5.04b Linear combinations: of normal distributions

4 The masses of a certain variety of potato are normally distributed with mean 180 g and variance \(1550 \mathrm {~g} ^ { 2 }\). Two potatoes of this variety are chosen at random. Find the probability that the mass of one of these potatoes is at least twice the mass of the other.

7 Kieran and Andreas are long-jumpers. They model the lengths, in metres, that they jump by the independent random variables \(K \sim \mathrm {~N} ( 5.64,0.0576 )\) and \(A \sim \mathrm {~N} ( 4.97,0.0441 )\) respectively. They each make a jump and measure the length. Find the probability that

1. the sum of the lengths of their jumps is less than 11 m ,
2. Kieran jumps more than 1.2 times as far as Andreas.

6 The lifetimes, in hours, of Longlive light bulbs and Enerlow light bulbs have the independent distributions \(\mathrm { N } \left( 1020,45 ^ { 2 } \right)\) and \(\mathrm { N } \left( 2800,52 ^ { 2 } \right)\) respectively.

1. Find the probability that the total of the lifetimes of 5 randomly chosen Longlive bulbs is less than 5200 hours.
2. Find the probability that the lifetime of a randomly chosen Enerlow bulb is at least 3 times that of a randomly chosen Longlive bulb.

1 The masses, in grams, of potatoes of types \(A\) and \(B\) have the distributions \(\mathrm { N } \left( 175,60 ^ { 2 } \right)\) and \(\mathrm { N } \left( 105,28 ^ { 2 } \right)\) respectively. Find the probability that a randomly chosen potato of type \(A\) has a mass that is at least twice the mass of a randomly chosen potato of type \(B\).

4 The masses, in grams, of tomatoes of type \(A\) and type \(B\) have the distributions \(\mathrm { N } \left( 125,30 ^ { 2 } \right)\) and \(\mathrm { N } \left( 130,32 ^ { 2 } \right)\) respectively.

1. Find the probability that the total mass of 4 randomly chosen tomatoes of type \(A\) and 6 randomly chosen tomatoes of type \(B\) is less than 1.5 kg .
2. Find the probability that a randomly chosen tomato of type \(A\) has a mass that is at least \(90 \%\) of the mass of a randomly chosen tomato of type \(B\).

6 The weights, in kilograms, of men and women have the distributions \(\mathrm { N } \left( 78,7 ^ { 2 } \right)\) and \(\mathrm { N } \left( 66,5 ^ { 2 } \right)\) respectively.

1. The maximum load that a certain cable car can carry safely is 1200 kg . If 9 randomly chosen men and 7 randomly chosen women enter the cable car, find the probability that the cable car can operate safely.
2. Find the probability that a randomly chosen woman weighs more than a randomly chosen man.

4 Each week a farmer sells \(X\) litres of milk and \(Y \mathrm {~kg}\) of cheese, where \(X\) and \(Y\) have the independent distributions \(\mathrm { N } \left( 1520,53 ^ { 2 } \right)\) and \(\mathrm { N } \left( 175,12 ^ { 2 } \right)\) respectively.

1. Find the mean and standard deviation of the total amount of milk that the farmer sells in 4 randomly chosen weeks. During a year when milk prices are low, the farmer makes a loss of 2 cents per litre on milk and makes a profit of 21 cents per kg on cheese, so the farmer's overall weekly profit is \(( 21 Y - 2 X )\) cents.
2. Find the probability that, in a randomly chosen week, the farmer's overall profit is positive.

3 A men's triathlon consists of three parts: swimming, cycling and running. Competitors' times, in minutes, for the three parts can be modelled by three independent normal variables with means 34.0, 87.1 and 56.9, and standard deviations 3.2, 4.1 and 3.8, respectively. For each competitor, the total of his three times is called the race time. Find the probability that the mean race time of a random sample of 15 competitors is less than 175 minutes.

1 Ellie is investigating the heights of two types of beech tree, \(A\) and \(B\), in a certain region. She has chosen a random sample of 60 beech trees of type \(A\) in the region, recorded their heights, \(x \mathrm {~m}\), and calculated unbiased estimates for the population mean and population variance as 35.6 m and \(4.95 \mathrm {~m} ^ { 2 }\) respectively. Ellie also chooses a random sample of 50 beech trees of type \(B\) in the region and records their heights, \(y \mathrm {~m}\). Her results are summarised as follows. $$\sum y = 1654 \quad \sum y ^ { 2 } = 54850$$ Find a \(95 \%\) confidence interval for the difference between the population mean heights of type \(A\) and type \(B\) beech trees in the region.

