2.04e Normal distribution: as model N(mu, sigma^2)

A garden centre sells canes of nominal length 150 cm. The canes are bought from a supplier who uses a machine to cut canes of length L where L ~ N(\(\mu\), 0.3²).

1. Find the value of \(\mu\), to the nearest 0.1 cm, such that there is only a 5\% chance that a cane supplied to the garden centre will have length less than 150 cm. [4]

A customer buys 10 of these canes from the garden centre.

1. Find the probability that at most 2 of the canes have length less than 150 cm. [3]

Another customer buys 500 canes.

1. Using a suitable approximation, find the probability that fewer than 35 of the canes will have length less than 150 cm. [6]

A farmer noticed that some of the eggs laid by his hens had double yolks. He estimated the probability of this happening to be 0.05. Eggs are packed in boxes of 12. Find the probability that in a box, the number of eggs with double yolks will be

1. exactly one, [3]
2. more than three. [2]

A customer bought three boxes.

1. Find the probability that only 2 of the boxes contained exactly 1 egg with a double yolk. [3]

The farmer delivered 10 boxes to a local shop.

1. Using a suitable approximation, find the probability that the delivery contained at least 9 eggs with double yolks. [4]

The weight of an individual egg can be modelled by a normal distribution with mean 65 g and standard deviation 2.4 g.

1. Find the probability that a randomly chosen egg weighs more than 68 g. [3]

1. Write down the condition needed to approximate a Poisson distribution by a Normal distribution. [1]

The random variable Y ~ Po(30).

1. Estimate P(Y > 28). [6]

A doctor expects to see, on average, 1 patient per week with a particular disease.

1. Suggest a suitable model for the distribution of the number of times per week that the doctor sees a patient with the disease. Give a reason for your answer. [3]
2. Using your model, find the probability that the doctor sees more than 3 patients with the disease in a 4 week period. [4]

The doctor decides to send information to his patients to try to reduce the number of patients he sees with the disease. In the first 6 weeks after the information is sent out, the doctor sees 2 patients with the disease.

1. Test, at the 5\% level of significance, whether or not there is reason to believe that sending the information has reduced the number of times the doctor sees patients with the disease. State your hypotheses clearly. [6]

Medical research into the nature of the disease discovers that it can be passed from one patient to another.

1. Explain whether or not this research supports your choice of model. Give a reason for your answer. [2]

A garden centre sells canes of nominal length 150 cm. The canes are bought from a supplier who uses a machine to cut canes of length \(L\) where \(L \sim \mathrm{N}(\mu, 0.3^2)\).

1. Find the value of \(\mu\), to the nearest 0.1 cm, such that there is only a 5\% chance that a cane supplied to the garden centre will have length less than 150 cm. [4]

A customer buys 10 of these canes from the garden centre.

1. Find the probability that at most 2 of the canes have length less than 150 cm. [3]

Another customer buys 500 canes.

1. Using a suitable approximation, find the probability that fewer than 35 of the canes will have length less than 150 cm. [6]

1. Write down the condition needed to approximate a Poisson distribution by a Normal distribution. [1]

The random variable \(Y \sim \text{Po}(30)\).

1. Estimate P(\(Y > 28\)). [6]

The continuous random variable \(L\) represents the error, in mm, made when a machine cuts rods to a target length. The distribution of \(L\) is continuous uniform over the interval \([-4.0, 4.0]\). Find

1. P\((L < -2.6)\), [1]
2. P\((L < -3.0 \text{ or } L > 3.0)\). [2]

A random sample of 20 rods cut by the machine was checked.

1. Find the probability that more than half of them were within 3.0 mm of the target length. [4]

A manufacturer produces large quantities of coloured mugs. It is known from previous records that 6\% of the production will be green. A random sample of 10 mugs was taken from the production line.

1. Define a suitable distribution to model the number of green mugs in this sample. [1]
2. Find the probability that there were exactly 3 green mugs in the sample. [3]

A random sample of 125 mugs was taken.

