5.04b Linear combinations: of normal distributions

5 The masses, in grams, of large and small packets of Maxwheat cereal have the independent distributions \(\mathrm { N } \left( 410.0,3.6 ^ { 2 } \right)\) and \(\mathrm { N } \left( 206.0,3.7 ^ { 2 } \right)\) respectively.

1. Find the probability that a randomly chosen large packet has a mass that is more than double the mass of a randomly chosen small packet.
   The packets are placed in boxes. The boxes are identical in appearance. \(60 \%\) of the boxes contain exactly 10 randomly chosen large packets. 40\% of the boxes contain exactly 20 randomly chosen small packets.
2. Find the probability that a randomly chosen box contains packets with a total mass of more than 4080 grams.

3 The masses, in kilograms, of large sacks of flour and small sacks of flour have the independent distributions \(\mathrm { N } \left( 40,1.5 ^ { 2 } \right)\) and \(\mathrm { N } \left( 12,0.7 ^ { 2 } \right)\) respectively.

1. Find the probability that the total mass of 6 randomly chosen large sacks of flour is more than 245 kg .
2. Find the probability that the mass of a randomly chosen large sack of flour is less than 4 times the mass of a randomly chosen small sack of flour.

1 The masses, in grams, of plums of a certain type have the distribution \(\mathrm { N } \left( 40.4,5.2 ^ { 2 } \right)\). The plums are packed in bags, with each bag containing 6 randomly chosen plums. If the total weight of the plums in a bag is less than 220 g the bag is rejected. Find the percentage of bags that are rejected.

3 Mary buys 3 packets of sugar and 5 packets of coffee and puts them in her shopping basket, together with her purse which weighs 350 g . Weights of packets of sugar are normally distributed with mean 500 g and standard deviation 20 g . Weights of packets of coffee are normally distributed with mean 200 g and standard deviation 12 g . Find the probability that the total weight in the shopping basket is less than 2900 g .

7 Machine \(A\) fills bags of fertiliser so that their weights follow a normal distribution with mean 20.05 kg and standard deviation 0.15 kg . Machine \(B\) fills bags of fertiliser so that their weights follow a normal distribution with mean 20.05 kg and standard deviation 0.27 kg .

1. Find the probability that the total weight of a random sample of 20 bags filled by machine \(A\) is at least 2 kg more than the total weight of a random sample of 20 bags filled by machine \(B\). [6]
2. A random sample of \(n\) bags filled by machine \(A\) is taken. The probability that the sample mean weight of the bags is greater than 20.07 kg is denoted by \(p\). Find the value of \(n\), given that \(p = 0.0250\) correct to 4 decimal places.

2 Each day at the gym, Sarah completes three runs. The distances, in metres, that she completes in the three runs have the independent distributions \(W \sim \mathrm {~N} ( 1520,450 ) , X \sim \mathrm {~N} ( 2250,720 )\) and \(Y \sim \mathrm {~N} ( 3860,1050 )\). Find the probability that, on a particular day, \(Y\) is less than the total of \(W\) and \(X\).

4 Wendy's journey to work consists of three parts: walking to the train station, riding on the train and then walking to the office. The times, in minutes, for the three parts of her journey are independent and have the distributions \(\mathrm { N } \left( 15.0,1.1 ^ { 2 } \right) , \mathrm { N } \left( 32.0,3.5 ^ { 2 } \right)\) and \(\mathrm { N } \left( 8.6,1.2 ^ { 2 } \right)\) respectively.

1. Find the mean and variance of the total time for Wendy's journey.
   If Wendy's journey takes more than 60 minutes, she is late for work.
2. Find the probability that, on a randomly chosen day, Wendy will be late for work.
3. Find the probability that the mean of Wendy's journey times over 15 randomly chosen days will be less than 54.5 minutes.

3 The lengths, in centimetres, of two types of insect, \(A\) and \(B\), are modelled by the random variables \(X \sim \mathrm {~N} ( 6.2,0.36 )\) and \(Y \sim \mathrm {~N} ( 2.4,0.25 )\) respectively. Find the probability that the length of a randomly chosen type \(A\) insect is greater than the sum of the lengths of 3 randomly chosen type \(B\) insects.

6 The masses, in kilograms, of large and small sacks of grain have the distributions \(\mathrm { N } ( 53,11 )\) and \(\mathrm { N } ( 14,3 )\) respectively.

