5.04b Linear combinations: of normal distributions

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]

A manufacturer produces two flavours of soft drink, cola and lemonade. The weights, \(C\) and \(L\), in grams, of randomly selected cola and lemonade cans are such that \(C \sim \text{N}(350, 8)\) and \(L \sim \text{N}(345, 17)\).

1. Find the probability that the weights of two randomly selected cans of cola will differ by more than 6 g. [6]

One can of each flavour is selected at random.

1. Find the probability that the can of cola weighs more than the can of lemonade. [6]

Cans are delivered to shops in boxes of 24 cans. The weights of empty boxes are normally distributed with mean 100 g and standard deviation 2 g.

1. Find the probability that a full box of cola cans weighs between 8.51 kg and 8.52 kg. [6]
2. State an assumption you made in your calculation in part (c). [1]

(Total 19 marks)

The workers in a large office block use a lift that can carry a maximum load of 1090 kg. The weights of the male workers are normally distributed with mean 78.5 kg and standard deviation 12.6 kg. The weights of the female workers are normally distributed with mean 62.0 kg and standard deviation 9.8 kg. Random samples of 7 males and 8 females can enter the lift.

1. Find the mean and variance of the total weight of the 15 people that enter the lift. [4]
2. Comment on any relationship you have assumed in part (a) between the two samples. [1]
3. Find the probability that the maximum load of the lift will be exceeded by the total weight of the 15 people. [4]

The random variable \(A\) is defined as $$A = 4X - 3Y$$ where \(X \sim \text{N}(30, 3^2)\), \(Y \sim \text{N}(20, 2^2)\) and \(X\) and \(Y\) are independent. Find

1. E(\(A\)), [2]
2. Var(\(A\)). [3]

The random variables \(Y_1\), \(Y_2\), \(Y_3\) and \(Y_4\) are independent and each has the same distribution as \(Y\). The random variable \(B\) is defined as $$B = \sum_{i=1}^{4} Y_i$$

1. Find P(\(B > A\)). [6]

The lifetimes of batteries from manufacturer \(A\) are normally distributed with mean 20 hours and standard deviation 5 hours when used in a camera.

1. Find the mean and standard deviation of the total lifetime of a pack of 6 batteries from manufacturer \(A\). [2]

Judy uses a camera that takes one battery at a time. She takes a pack of 6 batteries from manufacturer \(A\) to use in her camera on holiday.

1. Find the probability that the batteries will last for more than 110 hours on her holiday. [2]

The lifetimes of batteries from manufacturer \(B\) are normally distributed with mean 35 hours and standard deviation 8 hours when used in a camera.

1. Find the probability that the total lifetime of a pack of 6 batteries from manufacturer \(A\) is more than 4 times the lifetime of a single battery from manufacturer \(B\) when used in a camera. [6]

The weights of eggs are normally distributed with mean 60g and standard deviation 5g Sairah chooses 2 eggs at random.

1. Find the probability that the difference in weight of these 2 eggs is more than 2g (5) Sairah is packing eggs into cartons. The weight of an empty egg carton is normally distributed with mean 40g and standard deviation 1.5g
2. Find the distribution of the total weight of a carton filled with 12 randomly chosen eggs. (3)
3. Find the probability that a randomly chosen carton, filled with 12 randomly chosen eggs, weighs more than 800g (2)

The random variable \(R\) is defined as \(R = X + 4Y\) where \(X \sim \text{N}(8, 2^2)\), \(Y \sim \text{N}(14, 3^2)\) and \(X\) and \(Y\) are independent. Find

1. E\((R)\), [2]
2. Var\((R)\), [3]
3. P\((R < 41)\) [3]

The random variables \(Y_1\), \(Y_2\) and \(Y_3\) are independent and each has the same distribution as \(Y\). The random variable \(S\) is defined as $$S = \sum_{i=1}^{3} Y_i - \frac{1}{2}X.$$

1. Find Var\((S)\). [4]

The three tasks most frequently carried out in a garage are \(A\), \(B\) and \(C\). For each of the tasks the times, in minutes, taken by the garage mechanics are assumed to be normally distributed with means and standard deviations given in the following table. Assuming that the times for the three tasks are independent, calculate the probability that

1. the total time taken by a single randomly chosen mechanic to carry out all three tasks lies between 533 and 655 minutes, [5]
2. a randomly chosen mechanic takes longer to carry out task \(B\) than task \(C\). [5]

The number of cars passing a point on a single-track one-way road during a one-minute period is denoted by \(X\). Cars pass the point at random intervals and the expected value of \(X\) is denoted by \(\lambda\).

