5.04b Linear combinations: of normal distributions

2 John is observing butterflies being blown across a fence in a strong wind. He uses the Poisson distribution with mean 2.1 to model the number of butterflies he observes in one minute.

1. Find the probability that John observes
   (A) no butterflies in a minute,
   (B) at least 2 butterflies in a minute,
   (C) between 5 and 10 butterflies inclusive in a period of 5 minutes.
2. Use a suitable approximating distribution to find the probability that John observes at least 130 butterflies in a period of 1 hour. In fact some of the butterflies John observes being blown across the fence are being blown in pairs.
3. Explain why this invalidates one of the assumptions required for a Poisson distribution to be a suitable model. John decides to revise his model for the number of butterflies he observes in one minute. In this new model, the number of pairs of butterflies is modelled by the Poisson distribution with mean 0.2 , and the number of single butterflies is modelled by an independent Poisson distribution with mean 1.7.
4. Find the probability that John observes no more than 3 butterflies altogether in a period of one minute.

2 A particular genetic mutation occurs in one in every 300 births on average. A random sample of 1200 births is selected.

1. State the exact distribution of \(X\), the number of births in the sample which have the mutation.
2. Explain why \(X\) has, approximately, a Poisson distribution.
3. Use a Poisson approximating distribution to find
   (A) \(\mathrm { P } ( X = 1 )\),
   (B) \(\mathrm { P } ( X > 4 )\).
4. Twenty independent samples, each of 1200 births, are selected. State the mean and variance of a Normal approximating distribution suitable for modelling the total number of births with the mutation in the twenty samples.
5. Use this Normal approximating distribution to
   (A) find the probability that there are at least 90 births which have the mutation,
   ( \(B\) ) find the least value of \(k\) such that the probability that there are at most \(k\) births with this mutation is greater than 5\%.

2 It was stated in 2012 that \(3 \%\) of \(\pounds 1\) coins were fakes. Throughout this question, you should assume that this is still the case.

1. Find the probability that, in a random selection of \(25 \pounds 1\) coins, there is exactly one fake coin. A random sample of \(250 \pounds 1\) coins is selected.
2. Explain why a Poisson distribution is an appropriate approximating distribution for the number of fake coins in the sample.
3. Use a Poisson distribution to find the probability that, in this sample, there are
   (A) exactly 10 fake coins,
   (B) at least 10 fake coins.
4. Use a suitable approximating distribution to find the probability that there are at least 50 fake coins in a sample of 2000 coins. It is known that \(0.2 \%\) of another type of coin are fakes.
5. A random sample of size \(n\) of these coins is taken. Using a Poisson approximating distribution, show that the probability of at most one fake coin in the sample is equal to \(\mathrm { e } ^ { - \lambda } + \lambda \mathrm { e } ^ { - \lambda }\), where \(\lambda = 0.002 n\).
6. Use the approximation \(\mathrm { e } ^ { - \lambda } + \lambda \mathrm { e } ^ { - \lambda } \approx 1 - \frac { \lambda ^ { 2 } } { 2 }\) for small values of \(\lambda\) to estimate the value of \(n\) for which the probability in part ( \(\mathbf { v }\) ) is equal to 0.995 .

3 The random variable \(X\) represents the weight in kg of a randomly selected male dog of a particular breed. \(X\) is Normally distributed with mean 30.7 and standard deviation 3.5.

1. Find
   (A) \(\mathrm { P } ( X < 30 )\),
   (B) \(P ( 25 < X < 35 )\).
2. Five of these dogs are chosen at random. Find the probability that each of them weighs at least 30 kg .
3. The weights of females of the same breed of dog are Normally distributed with mean 26.8 kg . Given that \(5 \%\) of female dogs of this breed weigh more than 30 kg , find the standard deviation of their weights.
4. Sketch the distributions of the weights of male and female dogs of this breed on a single diagram.

2 When a genetic sequence of plant DNA is given a dose of radiation, some of the genes may mutate. The probability that a gene mutates is 0.012 . Mutations occur randomly and independently.

1. Explain the meanings of the terms 'randomly' and 'independently' in this context. A short stretch of DNA containing 20 genes is given a dose of radiation.
2. Find the probability that exactly 1 out of the 20 genes mutates. A longer stretch of DNA containing 500 genes is given a dose of radiation.
3. Explain why a Poisson distribution is an appropriate approximating distribution for the number of genes that mutate.
4. Use this Poisson distribution to find the probability that there are
   (A) exactly two genes that mutate,
   (B) at least two genes that mutate. A third stretch of DNA containing 50000 genes is given a dose of radiation.
5. Use a suitable approximating distribution to find the probability that there are at least 650 genes that mutate.