9

1. The random variable \(G\) has the distribution \(\mathrm { B } ( n , 0.75 )\). Find the set of values of \(n\) for which the distribution of \(G\) can be well approximated by a normal distribution.
2. The random variable \(H\) has the distribution \(\mathrm { B } ( n , p )\). It is given that, using a normal approximation, \(\mathrm { P } ( H \geqslant 71 ) = 0.0401\) and \(\mathrm { P } ( H \leqslant 46 ) = 0.0122\).
   1. Find the mean and standard deviation of the approximating normal distribution.
   2. Hence find the values of \(n\) and \(p\). 4

1 The marks obtained by a randomly chosen student in the two papers of an examination are denoted by the random variables \(X\) and \(Y\), where \(X \sim \mathrm {~N} ( 45,81 )\) and \(Y \sim \mathrm {~N} ( 33,63 )\). The student's overall mark for the examination, \(T\), is given by \(T = X + \lambda Y\), where the constant \(\lambda\) is chosen such that \(\mathrm { E } ( T ) = 100\).

1. Show that \(\lambda = \frac { 5 } { 3 }\).
2. Assuming that \(X\) and \(Y\) are independent, state the distribution of \(T\), giving the values of its parameters.
3. Comment on the assumption of independence.

4 Eezimix flour is sold in small bags of weight \(S\) grams, where \(S \sim \mathrm {~N} \left( 502.1,0.31 ^ { 2 } \right)\). It is also sold in large bags of weight \(L\) grams, where \(L \sim \mathrm {~N} \left( 1004.9,0.58 ^ { 2 } \right)\).

1. Find the probability that a randomly chosen large bag weighs at least 1 gram more than two randomly chosen small bags.
2. Find the probability that a randomly chosen large bag weighs less than twice the weight of a randomly chosen small bag.

3 Bill and Ben run their own gardening company. At regular intervals throughout the summer they come to work on my garden, mowing the lawns, hoeing the flower beds and pruning the bushes. From past experience it is known that the times, in minutes, spent on these tasks can be modelled by independent Normally distributed random variables as follows.

1. Find the probability that, on a randomly chosen visit, it takes less than 50 minutes to mow the lawns.
2. Find the probability that, on a randomly chosen visit, the total time for hoeing and pruning is less than 50 minutes.
3. If Bill mows the lawns while Ben does the hoeing and pruning, find the probability that, on a randomly chosen visit, Ben finishes first. Bill and Ben do my gardening twice a month and send me an invoice at the end of the month.
4. Write down the mean and variance of the total time (in minutes) they spend on mowing, hoeing and pruning per month.
5. The company charges for the total time spent at 15 pence per minute. There is also a fixed charge of \(\pounds 10\) per month. Find the probability that the total charge for a month does not exceed \(\pounds 40\).

2 A bus route runs from the centre of town A through the town's urban area to a point B on its boundary and then through the country to a small town C . Because of traffic congestion and general road conditions, delays occur on both the urban and the country sections. All delays may be considered independent. The scheduled time for the journey from A to B is 24 minutes. In fact, journey times over this section are given by the Normally distributed random variable \(X\) with mean 26 minutes and standard deviation 3 minutes. The scheduled time for the journey from B to C is 18 minutes. In fact, journey times over this section are given by the Normally distributed random variable \(Y\) with mean 15 minutes and standard deviation 2 minutes. Journey times on the two sections of route may be considered independent. The timetable published to the public does not show details of times at intermediate points; thus, if a bus is running early, it merely continues on its journey and is not required to wait.

1. Find the probability that a journey from A to B is completed in less than the scheduled time of 24 minutes.
2. Find the probability that a journey from A to C is completed in less than the scheduled time of 42 minutes.
3. It is proposed to introduce a system of bus lanes in the urban area. It is believed that this would mean that the journey time from A to B would be given by the random variable \(0.85 X\). Assuming this to be the case, find the probability that a journey from A to B would be completed in less than the currently scheduled time of 24 minutes.
4. An alternative proposal is to introduce an express service. This would leave out some bus stops on both sections of the route and its overall journey time from A to C would be given by the random variable \(0.9 X + 0.8 Y\). The scheduled time from A to C is to be given as a whole number of minutes. Find the least possible scheduled time such that, with probability 0.75 , buses would complete the journey on time or early.
5. A programme of minor road improvements is undertaken on the country section. After their completion, it is thought that the random variable giving the journey time from B to C is still Normally distributed with standard deviation 2 minutes. A random sample of 15 journeys is found to have a sample mean journey time from B to C of 13.4 minutes. Provide a two-sided \(95 \%\) confidence interval for the population mean journey time from B to C .

2 The operator of a section of motorway toll road records its weekly takings according to the types of vehicles using the motorway. For purposes of charging, there are three types of vehicle: cars, coaches, lorries. The weekly takings (in thousands of pounds) for each type are assumed to be Normally distributed. These distributions are independent of each other and are summarised in the table.

