5.04b Linear combinations: of normal distributions

5 Cans of drink are packed in boxes, each containing 4 cans. The weights of these cans are normally distributed with mean 510 g and standard deviation 14 g . The weights of the boxes, when empty, are independently normally distributed with mean 200 g and standard deviation 8 g .

1. Find the probability that the total weight of a full box of cans is between 2200 g and 2300 g .
2. Two cans of drink are chosen at random. Find the probability that they differ in weight by more than 20 g .

6 The random variable \(T\) denotes the time, in seconds, for 100 m races run by Tania. \(T\) is normally distributed with mean \(\mu\) and variance \(\sigma ^ { 2 }\). A random sample of 40 races run by Tania gave the following results. $$n = 40 \quad \Sigma t = 560 \quad \Sigma t ^ { 2 } = 7850$$

1. Calculate unbiased estimates of \(\mu\) and \(\sigma ^ { 2 }\).
   The random variable \(S\) denotes the time, in seconds, for 100 m races run by Suki. \(S\) has the independent distribution \(\mathrm { N } ( 14.2,0.3 )\).
2. Using your answers to part (a), find the probability that, in a randomly chosen 100 m race, Suki's time will be at least 0.1 s more than Tania's time.

4 Each month a company sells \(X \mathrm {~kg}\) of brown sugar and \(Y \mathrm {~kg}\) of white sugar, where \(X\) and \(Y\) have the independent distributions \(\mathrm { N } \left( 2500,120 ^ { 2 } \right)\) and \(\mathrm { N } \left( 3700,130 ^ { 2 } \right)\) respectively.

1. Find the mean and standard deviation of the total amount of sugar that the company sells in 3 randomly chosen months.
   The company makes a profit of \(\\) 1.50\( per kilogram of brown sugar sold and makes a loss of \)\\( 0.20\) per kilogram of white sugar sold.
2. Find the probability that, in a randomly chosen month, the total profit is less than \(\\) 3000$.

6 A factory makes loaves of bread in batches. One batch of loaves contains \(X\) kilograms of dried yeast and \(Y\) kilograms of flour, where \(X\) and \(Y\) have the independent distributions \(\mathrm { N } \left( 0.7,0.02 ^ { 2 } \right)\) and \(\mathrm { N } \left( 100.0,3.0 ^ { 2 } \right)\) respectively. Dried yeast costs \(\\) 13.50\( per kilogram and flour costs \)\\( 0.90\) per kilogram. For making one batch of bread the total of all other costs is \(\\) 55\(. The factory sells each batch of bread for \)\\( 200\). Find the probability that the profit made on one randomly chosen batch of bread is greater than \(\\) 40$. [7]

2 The masses, in kilograms, of small and large bags of wheat have the independent distributions \(\mathrm { N } ( 16.0,0.4 )\) and \(\mathrm { N } ( 51.0,0.9 )\) respectively. Find the probability that the total mass of 3 randomly chosen small bags is greater than the mass of one randomly chosen large bag. \includegraphics[max width=\textwidth, alt={}, center]{acd6f1c9-bbaf-40ca-b5cb-8466ddb9f596-04_2720_38_109_2010}

4 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 five 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 three times that of a randomly chosen Longlive bulb.

2 In athletics matches the triple jump event consists of a hop, followed by a step, followed by a jump. The lengths covered by Albert in each part are independent normal variables with means \(3.5 \mathrm {~m} , 2.9 \mathrm {~m}\), 3.1 m and standard deviations \(0.3 \mathrm {~m} , 0.25 \mathrm {~m} , 0.35 \mathrm {~m}\) respectively. The length of the triple jump is the sum of the three parts.

1. Find the mean and standard deviation of the length of Albert's triple jumps.
2. Find the probability that the mean of Albert's next four triple jumps is greater than 9 m .

5 A clock contains 4 new batteries each of which gives a voltage which is normally distributed with mean 1.54 volts and standard deviation 0.05 volts. The voltages of the batteries are independent. The clock will only work if the total voltage is greater than 5.95 volts.

1. Find the probability that the clock will work.
2. Find the probability that the average total voltage of the batteries of 20 clocks chosen at random exceeds 6.2 volts.