1. Find the probability that there were between 10 and 13 (inclusive) green mugs in this sample, using
   1. a Poisson approximation, [3]
   2. a Normal approximation. [6]

Bhim and Joe play each other at badminton and for each game, independently of all others, the probability that Bhim loses is 0.2 Find the probability that, in 9 games, Bhim loses

1. exactly 3 of the games, [3]
2. fewer than half of the games. [2]

Bhim attends coaching sessions for 2 months. After completing the coaching, the probability that he loses each game, independently of all others, is 0.05 Bhim and Joe agree to play a further 60 games.

1. Calculate the mean and variance for the number of these 60 games that Bhim loses. [2]
2. Using a suitable approximation calculate the probability that Bhim loses more than 4 games. [3]

In Manuel's restaurant the probability of a customer asking for a vegetarian meal is 0.30. During one particular day in a random sample of 20 customers at the restaurant 3 ordered a vegetarian meal.

1. Stating your hypotheses clearly, test, at the 5\% level of significance, whether or not the proportion of vegetarian meals ordered that day is unusually low. [5]

Manuel's chef believes that the probability of a customer ordering a vegetarian meal is 0.10. The chef proposes to take a random sample of 100 customers to test whether or not there is evidence that the proportion of vegetarian meals ordered is different from 0.10.

1. Stating your hypotheses clearly, use a suitable approximation to find the critical region for this test. The probability for each tail of the region should be as close as possible to 2.5\%. [6]
2. State the significance level of this test giving your answer to 2 significant figures. [1]

A biologist is studying the behaviour of sheep in a large field. The field is divided up into a number of equally sized squares and the average number of sheep per square is 2.25. The sheep are randomly spread throughout the field.

1. Suggest a suitable model for the number of sheep in a square and give a value for any parameter or parameters required. [1]

Calculate the probability that a randomly selected sample square contains

1. no sheep, [1]
2. more than 2 sheep. [4]

A sheepdog has been sent into the field to round up the sheep.

1. Explain why the model may no longer be applicable. [1]

In another field, the average number of sheep per square is 20 and the sheep are randomly scattered throughout the field.

1. Using a suitable approximation, find the probability that a randomly selected square contains fewer than 15 sheep. [7]

A farm produces potatoes. The potatoes are packed into sacks. The weight of a sack of potatoes is modelled by a normal distribution with mean 25.6 kg and standard deviation 0.24 kg

1. Find the probability that two randomly chosen sacks of potatoes differ in weight by more than 0.5 kg [6]

Sacks of potatoes are randomly selected and packed onto pallets. The weight of an empty pallet is modelled by a normal distribution with mean 20.0 kg and standard deviation 0.32 kg Each full pallet of potatoes holds 30 sacks of potatoes.

1. Find the probability that the total weight of a randomly chosen full pallet of potatoes is greater than 785 kg [5]

A fair six-sided die is labelled with the numbers 1, 2, 3, 4, 5 and 6. The die is rolled 40 times and the score, \(S\), for each roll is recorded.

1. Find the mean and the variance of \(S\). [2]
2. Find an approximation for the probability that the mean of the 40 scores is less than 3 [3]

The weights of a group of males are normally distributed with mean 80 kg and standard deviation 2.6 kg. A random sample of 10 of these males is selected.

1. Write down the distribution of \(M\), the mean weight, in kg, of this sample. [2]
2. Find P(\(M < 78.5\)). [3]

The weights of a group of females are normally distributed with mean 59 kg and standard deviation 1.9 kg. A random sample of 6 of the males and 4 of the females enters a lift that can carry a maximum load of 730 kg.

1. Find the probability that the maximum load will be exceeded when these 10 people enter the lift. [5]

The weights of tubs of margarine are known to be normally distributed. A random sample of 10 tubs of margarine were weighed, to the nearest gram, and the results were as follows. $$498 \quad 502 \quad 500 \quad 496 \quad 509 \quad 504 \quad 511 \quad 497 \quad 506 \quad 499$$

1. Find unbiased estimates of the mean and the variance of the population from which this sample was taken. [5]

Given that the population standard deviation is 5.0 g,

1. estimate limits, to 2 decimal places, between which 90\% of the weights of the tubs lie, [2]
2. find a 95\% confidence interval for the mean weight of the tubs. [5]

A second random sample of 15 tubs was found to have a mean weight of 501.9 g.

