5.04b Linear combinations: of normal distributions

3 The masses, in kilograms, of female and male animals of a certain species have the distributions \(\mathrm { N } \left( 102,27 ^ { 2 } \right)\) and \(\mathrm { N } \left( 170,55 ^ { 2 } \right)\) respectively. Find the probability that a randomly chosen female has a mass that is less than half the mass of a randomly chosen male. \includegraphics[max width=\textwidth, alt={}, center]{65b50bfb-5fd8-4cf3-ae3b-cffc12e23cd8-06_76_1659_484_244}

7 Before a certain type of book is published it is checked for errors, which are then corrected. For costing purposes each error is classified as either minor or major. The numbers of minor and major errors in a book are modelled by the independent distributions \(\mathrm { N } ( 380,140 )\) and \(\mathrm { N } ( 210,80 )\) respectively. You should assume that no continuity corrections are needed when using these models. A book of this type is chosen at random.

1. Find the probability that the number of minor errors is at least 200 more than the number of major errors.
   The costs of correcting a minor error and a major error are 20 cents and 50 cents respectively.
2. Find the probability that the total cost of correcting the errors in the book is less than \(\\) 190$.
   If you use the following lined page to complete the answer(s) to any question(s), the question number(s) must be clearly shown.

3 The masses, in kilograms, of female and male animals of a certain species have the distributions \(\mathrm { N } \left( 102,27 ^ { 2 } \right)\) and \(\mathrm { N } \left( 170,55 ^ { 2 } \right)\) respectively. Find the probability that a randomly chosen female has a mass that is less than half the mass of a randomly chosen male. \includegraphics[max width=\textwidth, alt={}, center]{6346fd4b-7bc9-4205-94db-67368b9415fe-06_76_1659_484_244}

6 The numbers of barrels of oil, in millions, extracted per day in two oil fields \(A\) and \(B\) are modelled by the independent random variables \(X\) and \(Y\) respectively, where \(X \sim \mathrm {~N} \left( 3.2,0.4 ^ { 2 } \right)\) and \(Y \sim \mathrm {~N} \left( 4.3,0.6 ^ { 2 } \right)\). The income generated by the oil from the two fields is \(\\) 90\( per barrel for \)A\( and \)\\( 95\) per barrel for \(B\).

1. Find the mean and variance of the daily income, in millions of dollars, generated by field \(A\). [3]
2. Find the probability that the total income produced by the two fields in a day is at least \(\\) 670$ million.

5 The marks in paper 1 and paper 2 of an examination are denoted by \(X\) and \(Y\) respectively, where \(X\) and \(Y\) have the independent continuous distributions \(\mathrm { N } \left( 56,6 ^ { 2 } \right)\) and \(\mathrm { N } \left( 43,5 ^ { 2 } \right)\) respectively.

1. Find the probability that a randomly chosen paper 1 mark is more than a randomly chosen paper 2 mark.
2. Each candidate's overall mark is \(M\) where \(M = X + 1.5 Y\). The minimum overall mark for grade A is 135 . Find the proportion of students who gain a grade A .

5 The times, in months, taken by a builder to build two types of house, \(P\) and \(Q\), are represented by the independent variables \(T _ { 1 } \sim \mathrm {~N} \left( 2.2,0.4 ^ { 2 } \right)\) and \(T _ { 2 } \sim \mathrm {~N} \left( 2.8,0.5 ^ { 2 } \right)\) respectively.

1. Find the probability that the total time taken to build one house of each type is less than 6 months.
2. Find the probability that the time taken to build a type \(Q\) house is more than 1.2 times the time taken to build a type \(P\) house.

3 Sugar and flour for making cakes are measured in cups. The mass, in grams, of one cup of sugar has the distribution \(\mathrm { N } ( 250,10 )\). The mass, in grams, of one cup of flour has the independent distribution \(\mathrm { N } ( 160,9 )\). Each cake contains 2 cups of sugar and 5 cups of flour. Find the probability that the total mass of sugar and flour in one cake exceeds 1310 grams.

5 The masses, in grams, of large boxes of chocolates and small boxes of chocolates have the distributions \(\mathrm { N } ( 325,6.1 )\) and \(\mathrm { N } ( 167,5.6 )\) respectively.