6 When Sunil travels from his home in England to visit his relatives in India, his journey is in four stages. The times, in hours, for the stages have independent normal distributions as follows. Bus from home to the airport: \(\quad \mathrm { N } ( 3.75,1.45 )\) Waiting in the airport: \(\quad \mathrm { N } ( 3.1,0.785 )\) Flight from England to India: \(\quad \mathrm { N } ( 11,1.3 )\) Car in India to relatives: \(\quad \mathrm { N } ( 3.2,0.81 )\)

1. Find the probability that the flight time is shorter than the total time for the other three stages.
2. Find the probability that, for 6 journeys to India, the mean time waiting in the airport is less than 4 hours.

4 The weekly distance in kilometres driven by Mr Parry has a normal distribution with mean 512 and standard deviation 62. Independently, the weekly distance in kilometres driven by Mrs Parry has a normal distribution with mean 89 and standard deviation 7.4.

1. Find the probability that, in a randomly chosen week, Mr Parry drives more than 5 times as far as Mrs Parry.
2. Find the mean and standard deviation of the total of the weekly distances in miles driven by Mr Parry and Mrs Parry. Use the approximation 8 kilometres \(= 5\) miles.

6 Yu Ming travels to work and returns home once each day. The times, in minutes, that he takes to travel to work and to return home are represented by the independent random variables \(W\) and \(H\) with distributions \(\mathrm { N } \left( 22.4,4.8 ^ { 2 } \right)\) and \(\mathrm { N } \left( 20.3,5.2 ^ { 2 } \right)\) respectively.

1. Find the probability that Yu Ming's total travelling time during a 5-day period is greater than 180 minutes.
2. Find the probability that, on a particular day, Yu Ming takes longer to return home than he takes to travel to work.

5 Each drink from a coffee machine contains \(X \mathrm {~cm} ^ { 3 }\) of coffee and \(Y \mathrm {~cm} ^ { 3 }\) of milk, where \(X\) and \(Y\) are independent variables with \(X \sim \mathrm {~N} \left( 184,15 ^ { 2 } \right)\) and \(Y \sim \mathrm {~N} \left( 50,8 ^ { 2 } \right)\). If the total volume of the drink is less than \(200 \mathrm {~cm} ^ { 3 }\) the customer receives the drink without charge.

1. Find the percentage of drinks which customers receive without charge.
2. Find the probability that, in a randomly chosen drink, the volume of coffee is more than 4 times the volume of milk.

5 Fiona and Jhoti each take one shower per day. The times, in minutes, taken by Fiona and Jhoti to take a shower are represented by the independent variables \(F \sim \mathrm {~N} \left( 12.2,2.8 ^ { 2 } \right)\) and \(J \sim \mathrm {~N} \left( 11.8,2.6 ^ { 2 } \right)\) respectively. Find the probability that, on a randomly chosen day,

1. the total time taken to shower by Fiona and Jhoti is less than 30 minutes,
2. Fiona takes at least twice as long as Jhoti to take a shower.

2 The independent random variables \(X\) and \(Y\) have the distributions \(\mathrm { N } ( 6.5,14 )\) and \(\mathrm { N } ( 7.4,15 )\) respectively. Find \(\mathrm { P } ( 3 X - Y < 20 )\).

3 Weights of cups have a normal distribution with mean 91 g and standard deviation 3.2 g . Weights of saucers have an independent normal distribution with mean 72 g and standard deviation 2.6 g . Cups and saucers are chosen at random to be packed in boxes, with 6 cups and 6 saucers in each box. Given that each empty box weighs 550 g , find the probability that the total weight of a box containing 6 cups and 6 saucers exceeds 1550 g .

5 Packets of cereal are packed in boxes, each containing 6 packets. The masses of the packets are normally distributed with mean 510 g and standard deviation 12 g . The masses of the empty boxes are normally distributed with mean 70 g and standard deviation 4 g .

1. Find the probability that the total mass of a full box containing 6 packets is between 3050 g and 3150 g .
2. A packet and an empty box are chosen at random. Find the probability that the mass of the packet is at least 8 times the mass of the empty box.

1 The masses, in grams, of apples of a certain type are normally distributed with mean 60.4 and standard deviation 8.2. The apples are packed in bags, with each bag containing 8 randomly chosen apples. The bags are checked by Quality Control and any bag containing apples with a total mass of less than 436 g is rejected. Find the proportion of bags that are rejected.