1. Find the probability that the mass of a randomly chosen large sack is greater than four times the mass of a randomly chosen small sack.
2. A lift can safely carry a maximum mass of 1000 kg . Find the probability that the lift can safely carry 12 randomly chosen large sacks and 25 randomly chosen small sacks. \(7 X\) is a random variable with distribution \(\operatorname { Po } ( 2.90 )\). A random sample of 100 values of \(X\) is taken. Find the probability that the sample mean is less than 2.88 .
   If you use the following lined page to complete the answer(s) to any question(s), the question number(s) must be clearly shown.

4 Each box of Seeds \& Raisins contains \(S\) grams of seeds and \(R\) grams of raisins. The weight of a box, when empty, is \(B\) grams. \(S , R\) and \(B\) are independent random variables, where \(S \sim \mathrm {~N} ( 300,45 )\), \(R \sim \mathrm {~N} ( 200,25 )\) and \(\mathrm { B } \sim \mathrm { N } ( 15,4 )\). A full box of Seeds \& Raisins is chosen at random.
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1. Find the probability that the total weight of the box and its contents is more than 500 grams. [5]
2. Find the probability that the weight of seeds in the box is less than 1.4 times the weight of raisins in the box.

5 Large packets of rice are packed in cartons, each containing 20 randomly chosen packets. The masses of these packets are normally distributed with mean 1010 g and standard deviation 3.4 g . The masses of the cartons, when empty, are independently normally distributed with mean 50 g and standard deviation 2.0 g .

1. Find the variance of the masses of full cartons.
   Small packets of rice are packed in boxes. The total masses of full boxes are normally distributed with mean 6730 g and standard deviation 15.0 g . The masses of the boxes and cartons are distributed independently of each other.
2. Find the probability that the mass of a randomly chosen full carton is more than three times the mass of a randomly chosen full box.

5

1. Two random variables \(X\) and \(Y\) have the independent distributions \(\mathrm { N } ( 7,3 )\) and \(\mathrm { N } ( 6,2 )\) respectively. A random value of each variable is taken. Find the probability that the two values differ by more than 2 .
2. Each candidate's overall score in a science test is calculated as follows. The mark for theory is denoted by \(T\), the mark for practical is denoted by \(P\), and the overall score is given by \(T + 1.5 P\). The variables \(T\) and \(P\) are assumed to be independent with distributions \(\mathrm { N } ( 62,158 )\) and \(\mathrm { N } ( 42,108 )\) respectively. You should assume that no continuity corrections are needed when using these distributions.
   1. A pass is awarded to candidates whose overall score is at least 90 . Find the proportion of candidates who pass.
   2. Comment on the assumption that the variables \(T\) and \(P\) are independent.

2 The random variable \(X\) has the distribution \(\mathrm { N } \left( 31.2,10.4 ^ { 2 } \right)\). Two independent random values of \(X\), denoted by \(X _ { 1 }\) and \(X _ { 2 }\), are chosen. Find \(\mathrm { P } \left( X _ { 1 } > 3 X _ { 2 } \right)\).

4 A random variable \(X\) has the distribution \(\mathrm { N } ( 10,12 )\). Two independent values of \(X\), denoted by \(X _ { 1 }\) and \(X _ { 2 }\), are chosen at random.

1. Write down the value of \(\mathrm { P } \left( X _ { 1 } > X _ { 2 } \right)\).
2. Find \(\mathrm { P } \left( X _ { 1 } > 2 X _ { 2 } - 3 \right)\).

3 The masses in kilograms of large and small bags of cement have the independent distributions \(\mathrm { N } ( 50,2.4 )\) and \(\mathrm { N } ( 26,1.8 )\) respectively. Find the probability that the total mass of 5 randomly chosen large bags of cement is greater than the total mass of 10 randomly chosen small bags of cement. \includegraphics[max width=\textwidth, alt={}, center]{7c078a14-98f9-4292-ae76-a2642238176f-04_2714_34_143_2012} \includegraphics[max width=\textwidth, alt={}, center]{7c078a14-98f9-4292-ae76-a2642238176f-05_2724_35_136_20}

6 The volumes, in millilitres, of large and small cups of tea are modelled by the distributions \(\mathrm { N } ( 200,30 )\) and \(\mathrm { N } ( 110,20 )\) respectively.