1. Find the probability that the total mass of 10 randomly chosen large boxes of chocolates is less than 3240 g .
2. Find the probability that the mass of a randomly chosen large box of chocolates is more than twice the mass of a randomly chosen small box of chocolates.

3 The masses, in kilograms, of female and male animals of a certain species have the distributions \(\mathrm { N } \left( 102,27 ^ { 2 } \right)\) and \(\mathrm { N } \left( 170,55 ^ { 2 } \right)\) respectively. Find the probability that a randomly chosen female has a mass that is less than half the mass of a randomly chosen male. \includegraphics[max width=\textwidth, alt={}, center]{fb305858-2d96-4a5d-b1a9-a965c248fb8d-06_76_1659_484_244}

3 The masses, in kilograms, of female and male animals of a certain species have the distributions \(\mathrm { N } \left( 102,27 ^ { 2 } \right)\) and \(\mathrm { N } \left( 170,55 ^ { 2 } \right)\) respectively. Find the probability that a randomly chosen female has a mass that is less than half the mass of a randomly chosen male. \includegraphics[max width=\textwidth, alt={}, center]{937c15d2-fb12-4af8-96d3-c54c81d771ba-06_76_1659_484_244}

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]

2 The independent random variables \(X\) and \(Y\) have the distributions \(\mathrm { N } ( 9.2,12.1 )\) and \(\mathrm { N } ( 3.0,8.6 )\) respectively. Find \(\mathrm { P } ( X > 3 Y )\).

3 Tien throws a ball. The distance it travels can be modelled by a normal distribution with mean 20 m and variance \(9 \mathrm {~m} ^ { 2 }\). His younger sister Su Chen also throws a ball and the distance her ball travels can be modelled by a normal distribution with mean 14 m and variance \(12 \mathrm {~m} ^ { 2 }\). Su Chen is allowed to add 5 metres on to her distance and call it her 'upgraded distance'. Find the probability that Tien's distance is larger than Su Chen's upgraded distance.

4 The weights of men follow a normal distribution with mean 71 kg and standard deviation 7 kg . The weights of women follow a normal distribution with mean 57 kg and standard deviation 5 kg . The total weight of 5 men and 2 women chosen randomly is denoted by \(X \mathrm {~kg}\).

1. Show that \(\mathrm { E } ( X ) = 469\) and \(\operatorname { Var } ( X ) = 295\).
2. The total weight of 4 men and 3 women chosen randomly is denoted by \(Y \mathrm {~kg}\). Find the mean and standard deviation of \(X - Y\) and hence find \(\mathrm { P } ( X - Y > 22 )\).

7 A journey in a certain car consists of two stages with a stop for filling up with fuel after the first stage. The length of time, \(T\) minutes, taken for each stage has a normal distribution with mean 74 and standard deviation 7.3. The length of time, \(F\) minutes, it takes to fill up with fuel has a normal distribution with mean 5 and standard deviation 1.7. The length of time it takes to pay for the fuel is exactly 4 minutes. The variables \(T\) and \(F\) are independent and the times for the two stages are independent of each other.

1. Find the probability that the total time for the journey is less than 154 minutes.
2. A second car has a fuel tank with exactly twice the capacity of the first car. Find the mean and variance of this car's fuel fill-up time.
3. This second car's time for each stage of the journey follows a normal distribution with mean 69 minutes and standard deviation 5.2 minutes. The length of time it takes to pay for the fuel for this car is also exactly 4 minutes. Find the probability that the total time for the journey taken by the first car is more than the total time taken by the second car.

4 In summer, wasps' nests occur randomly in the south of England at an average rate of 3 nests for every 500 houses.

1. Find the probability that two villages in the south of England, with 600 houses and 700 houses, have a total of exactly 3 wasps' nests.
2. Use a suitable approximation to estimate the probability of there being fewer than 369 wasps' nests in a town with 64000 houses.

5 Climbing ropes produced by a manufacturer have breaking strengths which are normally distributed with mean 160 kg and standard deviation 11.3 kg . A group of climbers have weights which are normally distributed with mean 66.3 kg and standard deviation 7.1 kg .