6 A mathematics examination is taken by 29 boys and 26 girls. Experience has shown that the probability that any boy forgets to bring a calculator to the examination is 0.3 , and that any girl forgets is 0.2 . Whether or not any student forgets to bring a calculator is independent of all other students. The numbers of boys and girls who forget to bring a calculator are denoted by \(B\) and \(G\) respectively, and \(F = B + G\).

1. Find \(\mathrm { E } ( F )\) and \(\operatorname { Var } ( F )\).
2. Using suitable approximations to the distributions of \(B\) and \(G\), which should be justified, find the smallest number of spare calculators that should be available in order to be at least \(99 \%\) certain that all 55 students will have a calculator.

2 The amount of tomato juice, \(X \mathrm { ml }\), dispensed into cartons of a particular brand has a normal distribution with mean 504 and standard deviation 3 . The juice is sold in packs of 4 cartons, filled independently. The total amount of juice in one pack is \(Y \mathrm { ml }\).

1. Find \(\mathrm { P } ( Y < 2000 )\). The random variable \(V\) is defined as \(Y - 4 X\).
2. Find \(\mathrm { E } ( V )\) and \(\operatorname { Var } ( V )\).
3. What is the probability that the amount of juice in a randomly chosen pack is more than 4 times the amount of juice in a randomly chosen carton?

1 The independent random variables \(X\) and \(Y\) have the distributions \(\mathrm { N } \left( 10 , \sigma ^ { 2 } \right)\) and \(\operatorname { Po } ( 2 )\) respectively. The random variable \(S\) is given by \(S = 5 X - 2 Y + c\), where \(c\) is a constant.
It is given that \(\mathrm { E } ( S ) = \operatorname { Var } ( S ) = 408\).

1. Find the value of \(c\) and show that \(\sigma ^ { 2 } = 16\).
2. Find \(\mathrm { P } ( X \geqslant \mathrm { E } ( Y ) )\).

7 The random variable \(X\) has distribution \(\mathrm { N } ( \mu , 1 )\). A random sample of 4 observations of \(X\) is taken. The sample mean is denoted by \(\bar { X }\).

1. Find the value of the constant \(a\) for which ( \(\bar { X } - a , \bar { X } + a\) ) is a \(98 \%\) confidence interval for \(\mu\). The independent random variable \(Y\) has distribution \(\mathrm { N } ( \mu , 9 )\). A random sample of 16 observations of \(Y\) is taken. The sample mean is denoted by \(\bar { Y }\).
2. Write down the distribution of \(\bar { X } - \bar { Y }\).
3. A \(90 \%\) confidence interval for \(\mu\) based on \(\bar { Y }\) is given by ( \(\bar { Y } - 1.234 , \bar { Y } + 1.234\) ). Find the probability that this interval does not overlap with the interval in part (i).

7 The employees of a certain company have masses which are normally distributed. Female employees have a mean of 66.7 kg and standard deviation 9.3 kg , and male employees have a mean of 78.3 kg and standard deviation 8.5 kg . It may be assumed that all employees' masses are independent. On the ground floor 6 women and 9 men enter the empty staff lift for which it is stated that the maximum load is 1150 kg .

1. Calculate the probability that the maximum load is exceeded. At the first floor all 15 passengers leave and 6 women, 8 men and an unknown employee enter.
2. Assuming that the unknown employee is equally likely to be a woman or a man, calculate the probability that the maximum load is now exceeded.

5 The discrete random variables \(X\) and \(Y\) are independent with \(X \sim \mathrm {~B} \left( 32 , \frac { 1 } { 2 } \right)\) and \(Y \sim \operatorname { Po } ( 28 )\).

1. Find the values of \(\mathrm { E } ( Y - X )\) and \(\operatorname { Var } ( Y - X )\).
2. State, with justification, an approximate distribution for \(Y - X\).
3. Hence find \(\mathrm { P } ( | Y - X | \geqslant 3 )\).

1 The blood-test procedure at a clinic is that a person arrives, takes a numbered ticket and waits for that number to be called. The waiting times between the numbers called have independent normal distributions with mean 3.5 minutes and standard deviation 0.9 minutes. My ticket is number 39 and as I take my ticket number 1 is being called, so that I have to wait for 38 numbers to be called. Find the probability that I will have to wait between 120 minutes and 140 minutes.