8 In an examination, the marks in the theory paper and the marks in the practical paper are denoted by the random variables \(X\) and \(Y\) respectively, where \(X \sim \mathrm {~N} ( 57,13 )\) and \(Y \sim \mathrm {~N} ( 28,5 )\). You may assume that each candidate's marks in the two papers are independent. The final score of each candidate is found by calculating \(X + 2.5 Y\). A candidate is chosen at random. Without using a continuity correction, find the probability that this candidate

1. has a final score that is greater than 140 ,
2. obtains at least 20 more marks in the theory paper than in the practical paper.

3 The daily times, in minutes, that Yu Ming takes showering, getting dressed and having breakfast are independent and have the distributions \(\mathrm { N } \left( 9,2.2 ^ { 2 } \right) , \mathrm { N } \left( 8,1.3 ^ { 2 } \right)\) and \(\mathrm { N } \left( 17,2.6 ^ { 2 } \right)\) respectively. The total daily time that Yu Ming takes for all three activities is denoted by \(T\) minutes.

1. Find the mean and variance of \(T\).
2. Yu Ming notes the value of \(T\) on each day in a random sample of 70 days and calculates the sample mean. Find the probability that the sample mean is between 33 and 35 .

4 The volume, in millilitres, of a small cup of coffee has the distribution \(\mathrm { N } ( 103.4,10.2 )\). The volume of a large cup of coffee is 1.5 times the volume of a small cup of coffee.

1. Find the mean and standard deviation of the volume of a large cup of coffee.
2. Find the probability that the total volume of a randomly chosen small cup of coffee and a randomly chosen large cup of coffee is greater than 250 ml .

2 The random variable \(X\) has the distribution \(\mathrm { N } ( 3,1.2 )\). The random variable \(A\) is defined by \(A = 2 X\). The random variable \(B\) is defined by \(B = X _ { 1 } + X _ { 2 }\), where \(X _ { 1 }\) and \(X _ { 2 }\) are independent random values of \(X\). Describe fully the distribution of \(A\) and the distribution of \(B\). Distribution of \(A\) : \(\_\_\_\_\) Distribution of \(B\) : \(\_\_\_\_\)

4 The heights of a certain variety of plant are normally distributed with mean 110 cm and variance \(1050 \mathrm {~cm} ^ { 2 }\). Two plants of this variety are chosen at random. Find the probability that the height of one of these plants is at least 1.5 times the height of the other.

4 A factory supplies boxes of children's bricks. Each box contains 10 randomly chosen large bricks and 20 randomly chosen small bricks. The masses, in grams, of large and small bricks have the distributions \(\mathrm { N } ( 60,1.2 )\) and \(\mathrm { N } ( 30,0.7 )\) respectively. The mass of an empty box is 8 g . Find the probability that the total weight of a box and its contents is less than 1200 g .

6 The masses, in kilograms, of cartons of sugar and cartons of flour have the distributions \(\mathrm { N } \left( 78.8,12.6 ^ { 2 } \right)\) and \(\mathrm { N } \left( 62.0,10.0 ^ { 2 } \right)\) respectively.

1. The standard load for a certain crane is 8 cartons of sugar and 3 cartons of flour. The maximum load that can be carried safely by the crane is 900 kg . Stating a necessary assumption, find the percentage of standard loads that will exceed the maximum safe load.
2. Find the probability that a randomly chosen carton of sugar has a smaller mass than a randomly chosen carton of flour.

4 Each year a transport firm uses \(X\) litres of gasoline and \(Y\) litres of diesel fuel, where \(X\) and \(Y\) have the independent distributions \(X \sim \mathrm {~N} ( 10700,950 ) ^ { 2 }\) and \(Y \sim \mathrm {~N} \left( 13400,1210 ^ { 2 } \right)\).

1. Find the probability that in a randomly chosen year the firm uses more gasoline than diesel fuel.
   The costs per litre of gasoline and diesel fuel are \\(0.80 and \\)0.85 respectively.
2. Find the probability that the total cost of gasoline and diesel fuel in a randomly chosen year is between \(\\) 20000\( and \)\\( 22000\).