1. Find the probability that a rope chosen randomly will break under the combined weight of 2 climbers chosen randomly. Each climber carries, in a rucksack, equipment amounting to half his own weight.
2. Find the mean and variance of the combined weight of a climber and his rucksack.
3. Find the probability that the combined weight of a climber and his rucksack is greater than 87 kg .

3 Weights of garden tables are normally distributed with mean 36 kg and standard deviation 1.6 kg . Weights of garden chairs are normally distributed with mean 7.3 kg and standard deviation 0.4 kg . Find the probability that the total weight of 2 randomly chosen tables is more than the total weight of 10 randomly chosen chairs.

7 The volume of liquid in cans of cola is normally distributed with mean 330 millilitres and standard deviation 5.2 millilitres. The volume of liquid in bottles of tonic water is normally distributed with mean 500 millilitres and standard deviation 7.1 millilitres.

1. Find the probability that 3 randomly chosen cans of cola contain less liquid than 2 randomly chosen bottles of tonic water.
2. A new drink is made by mixing the contents of 2 cans of cola with half a bottle of tonic water. Find the probability that the volume of the new drink is more than 900 millilitres.

7

1. Random variables \(Y\) and \(X\) are related by \(Y = a + b X\), where \(a\) and \(b\) are constants and \(b > 0\). The standard deviation of \(Y\) is twice the standard deviation of \(X\). The mean of \(Y\) is 7.92 and is 0.8 more than the mean of \(X\). Find the values of \(a\) and \(b\).
2. Random variables \(R\) and \(S\) are such that \(R \sim \mathrm {~N} \left( \mu , 2 ^ { 2 } \right)\) and \(S \sim \mathrm {~N} \left( 2 \mu , 3 ^ { 2 } \right)\). It is given that \(\mathrm { P } ( R + S > 1 ) = 0.9\).
   1. Find \(\mu\).
   2. Hence find \(\mathrm { P } ( S > R )\).

5 The marks of candidates in Mathematics and English in 2009 were represented by the independent random variables \(X\) and \(Y\) with distributions \(\mathrm { N } \left( 28,5.6 ^ { 2 } \right)\) and \(\mathrm { N } \left( 52,12.4 ^ { 2 } \right)\) respectively. Each candidate's marks were combined to give a final mark \(F\), where \(F = X + \frac { 1 } { 2 } Y\).

1. Find \(\mathrm { E } ( F )\) and \(\operatorname { Var } ( F )\).
2. The final marks of a random sample of 10 candidates from Grinford in 2009 had a mean of 49. Test at the 5\% significance level whether this result suggests that the mean final mark of all candidates from Grinford in 2009 was lower than elsewhere.

4 The masses, in milligrams, of three minerals found in 1 tonne of a certain kind of rock are modelled by three independent random variables \(P , Q\) and \(R\), where \(P \sim \mathrm {~N} \left( 46,19 ^ { 2 } \right) , Q \sim \mathrm {~N} \left( 53,23 ^ { 2 } \right)\) and \(R \sim \mathrm {~N} \left( 25,10 ^ { 2 } \right)\). The total value of the minerals found in 1 tonne of rock is modelled by the random variable \(V\), where \(V = P + Q + 2 R\). Use the model to find the probability of finding minerals with a value of at least 93 in a randomly chosen tonne of rock.

1 The random variable \(X\) has the distribution \(\operatorname { Po } ( 1.3 )\). The random variable \(Y\) is defined by \(Y = 2 X\).

1. Find the mean and variance of \(Y\).
2. Give a reason why the variable \(Y\) does not have a Poisson distribution.

3 Three coats of paint are sprayed onto a surface. The thicknesses, in millimetres, of the three coats have independent distributions \(\mathrm { N } \left( 0.13,0.02 ^ { 2 } \right) , \mathrm { N } \left( 0.14,0.03 ^ { 2 } \right)\) and \(\mathrm { N } \left( 0.10,0.01 ^ { 2 } \right)\). Find the probability that, at a randomly chosen place on the surface, the total thickness of the three coats of paint is less than 0.30 millimetres.

1 The lengths of logs are normally distributed with mean 3.5 m and standard deviation 0.12 m . Describe fully the distribution of the total length of 8 randomly chosen logs.