  .

Using the answer to part (c), find an estimate for the probability that the mean height of the flowers in the new sample is between 20 and 30 cm.

Jessamy is interested in the quality of the soil in her local area and decides to test 100 soil samples for levels of nitrogen. From her past research Jessamy knows that the level of nitrogen in an individual sample, , has a mean of 41 ppm and a standard deviation of 7 ppm.

Let  be the total of the levels of nitrogen in Jessamy’s batch of samples.

Find 

Explain why a normal distribution can be used to give an approximate model for X.

Use the model to find the an estimate for the value of such that .

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.