1. Find the probability that the weekly takings for coaches are less than \(\pounds 40000\).
2. Find the probability that the weekly takings for lorries exceed the weekly takings for cars.
3. Find the probability that over a 4 -week period the total takings for cars exceed \(\pounds 225000\). What assumption must be made about the four weeks?
4. Each week the operator allocates part of the takings for repairs. This is determined for each type of vehicle according to estimates of the long-term damage caused. It is calculated as follows: \(5 \%\) of takings for cars, \(10 \%\) for coaches and \(20 \%\) for lorries. Find the probability that in any given week the total amount allocated for repairs will exceed \(\pounds 20000\).

2 The random variable \(Z\) has the standard Normal distribution. The random variable \(Y\) is defined by \(Y = Z ^ { 2 }\).
You are given that \(Y\) has the following probability density function. $$\mathrm { f } ( y ) = \frac { 1 } { \sqrt { 2 \pi y } } \mathrm { e } ^ { - \frac { 1 } { 2 } y } , \quad y > 0$$

1. Show that the moment generating function (mgf) of \(Y\) is given by $$\mathrm { M } _ { Y } ( \theta ) = ( 1 - 2 \theta ) ^ { - \frac { 1 } { 2 } }$$
2. Use the mgf to obtain \(\mathrm { E } ( Y )\) and \(\operatorname { Var } ( Y )\). The random variable \(U\) is defined by $$U = Z _ { 1 } ^ { 2 } + Z _ { 2 } ^ { 2 } + \ldots + Z _ { n } ^ { 2 } ,$$ where \(Z _ { 1 } , Z _ { 2 } , \ldots , Z _ { n }\) are independent standard Normal random variables.
3. State an appropriate general theorem for mgfs and hence write down the mgf of \(U\). State the values of \(\mathrm { E } ( U )\) and \(\operatorname { Var } ( U )\). The random variable \(W\) is defined by $$W = \frac { U - n } { \sqrt { 2 n } }$$
4. Show that the logarithm of the \(\operatorname { mgf }\) of \(W\) is $$- \sqrt { \frac { n } { 2 } } \theta - \frac { n } { 2 } \ln \left( 1 - \sqrt { \frac { 2 } { n } } \theta \right) .$$ Use the series expansion of \(\ln ( 1 - t )\) to show that, as \(n \rightarrow \infty\), this expression tends to \(\frac { 1 } { 2 } \theta ^ { 2 }\).
   State what this implies about the distribution of \(W\) for large \(n\).

2 The random variable \(X\) has probability density function \(\mathrm { f } ( x )\) where $$\mathrm { f } ( x ) = \lambda \mathrm { e } ^ { - \lambda x } , \quad x > 0 .$$

1. Obtain the moment generating function (mgf) of \(X\).
2. Use the mgf to find \(\mathrm { E } ( X )\) and \(\operatorname { Var } ( X )\). The random variable \(Y\) is defined as follows: $$Y = X _ { 1 } + \ldots + X _ { n } ,$$ where the \(X _ { i }\) are independently and identically distributed as \(X\).
3. Write down expressions for \(\mathrm { E } ( Y )\) and \(\operatorname { Var } ( Y )\). Obtain the \(\operatorname { mgf }\) of \(Y\).
4. Find the \(\operatorname { mgf }\) of \(Z\) where \(Z = \frac { Y - \frac { n } { \lambda } } { \frac { \sqrt { n } } { \lambda } }\).
5. By considering the logarithm of the mgf of \(Z\), show that the distribution of \(Z\) tends to the standard Normal distribution as \(n\) tends to infinity.

4 Cola is sold in bottles and cans. The volume of cola in a bottle is normally distributed with mean 500 ml and standard deviation 10 ml . The volume of cola in a can is normally distributed with mean 330 ml and standard deviation 8 ml . Find the probability that the total volume of cola in 2 randomly selected bottles is greater than 3 times the volume of cola in a randomly selected can.

1 A laminate consists of 4 layers of material \(C\) and 3 layers of material \(D\). The thickness of a layer of material \(C\) has a normal distribution with mean 1 mm and standard deviation 0.1 mm , and the thickness of a layer of material \(D\) has a normal distribution with mean 8 mm and standard deviation 0.2 mm . The layers are independent of one another.

1. Find the mean and variance of the total thickness of the laminate.
2. What total thickness is exceeded by \(1 \%\) of the laminates?

7 Past experience shows that \(35 \%\) of the senior pupils in a large school know the regulations about bringing cars to school. The head teacher addresses this subject in an assembly, and afterwards a random sample of 120 senior pupils is selected. In this sample it is found that 50 of these pupils know the regulations. Use a suitable approximation to test, at the \(10 \%\) significance level, whether there is evidence that the proportion of senior pupils who know the regulations has increased. Justify your approximation.