1. State, in the context of the question, two conditions needed for \(X\) to be well modelled by a Poisson distribution. [2]
2. At a quiet time of the day, \(\lambda = 6.50\). Assuming that a Poisson distribution is valid, calculate P\((4 \leq X < 8)\). [3]
3. At a busy time of the day, \(\lambda = 30\).
   1. Assuming that a Poisson distribution is valid, use a suitable approximation to find P\((X > 35)\). Justify your approximation. [6]
   2. Give a reason why a Poisson distribution might not be valid in this context when \(\lambda = 30\). [1]

A charity receives donations of more than £10000 at an average rate of 25 per year. Find the probability that the charity receives

1. exactly 30 such donations in one year, [3]
2. less than 3 such donations in one month. [5]
3. Using a suitable approximation, find the probability that the charity receives more than 45 donations of more than £10000 in the next two years. [5]

Ben is a fencing contractor who is often required to repair a garden fence by replacing a broken post between fence panels, as illustrated. \includegraphics{figure_4} The tasks involved are as follows. \(U\): detach the two fence panels from the broken post \(V\): remove the broken post \(W\): insert a new post \(X\): attach the two fence panels to the new post The mean and the standard deviation of the time, in minutes, for each of these tasks are shown in the table.

Task	Mean	Standard deviation
\(U\)	15	5
\(V\)	40	15
\(W\)	75	20
\(X\)	20	10

The random variables \(U\), \(V\), \(W\) and \(X\) are pairwise independent, except for \(V\) and \(W\) for which \(\rho_{VW} = 0.25\).

1. Determine values for the mean and the variance of:
   1. \(R = U + X\);
   2. \(F = V + W\);
   3. \(T = R + F\);
   4. \(D = W - V\).
   [8 marks]
2. Assuming that each of \(R\), \(F\), \(T\) and \(D\) is approximately normally distributed, determine the probability that:
   1. the total time taken by Ben to repair a garden fence is less than 3 hours;
   2. the time taken by Ben to insert a new post is at least 1 hour more than the time taken by him to remove the broken post.
   [5 marks]

Geoffrey is a university lecturer. He has to prepare five questions for an examination. He knows by experience that it takes about 3 hours to prepare a question, and he models the time (in minutes) taken to prepare one by the Normally distributed random variable \(X\) with mean 180 and standard deviation 12, independently for all questions.

1. One morning, Geoffrey has a gap of 2 hours 50 minutes (170 minutes) between other activities. Find the probability that he can prepare a question in this time. [3]
2. One weekend, Geoffrey can devote 14 hours to preparing the complete examination paper. Find the probability that he can prepare all five questions in this time. [3]

A colleague, Helen, has to check the questions.

1. She models the time (in minutes) to check a question by the Normally distributed random variable \(Y\) with mean 50 and standard deviation 6, independently for all questions and independently of \(X\). Find the probability that the total time for Geoffrey to prepare a question and Helen to check it exceeds 4 hours. [3]
2. When working under pressure of deadlines, Helen models the time to check a question in a different way. She uses the Normally distributed random variable \(\frac{1}{2}X\), where \(X\) is as above. Find the length of time, as given by this model, which Helen needs to ensure that, with probability 0.9, she has time to check a question. [4]

Ian, an educational researcher, suggests that a better model for the time taken to prepare a question would be a constant \(k\) representing "thinking time" plus a random variable \(T\) representing the time required to write the question itself, independently for all questions.

1. Taking \(k\) as 45 and \(T\) as Normally distributed with mean 120 and standard deviation 10 (all units are minutes), find the probability according to Ian's model that a question can be prepared in less than 2 hours 30 minutes. [2]

Juliet, an administrator, proposes that the examination should be reduced in time and shorter questions should be used.