5 The independent random variables \(X\) and \(Y\) have distributions \(\mathrm { N } \left( 30 , \sigma ^ { 2 } \right)\) and \(\mathrm { N } \left( 20 , \sigma ^ { 2 } \right)\) respectively. The random variable \(a X + b Y\), where \(a\) and \(b\) are constants, has the distribution \(\mathrm { N } \left( 410,130 \sigma ^ { 2 } \right)\).

1. Given that \(a\) and \(b\) are integers, find the value of \(a\) and the value of \(b\).
2. Given that \(\mathrm { P } ( X > Y ) = 0.966\), find \(\sigma ^ { 2 }\).

2 A factory manufactures paperweights consisting of glass mounted on a wooden base. The volume of glass, in \(\mathrm { cm } ^ { 3 }\), in a paperweight has a Normal distribution with mean 56.5 and standard deviation 2.9. The volume of wood, in \(\mathrm { cm } ^ { 3 }\), also has a Normal distribution with mean 38.4 and standard deviation 1.1. These volumes are independent of each other. For the purpose of quality control, paperweights for testing are chosen at random from the factory's output.

1. Find the probability that the volume of glass in a randomly chosen paperweight is less than \(60 \mathrm {~cm} ^ { 3 }\).
2. Find the probability that the total volume of a randomly chosen paperweight is more than \(100 \mathrm {~cm} ^ { 3 }\). The glass has a mass of 3.1 grams per \(\mathrm { cm } ^ { 3 }\) and the wood has a mass of 0.8 grams per \(\mathrm { cm } ^ { 3 }\).
3. Find the probability that the total mass of a randomly chosen paperweight is between 200 and 220 grams.
4. The factory manager introduces some modifications intended to reduce the mean mass of the paperweights to 200 grams or less. The variance is also affected but not the Normality. Subsequently, for a random sample of 10 paperweights, the sample mean mass is 205.6 grams and the sample standard deviation is 8.51 grams. Is there evidence, at the \(5 \%\) level of significance, that the intended reduction of the mean mass has not been achieved?

4 The weights of a particular variety (A) of tomato are known to be Normally distributed with mean 80 grams and standard deviation 11 grams.

1. Find the probability that a randomly chosen tomato of variety A weighs less than 90 grams. The weights of another variety (B) of tomato are known to be Normally distributed with mean 70 grams. These tomatoes are packed in sixes using packaging that weighs 15 grams.
2. The probability that a randomly chosen pack of 6 tomatoes of variety B , including packaging, weighs less than 450 grams is 0.8463 . Show that the standard deviation of the weight of single tomatoes of variety B is 6 grams, to the nearest gram.
3. Tomatoes of variety A are packed in fives using packaging that weighs 25 grams. Find the probability that the total weight of a randomly chosen pack of variety A is greater than the total weight of a randomly chosen pack of variety B .
4. A new variety (C) of tomato is introduced. The weights, \(c\) grams, of a random sample of 60 of these tomatoes are measured giving the following results. $$\Sigma c = 3126.0 \quad \Sigma c ^ { 2 } = 164223.96$$ Find a \(95 \%\) confidence interval for the true mean weight of these tomatoes.

1 Each month the amount of electricity, measured in kilowatt-hours ( kWh ), used by a particular household is Normally distributed with mean 406 and standard deviation 12.

1. Find the probability that, in a randomly chosen month, less than 420 kWh is used. The charge for electricity used is 14.6 pence per kWh .
2. Write down the distribution of the total charge for the amount of electricity used in any one month. Hence find the probability that, in a randomly chosen month, the total charge is more than \(\pounds 60\).
3. The household receives a bill every three months. Assume that successive months may be regarded as independent of each other. Find the value of \(b\) such that the probability that a randomly chosen bill is less than \(\pounds b\) is 0.99 . In a different household, the amount of electricity used per month was Normally distributed with mean 432 kWh . This household buys a new washing machine that is claimed to be cheaper to run than the old one. Over the next six months the amounts of electricity used, in kWh , are as follows. $$\begin{array} { l l l l l l } 404 & 433 & 420 & 423 & 413 & 440 \end{array}$$
4. Treating this as a random sample, carry out an appropriate test, with a \(5 \%\) significance level, to see if there is any evidence to suggest that the amount of electricity used per month by this household has decreased on average.

2 In a particular chain of supermarkets, one brand of pasta shapes is sold in small packets and large packets. Small packets have a mean weight of 505 g and a standard deviation of 11 g . Large packets have a mean weight of 1005 g and a standard deviation of 17 g . It is assumed that the weights of packets are Normally distributed and are independent of each other.