1. Find the probability that the total volume of a randomly chosen large cup of tea and a randomly chosen small cup of tea is less than 300 ml .
2. Find the probability that the volume of a randomly chosen large cup of tea is more than twice the volume of a randomly chosen small cup of tea.

5 The volumes, in litres, of juice in large and small bottles have the distributions \(\mathrm { N } ( 5.10,0.0102 )\) and \(\mathrm { N } ( 2.51,0.0036 )\) respectively.

1. Find the probability that the total volume of juice in 3 randomly chosen large bottles and 4 randomly chosen small bottles is less than 25.5 litres.
2. Find the probability that the volume of juice in a randomly chosen large bottle is at least twice the volume of juice in a randomly chosen small bottle.

5 The heights of buildings in a large city are normally distributed with mean 18.3 m and standard deviation 2.5 m .

1. Find the probability that the total height of 5 randomly chosen buildings in the city is more than 95 m .
2. Find the probability that the difference between the heights of two randomly chosen buildings in the city is less than 1 m .

2 Each day Samuel travels from \(A\) to \(B\) and from \(B\) to \(C\). He then returns directly from \(C\) to \(A\). The times, in minutes, for these three journeys have the independent distributions \(\mathrm { N } \left( 20,2 ^ { 2 } \right) , \mathrm { N } \left( 18,1.5 ^ { 2 } \right)\) and \(\mathrm { N } \left( 30,1.8 ^ { 2 } \right)\), respectively. Find the probability that, on a randomly chosen day, the total time for his two journeys from \(A\) to \(B\) and \(B\) to \(C\) is less than the time for his return journey from \(C\) to \(A\). [5]

5 The thickness of books in a large library is normally distributed with mean 2.4 cm and standard deviation 0.3 cm .

1. Find the probability that the total thickness of 6 randomly chosen books is more than 16 cm .
2. Find the probability that the thickness of a book chosen at random is less than 1.1 times the thickness of a second book chosen at random.

5 Each box of Fruity Flakes contains \(X\) grams of flakes and \(Y\) grams of fruit, where \(X\) and \(Y\) are independent random variables, having distributions \(\mathrm { N } ( 400,50 )\) and \(\mathrm { N } ( 100,20 )\) respectively. The weight of each box, when empty, is exactly 20 grams. A full box of Fruity Flakes is chosen at random.

1. Find the probability that the total weight of the box and its contents is less than 530 grams.
2. Find the probability that the weight of flakes in the box is more than 4.1 times the weight of fruit in the box.

5 Large packets of sugar are packed in cartons, each containing 12 packets. The weights of these packets are normally distributed with mean 505 g and standard deviation 3.2 g . The weights of the cartons, when empty, are independently normally distributed with mean 150 g and standard deviation 7 g .

1. Find the probability that the total weight of a full carton is less than 6200 g .
   Small packets of sugar are packed in boxes. The total weight of a full box has a normal distribution with mean 3130 g and standard deviation 12.1 g .
2. Find the probability that the weight of a randomly chosen full carton is less than double the weight of a randomly chosen full box.

7

1. A random variable \(X\) is normally distributed with mean 4.2 and standard deviation 1.1. Find the probability that the sum of two randomly chosen values of \(X\) is greater than 10 .
2. Each candidate's overall score for an essay is calculated as follows. The mark for creativity is denoted by \(C\), the penalty mark for spelling errors is denoted by \(S\) and the overall score is defined by \(C - \frac { 1 } { 2 } S\). The variables \(C\) and \(S\) are independent and have distributions \(\mathrm { N } ( 29,105 )\) and \(\mathrm { N } ( 17,15 )\) respectively. Find the proportion of candidates receiving a negative overall score.

3 The mass, in tonnes, of iron ore produced per day at a mine is normally distributed with mean 7.0 and standard deviation 0.46. Find the probability that the total amount of iron ore produced in 10 randomly chosen days is more than 71 tonnes.

6 The times, in minutes, taken to complete the two parts of a task are normally distributed with means 4.5 and 2.3 respectively and standard deviations 1.1 and 0.7 respectively.

1. Find the probability that the total time taken for the task is less than 8.5 minutes.
2. Find the probability that the time taken for the first part of the task is more than twice the time taken for the second part.