Sum versus sum comparison

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.

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.
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6 The masses, in grams, of small and large bags of flour have the distributions \(\mathrm { N } ( 510,100 )\) and \(\mathrm { N } ( 1015,324 )\) respectively. André selects 4 small bags of flour and 2 large bags of flour at random.

1. Find the probability that the total mass of these 6 bags of flour is less than 4130 g .
2. Find the probability that the total mass of the 4 small bags is more than the total mass of the 2 large bags.

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.
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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.
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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.

4 A machine manufactures batches of 100 titanium sheets. The thickness of every sheet in a batch is Normally distributed with mean \(\mu \mathrm { mm }\) and standard deviation 0.03 mm . You should assume that each sheet is of uniform thickness and that the thicknesses of different sheets are independent of each other. The values of \(\mu\) for three different batches, A, B and C, are 3.125, 3.117 and 3.109 respectively.

1. Determine the probability that the total thickness of 10 sheets from Batch A is less than 31.0 mm .
2. Determine the probability that, if a single sheet from Batch A is cut into pieces and 10 of the pieces are stacked together, the total thickness of the stack is less than 31.0 mm .
3. Determine the probability that, if one sheet from each of Batches A, B and C are stacked together, the total thickness of the stack is at least 9.4 mm .
4. Determine the probability that the total thickness of 10 sheets from Batch A is less than the total thickness of 10 sheets from Batch B.

7 Fence panels come in two sizes, large and small. The lengths of the large panels are normally distributed with mean 198 cm and standard deviation 5 cm . The lengths of the small panels are normally distributed with mean 74 cm and standard deviation 3 cm .

1. Find the probability that the total length of a random sample of 3 large panels is greater than the total length of a random sample of 8 small panels. One large panel and one small panel are selected at random.
2. Find the probability that the length of the large panel is more than \(\frac { 8 } { 3 }\) times the length of the small panel. Rosa needs 1000 cm of fencing. The large panels cost \(\pounds 80\) each and the small panels cost \(\pounds 30\) each. Rosa's plan is to buy 5 large panels and measure the total length. If the total length is less than 1000 cm she will then buy one small panel as well.
3. Calculate whether or not the expected cost of Rosa's plan is cheaper than simply buying 14 small panels.

3. A shop sells carrots and broccoli. The weights of carrots can be modelled by a normal distribution with mean 130 grams and variance 25 grams \(^ { 2 }\) and the weights of broccoli can be modelled by a normal distribution with mean 400 grams and variance 80 grams \({ } ^ { 2 }\). Find the probability that the weight of six randomly chosen carrots is more than two times the weight of one randomly chosen broccoli.

5 The duration, \(X\) minutes, of a timetabled 1-hour lesson may be assumed to be normally distributed with mean 54 and standard deviation 2. The duration, \(Y\) minutes, of a timetabled \(1 \frac { 1 } { 2 }\)-hour lesson may be assumed to be normally distributed with mean 83 and standard deviation 3. Assuming the durations of lessons to be independent, determine the probability that the total duration of a random sample of three 1 -hour lessons is less than the total duration of a random sample of two \(1 \frac { 1 } { 2 }\)-hour lessons.
(7 marks)

2. Geraint is a beekeeper. The amounts of honey, \(X \mathrm {~kg}\), that he collects annually, from each hive are modelled by the normal distribution \(\mathrm { N } \left( 15,5 ^ { 2 } \right)\). At location \(A\), Geraint has three hives and at location \(B\) he has five hives. You may assume that the amounts of honey collected from the eight hives are independent of each other.

1. 1. Find the probability that Geraint collects more than 14 kg of honey from the first hive at location \(A\).
   2. Find the probability that he collects more than 14 kg of honey from exactly two out of the three hives at location \(A\).
2. Find the probability that the total amount of honey that Geraint collects from all eight hives is more than 160 kg .
3. Find the probability that Geraint collects at least twice as much honey from location B as from location A.