1. Juliet suggests that Ian's model should be used for the time taken to prepare such shorter questions but with \(k = 30\) and \(T\) replaced by \(\frac{2}{3}T\). Find the probability as given by this model that a question can be prepared in less than \(1\frac{1}{4}\) hours. [3]

An electronics company purchases two types of resistor from a manufacturer. The resistances of the resistors (in ohms) are known to be Normally distributed. Type A have a mean of 100 ohms and standard deviation of 1.9 ohms. Type B have a mean of 50 ohms and standard deviation of 1.3 ohms.

1. Find the probability that the resistance of a randomly chosen resistor of type A is less than 103 ohms. [3]
2. Three resistors of type A are chosen at random. Find the probability that their total resistance is more than 306 ohms. [3]
3. One resistor of type A and one resistor of type B are chosen at random. Find the probability that their total resistance is more than 147 ohms. [3]
4. Find the probability that the total resistance of two randomly chosen type B resistors is within 3 ohms of one randomly chosen type A resistor. [5]
5. The manufacturer now offers type C resistors which are specified as having a mean resistance of 300 ohms. The resistances of a random sample of 100 resistors from the first batch supplied have sample mean 302.3 ohms and sample standard deviation 3.7 ohms. Find a 95\% confidence interval for the true mean resistance of the resistors in the batch. Hence explain whether the batch appears to be as specified. [4]

1. The manager of a company that employs 250 travelling sales representatives wishes to carry out a detailed analysis of the expenses claimed by the representatives. He has an alphabetical (by surname) list of the representatives. He chooses a sample of representatives by selecting the 10th, 20th, 30th and so on. Name the type of sampling the manager is attempting to use. Describe a weakness in his method of using it, and explain how he might overcome this weakness. [3]

The representatives each use their own cars to drive to meetings with customers. The total distance, in miles, travelled by a representative in a month is Normally distributed with mean 2018 and standard deviation 96.

1. Find the probability that, in a randomly chosen month, a randomly chosen representative travels more than 2100 miles. [3]
2. Find the probability that, in a randomly chosen 3-month period, a randomly chosen representative travels less than 6000 miles. What assumption is needed here? Give a reason why it may not be realistic. [5]
3. Each month every representative submits a claim for travelling expenses plus commission. Travelling expenses are paid at the rate of 45 pence per mile. The commission is 10\% of the value of sales in that month. The value, in £, of the monthly sales has the distribution N(21200, 1100²). Find the probability that a randomly chosen claim lies between £3000 and £3300. [7]

An examiner believes that once she has marked the first 20 papers the time it takes her to mark one paper for a particular exam follows a Normal distribution. Having already marked more than 20 papers for each of the \(P1\), \(M1\) and \(S1\) modules set one summer, the mean and standard deviation, in seconds, of the time it takes her to mark a paper for each module are as shown in the table below.

1. Find the probability that the difference in the time it takes her to mark two randomly chosen \(P1\) papers is less than 5 seconds. [6]
2. Find the probability that it takes her less than 10 hours to mark 45 \(M1\) and 80 \(S1\) papers. [7]

The length of time that registered customers spend on each visit to a supermarket's website is normally distributed with a mean of 28.5 minutes and a standard deviation of 7.2 minutes. Eight visitors to the site are selected at random and the length of time, \(T\) minutes, that each stays is recorded.

1. Write down the distribution of \(\overline{T}\), the mean time spent at the site by these eight visitors. [2 marks]
2. Find \(P(25 < \overline{T} < 30)\). [4 marks]

The mass of waste in filled large dustbin bags is normally distributed with a mean of 6.8 kg and a standard deviation of 1.5 kg. The mass of waste in filled small dustbin bags is normally distributed with a mean of 3.2 kg and a standard deviation of 0.6 kg. One week there are 8 large and 3 small dustbin bags left for collection outside a block of flats. Find the probability that this waste has a total mass of more than 70 kg. [7 marks]

An organic farm produces eggs which it sells through a local shop. The weight of the eggs produced on the farm are normally distributed with a mean of 55 grams and a standard deviation of 3.9 grams.

1. Find the probability that two of the farm's eggs chosen at random differ in weight by more than 4 grams. [5]

The farm sells boxes of six eggs selected at random. The weight of the boxes used are normally distributed with a mean of 28 grams and a standard deviation of 1.2 grams.