2 Clover stems usually have three leaves. Occasionally a clover stem has four leaves. This is considered by some to be lucky and is known as a four-leaf clover. On average 1 in 10000 clover stems is a four-leaf clover. You may assume that four-leaf clovers occur randomly and independently. A random sample of 5000 clover stems is selected.

1. State the exact distribution of \(X\), the number of four-leaf clovers in the sample.
2. Explain why \(X\) may be approximated by a Poisson distribution. Write down the mean of this Poisson distribution.
3. Use this Poisson distribution to find the probability that the sample contains at least one four-leaf clover.
4. Find the probability that in 20 samples, each of 5000 clover stems, there are exactly 9 samples which contain at least one four-leaf clover.
5. Find the expected number of these 20 samples which contain at least one four-leaf clover. The table shows the numbers of four-leaf clovers in these 20 samples.

   Number of four-leaf clovers	0	1	2	\(> 2\)
   Number of samples	11	7	2	0

6. Calculate the mean and variance of the data in the table.
7. Briefly comment on whether your answers to parts (v) and (vi) support the use of the Poisson approximating distribution in part (iii).

2 On average 2\% of a particular model of laptop computer are faulty. Faults occur independently and randomly.

1. Find the probability that exactly 1 of a batch of 10 laptops is faulty.
2. State the conditions under which the use of a Poisson distribution is appropriate as an approximation to a binomial distribution.
3. A school buys a batch of 150 of these laptops. Use a Poisson approximating distribution to find the probability that
   (A) there are no faulty laptops in the batch,
   (B) there are more than the expected number of faulty laptops in the batch.
4. A large company buys a batch of 2000 of these laptops for its staff.
   (A) State the exact distribution of the number of faulty laptops in this batch.
   (B) Use a suitable approximating distribution to find the probability that there are at most 50 faulty laptops in this batch.

2 A student is investigating the numbers of sultanas in a particular brand of biscuit. The data in the table show the numbers of sultanas in a random sample of 50 of these biscuits.

1. Show that the sample mean is 1.84 and calculate the sample variance.
2. Explain why these results support a suggestion that a Poisson distribution may be a suitable model for the distribution of the numbers of sultanas in this brand of biscuit. For the remainder of the question you should assume that a Poisson distribution with mean 1.84 is a suitable model for the distribution of the numbers of sultanas in these biscuits.
3. Find the probability of
   (A) no sultanas in a biscuit,
   (B) at least two sultanas in a biscuit.
4. Show that the probability that there are at least 10 sultanas in total in a packet containing 5 biscuits is 0.4389 .
5. Six packets each containing 5 biscuits are selected at random. Find the probability that exactly 2 of the six packets contain at least 10 sultanas.
6. Sixty packets each containing 5 biscuits are selected at random. Use a suitable approximating distribution to find the probability that more than half of the sixty packets contain at least 10 sultanas.

2 The number of printing errors per page in a book is modelled by a Poisson distribution with a mean of 0.85 .

1. State conditions for a Poisson distribution to be a suitable model for the number of printing errors per page.
2. A page is chosen at random. Find the probability of
   (A) exactly 1 error on this page,
   (B) at least 2 errors on this page. 10 pages are chosen at random.
3. Find the probability of exactly 10 errors in these 10 pages.
4. Find the least integer \(k\) such that the probability of there being \(k\) or more errors in these 10 pages is less than \(1 \%\). 30 pages are chosen at random.
5. Use a suitable approximating distribution to find the probability of no more than 30 errors in these 30 pages.

3 The lifetime of a particular type of light bulb is \(X\) hours, where \(X\) is Normally distributed with mean 1100 and variance 2000.

1. Find \(\mathrm { P } ( 1100 < X < 1200 )\).
2. Use a suitable approximating distribution to find the probability that, in a random sample of 100 of these light bulbs, no more than 40 have a lifetime between 1100 and 1200 hours.
3. A factory has a large number of these light bulbs installed. As soon as \(1 \%\) of the bulbs have come to the end of their lifetimes, it is company policy to replace all of the bulbs. After how many hours should the bulbs need to be replaced?
4. The bulbs are to be replaced by low-energy bulbs. The lifetime of these bulbs is Normally distributed and the mean is claimed by the manufacturer to be 7000 hours. The standard deviation is known to be 100 hours. A random sample of 25 low-energy bulbs is selected. Their mean lifetime is found to be 6972 hours. Carry out a 2 -tail test at the \(10 \%\) level to investigate the claim.
   [0pt] [Question 4 is printed overleaf.]