1. Stating your hypotheses clearly and using a 1\% level of significance, test whether or not the mean weight of these tubs is greater than 500 g. [5]

The weights of a group of males are normally distributed with mean 80 kg and standard deviation 2.6 kg. A random sample of 10 of these males is selected.

1. Write down the distribution of \(\bar{M}\), the mean weight, in kg, of this sample. [2]
2. Find P(\(\bar{M} < 78.5\)). [3]

The weights of a group of females are normally distributed with mean 59 kg and standard deviation 1.9 kg. A random sample of 6 of the males and 4 of the females enters a lift that can carry a maximum load of 730 kg.

1. Find the probability that the maximum load will be exceeded when these 10 people enter the lift. [5]

The random variable \(X\) is normally distributed with mean \(\mu\) and variance \(\sigma^2\).

1. If \(2\mu = 3\sigma\), find P\((X < 2\mu)\). [5 marks]
2. If, instead, P\((X < 3\mu) = 0.86\),
   1. find \(\mu\) in terms of \(\sigma\), [4 marks]
   2. calculate P\((X > 0)\). [4 marks]

The heights of the students at a university are assumed to follow a normal distribution. 1% of the students are over 200 cm tall and 76% are between 165 cm and 200 cm tall. Find

1. the mean and the variance of the distribution, [9 marks]
2. the percentage of the students who are under 158 cm tall. [3 marks]
3. Comment briefly on the suitability of a normal distribution to model such a population. [2 marks]

The times taken by a group of people to complete a task are modelled by a normal distribution with mean 8 hours and standard deviation 2 hours. Use this model to calculate

1. the probability that a person chosen at random took between 5 and 9 hours to complete the task, [4 marks]
2. the range, symmetrical about the mean, within which 80% of the people's times lie. [5 marks]

It is found that, in fact, 80% of the people take more than 5 hours. The model is modified so that the mean is still 8 hours but the standard deviation is no longer 2 hours.

1. Find the standard deviation of the times in the modified model. [3 marks]

The random variable \(X\) has the normal distribution \(N(2, 1.7^2)\).

1. State the standard deviation of \(X\). [1 mark]
2. Find \(P(X < 0)\). [2 marks]
3. Find \(P(0.6 < X < 3.4)\). [4 marks]

The ages of the residents of a retirement community are assumed to be normally distributed. 15% of the residents are under 60 years old and 5% are over 90 years old.

1. Using this information, find the mean and the standard deviation of the ages. [7 marks]
2. If there are 200 residents, find how many are over 80 years old. [3 marks]

A company makes two cars, model \(A\) and model \(B\). The distance that model \(A\) travels on 10 litres of petrol is normally distributed with mean 109 km and variance 72.25 km\(^2\). The distance that model \(B\) travels on 10 litres of petrol is normally distributed with mean 108.5 km and variance 169 km\(^2\). In a trial, one of each model is filled with 10 litres of petrol and sent on a journey of 110 km. Find which model has the greater probability of completing this journey, and state the value of this probability. [8 marks]

The entrance to a car park is \(1.9\) m wide. It is found that this is too narrow for \(2\%\) of the vehicles which need to use the car park. The widths of these vehicles are modelled by a normal distribution with mean \(1.6\) m.

1. Find the standard deviation of the distribution. [4 marks]

It is decided to widen the entrance so that \(99.5\%\) of vehicles will be able to use it.

1. Find the minimum width needed to achieve this. [4 marks]

The rainfall at a weather station was recorded every day of the twentieth century. One year is selected at random from the records and the total rainfall, in cm, in January of that year is denoted by \(R\). Assuming that \(R\) can be modelled by a normal distribution with standard deviation \(12.6\), and given that P\((R > 100) = 0.0764\),

1. find the mean of \(R\), [4 marks]
2. calculate P\((75 < R < 80)\). [5 marks]