1. Find the probability that a randomly chosen large packet weighs between 995 g and 1020 g .
2. Find the probability that the weights of two randomly chosen small packets differ by less than 25 g .
3. Find the probability that the total weight of two randomly chosen small packets exceeds the weight of a randomly chosen large packet.
4. Find the probability that the weight of one randomly chosen small packet exceeds half the weight of a randomly chosen large packet by at least 5 g .
5. A different brand of pasta shapes is sold in packets of which the weights are assumed to be Normally distributed with standard deviation 14 g . A random sample of 20 packets of this pasta is found to have a mean weight of 246 g . Find a \(95 \%\) confidence interval for the population mean weight of these packets.

3 In the manufacture of child car seats, a resin made up of three ingredients is used. The ingredients are two polymers and an impact modifier. The resin is prepared in batches. Each ingredient is supplied by a separate feeder and the amount supplied to each batch, in kg, is assumed to be Normally distributed with mean and standard deviation as shown in the table below. The three feeders are also assumed to operate independently of each other.

1. Find the probability that, in a randomly chosen batch of resin, there is no more than 2100 kg of polymer 1.
2. Find the probability that, in a randomly chosen batch of resin, the amount of polymer 1 exceeds the amount of polymer 2 by at least 400 kg .
3. Find the value of \(b\) such that the total amount of the ingredients in a randomly chosen batch exceeds \(b \mathrm {~kg} 95 \%\) of the time.
4. Polymer 1 costs \(\pounds 1.20\) per kg, polymer 2 costs \(\pounds 1.30\) per kg and the impact modifier costs \(\pounds 0.80\) per kg. Find the mean and variance of the total cost of a batch of resin.
5. Each batch of resin is used to make a large number of car seats from which a random sample of 50 seats is selected in order that the tensile strength (in suitable units) of the resin can be measured. From one such sample, the \(99 \%\) confidence interval for the true mean tensile strength of the resin in that batch was calculated as \(( 123.72,127.38 )\). Find the mean and standard deviation of the sample.

1 Andy, a carpenter, constructs wooden shelf units for storing CDs. The wood used for the shelves has a thickness which is Normally distributed with mean 14 mm and standard deviation 0.55 mm . Andy works to a design which allows a gap of 145 mm between the shelves, but past experience has shown that the gap is Normally distributed with mean 144 mm and standard deviation 0.9 mm . Dimensions of shelves and gaps are assumed to be independent of each other.

1. Find the probability that a randomly chosen gap is less than 145 mm .
2. Find the probability that the combined height of a gap and a shelf is more than 160 mm . A complete unit has 7 shelves and 6 gaps.
3. Find the probability that the overall height of a unit lies between 960 mm and 965 mm . Hence find the probability that at least 3 out of 4 randomly chosen units are between 960 mm and 965 mm high.
4. I buy two randomly chosen CD units made by Andy. The probability that the difference in their heights is less than \(h \mathrm {~mm}\) is 0.95 . Find \(h\).

4 The weights of Avonley Blue cheeses made by a small producer are found to be Normally distributed with mean 10 kg and standard deviation 0.4 kg .

1. Find the probability that a randomly chosen cheese weighs less than 9.5 kg . One particular shop orders four Avonley Blue cheeses each week from the producer. From experience, the shopkeeper knows that the weekly demand from customers for Avonley Blue cheese is Normally distributed with mean 35 kg and standard deviation 3.5 kg . In the interests of food hygiene, no cheese is kept by the shopkeeper from one week to the next.
2. Find the probability that, in a randomly chosen week, demand from customers for Avonley Blue will exceed the supply. Following a campaign to promote Avonley Blue cheese, the shopkeeper finds that the weekly demand for it has increased by \(30 \%\) (i.e. the mean and standard deviation are both increased by \(30 \%\) ). Therefore the shopkeeper increases his weekly order by one cheese.
3. Find the probability that, in a randomly chosen week, demand will now exceed supply.
4. Following complaints, the cheese producer decides to check the mean weight of the Avonley Blue cheeses. For a random sample of 12 cheeses, she finds that the mean weight is 9.73 kg . Assuming that the population standard deviation of the weights is still 0.4 kg , find a \(95 \%\) confidence interval for the true mean weight of the cheeses and comment on the result. Explain what is meant by a 95\% confidence interval. RECOGNISING ACHIEVEMENT

3 The triathlon is a sports event in which competitors take part in three stages, swimming, cycling and running, one straight after the other. The winner is the competitor with the shortest overall time. In this question the times for the separate stages are assumed to be Normally distributed and independent of each other. For a particular triathlon event in which there was a very large number of competitors, the mean and standard deviation of the times, measured in minutes, for each stage were as follows.