1. Find the probability that a randomly chosen box with six eggs in weighs less than 350 grams. [6]

The rules for the weight of a cricket ball state: ``The ball, when new, shall weigh not less than 155.9 g, nor more than 163 g.'' A company produces cricket balls whose weights are normally distributed. It wants 99\% of the balls it produces to be an acceptable weight.

1. What is the largest acceptable standard deviation? [3]

The weights of the cricket balls are in fact normally distributed with mean 159.5 grams and standard deviation 1.2 grams. The company also produces tennis balls. The weights of the tennis balls are normally distributed with mean 58.5 grams and standard deviation 1.3 grams.

1. Find the probability that the weight of a randomly chosen cricket ball is more than three times the weight of a randomly chosen tennis ball. [6]

A farmer uses many identical containers to store four different types of grain: wheat, corn, einkorn and emmer.

1. The mass \(W\), in kg, of wheat stored in each individual container is normally distributed with mean \(\mu\) and standard deviation 0.6. Given that, for containers of wheat, 10\% store less than 19 kg, find the value of \(\mu\). [3]

The mass \(X\), in kg, of corn stored in each individual container is normally distributed with mean 20.1 and standard deviation 1.2.

1. Find the probability that the mean mass of corn in a random sample of 8 containers of corn will be greater than 20 kg. [3]

The mass \(Y\), in kg, of einkorn stored in each individual container is normally distributed with mean 22.2 and standard deviation 1.5. The farmer and his wife need to move two identical wheelbarrows, one of which is loaded with 3 containers of corn, and the other of which is loaded with 3 containers of einkorn. They agree that the farmer's wife will move the heavier wheelbarrow.

1. Calculate the probability that the farmer's wife will move
   1. the einkorn,
   2. the corn. [5]
2. The mass \(E\), in kg, of emmer stored in each individual container is normally distributed with mean 10.5 and standard deviation \(\sigma\). The farmer's son tries to calculate the probability that the mass of corn in a single container will be more than three times the mass of emmer in a single container. He obtains an answer of 0.35208.
   1. Find the value of \(\sigma\) that the farmer's son used.
   2. Explain why the value of \(\sigma\) that he used is unreasonable. [8]

Alun does the crossword in the Daily Bugle every day. The time that he takes to complete the crossword, \(X\) minutes, is modelled by the normal distribution \(\mathrm{N}(32, 4^2)\). You may assume that the times taken to complete the crossword on successive days are independent.

1. 1. Find the upper quartile of \(X\) and explain its meaning in context.
   2. Find the probability that the total time taken by Alun to complete the crosswords on five randomly chosen days is greater than 170 minutes. [7]
2. Belle also does the crossword every day and the time that she takes to complete the crossword, \(Y\) minutes, is modelled by the normal distribution \(\mathrm{N}(18, 2^2)\). Find the probability that, on a randomly chosen day, the time taken by Alun to complete the crossword is more than twice the time taken by Belle to complete the crossword. [6]

The mass \(J\) kg of a bag of randomly chosen Jersey potatoes is a normally distributed random variable with mean 1.00 and standard deviation 0.06. The mass \(K\) kg of a bag of randomly chosen King Edward potatoes is an independent normally distributed random variable with mean 0.80 and standard deviation 0.04.

1. Find the probability that the total mass of 6 bags of Jersey potatoes and 8 bags of King Edward potatoes is greater than 12.70 kg. [3]
2. Find the probability that the mass of one bag of King Edward potatoes is more than 75\% of the mass of one bag of Jersey potatoes. [3]

A machine is selecting independently and at random long rods and short rods. The length of the long rods, \(X\) cm, is normally distributed with mean 25 cm and variance 3 cm\(^2\) and the length of the short rods, \(Y\) cm, is normally distributed with mean 15 cm and variance 2 cm\(^2\). Assume that \(X\) and \(Y\) are independent random variables.

1. One long rod and one short rod are chosen at random. Find the probability that the difference in the lengths, \(X - Y\), is between 8 cm and 11 cm. [4]
2. Two long rods and two short rods are chosen at random and are assembled into an approximately rectangular frame. Find the probability that the perimeter of the resulting frame is more than 75 cm. [4]