1. For a randomly chosen competitor, find the probability that the swimming time is between 10 and 13 minutes.
2. For a randomly chosen competitor, find the probability that the running time exceeds the swimming time by more than 10 minutes.
3. For a randomly chosen competitor, find the probability that the swimming and running times combined exceed \(\frac { 2 } { 3 }\) of the cycling time.
4. In a different triathlon event the total times, in minutes, for a random sample of 12 competitors were as follows. $$\begin{array} { l l l l l l l l l l l l } 103.59 & 99.04 & 85.03 & 81.34 & 106.79 & 89.14 & 98.55 & 98.22 & 108.87 & 116.29 & 102.51 & 92.44 \end{array}$$ Find a 95\% confidence interval for the mean time of all competitors in this event.
5. Discuss briefly whether the assumptions of Normality and independence for the stages of triathlon events are reasonable.

4 A company that makes meat pies includes a "small" size in its product range. These pies consist of a pastry case and meat filling, the weights of which are independent of each other. The weight of the pastry case, \(C\), is Normally distributed with mean 96 g and variance \(21 \mathrm {~g} ^ { 2 }\). The weight of the meat filling, \(M\), is Normally distributed with mean 57 g and variance \(14 \mathrm {~g} ^ { 2 }\).

1. Find the probability that, in a randomly chosen pie, the weight of the pastry case is between 90 and 100 g .
2. The wrappers on the pies state that the weight is 145 g . Find the proportion of pies that are underweight.
3. The pies are sold in packs of 4 . Find the value of \(w\) such that, in \(95 \%\) of packs, the total weight of the 4 pies in a randomly chosen pack exceeds \(w \mathrm {~g}\).
4. It is required that the weight of the meat filling in a pie should be at least \(35 \%\) of the total weight. Show that this means that \(0.65 M - 0.35 C \geqslant 0\). Hence find the probability that, in a randomly chosen pie, this requirement is met.

1

1. Let \(X\) be a random variable with variance \(\sigma ^ { 2 }\). The independent random variables \(X _ { 1 }\) and \(X _ { 2 }\) are both distributed as \(X\). Write down the variances of \(X _ { 1 } + X _ { 2 }\) and \(2 X\); explain why they are different. A large company has produced an aptitude test which consists of three parts. The parts are called mathematical ability, spatial awareness and communication. The scores obtained by candidates in the three parts are continuous random variables \(X , Y\) and \(W\) which have been found to have independent Normal distributions with means and standard deviations as shown in the table.

   	Mean	Standard deviation
   Mathematical ability, \(X\)	30.1	5.1
   Spatial awareness, \(Y\)	25.4	4.2
   Communication, \(W\)	28.2	3.9

2. Find the probability that a randomly selected candidate obtains a score of less than 22 in the mathematical ability part of the test.
3. Find the probability that a randomly selected candidate obtains a total score of at least 100 in the whole test.
4. For a particular role in the company, the score \(2 X + Y\) is calculated. Find the score that is exceeded by only \(2 \%\) of candidates.
5. For a different role, a candidate must achieve a score in communication which is at least \(60 \%\) of the score obtained in mathematical ability. What proportion of candidates do not achieve this?

1 A game consists of 20 rounds. Each round is denoted as either a starter, middle or final round. The times taken for each round are independently and Normally distributed with the following parameters (given in seconds). The game consists of 4 starter, 12 middle and 4 final rounds. Find the probability that

1. the mean time per round for the 4 final rounds will exceed 260 seconds,
2. all 20 rounds will be completed in a total time of 75 minutes or less,
3. the 12 middle rounds will take at least 3.5 times as long in total as the 4 starter rounds,
4. the mean time per round for the 12 middle rounds will be at least 25 seconds less than the mean time per round for the 4 final rounds.

4 The moment generating function of a continuous random variable \(Y\), which has a \(\chi ^ { 2 }\) distribution with \(n\) degrees of freedom, is \(( 1 - 2 t ) ^ { - \frac { 1 } { 2 } n }\), where \(0 \leqslant t < \frac { 1 } { 2 }\).

1. Find \(\mathrm { E } ( Y )\) and \(\operatorname { Var } ( Y )\). For the case \(n = 1\), the sum of 60 independent observations of \(Y\) is denoted by \(S\).
2. Write down the moment generating function of \(S\) and hence identify the distribution of \(S\).
3. Use a normal approximation to estimate \(\mathrm { P } ( S \geqslant 70 )\).