Sum or total of normal variables

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.

2 The random variable \(X\) has the distribution \(\mathrm { N } \left( 3.2,1.2 ^ { 2 } \right)\). The sum of 60 independent observations of \(X\) is denoted by \(S\). Find \(\mathrm { P } ( S > 200 )\).

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.

2 The amount of tomato juice, \(X \mathrm { ml }\), dispensed into cartons of a particular brand has a normal distribution with mean 504 and standard deviation 3 . The juice is sold in packs of 4 cartons, filled independently. The total amount of juice in one pack is \(Y \mathrm { ml }\).

1. Find \(\mathrm { P } ( Y < 2000 )\). The random variable \(V\) is defined as \(Y - 4 X\).
2. Find \(\mathrm { E } ( V )\) and \(\operatorname { Var } ( V )\).
3. What is the probability that the amount of juice in a randomly chosen pack is more than 4 times the amount of juice in a randomly chosen carton?

1. A market stall sells vegetables. Two of the vegetables sold are broccoli heads and cabbages.

The weights of these broccoli heads, \(B\) kilograms, follow a normal distribution $$B \sim \mathrm {~N} \left( 0.588,0.084 ^ { 2 } \right)$$ The weights of these cabbages, \(C\) kilograms, follow a normal distribution $$C \sim \mathrm {~N} \left( 0.908,0.039 ^ { 2 } \right)$$

1. Find the probability that the total weight of two randomly chosen broccoli heads is less than the weight of a randomly chosen cabbage. Broccoli heads cost \(\pounds 2.50\) per kg and cabbages cost \(\pounds 3.00\) per kg. Jaymini buys 1 broccoli head and 2 cabbages, chosen randomly.
2. Find the probability that she pays more than £7 The market stall offers a discount for buying 5 or more broccoli heads. The price with the discount is \(\pounds w\) per kg. Let \(\pounds D\) be the price with the discount of 5 broccoli heads.
3. Find, in terms of \(w\), the mean and standard deviation of \(D\) Given that \(\mathrm { P } ( D < 6 ) < 0.1\)
4. find the smallest possible value of \(w\), giving your answer to 2 decimal places.

1. The weights of bags of carrots, \(C \mathrm {~kg}\), are such that \(C \sim \mathrm {~N} \left( 1.2,0.03 ^ { 2 } \right)\)

Three bags of carrots are selected at random.

1. Calculate the probability that their total weight is more than 3.5 kg . The weights of bags of potatoes, \(R \mathrm {~kg}\), are such that \(R \sim \mathrm {~N} \left( 2.3,0.03 ^ { 2 } \right)\)
   Two bags of potatoes are selected at random.
2. Calculate the probability that the difference in their weights is more than 0.05 kg . The weights of trays, \(T \mathrm {~kg}\), are such that \(T \sim \mathrm {~N} \left( 2.5 , \sqrt { 0.1 } ^ { 2 } \right)\)
   The random variable \(G\) represents the total weight, in kg, of a single tray packed with 10 bags of potatoes where \(G\) and \(T\) are independent.
3. Calculate \(\mathrm { P } ( G < 2 T + 20 )\)

2 A supermarket sells oranges. Their weights are modelled by the random variable \(X\) which has a Normal distribution with mean 345 grams and standard deviation 15 grams. When the oranges have been peeled, their weights in grams, \(Y\), are modelled by \(Y = 0.7 X\).

1. Find the probability that a randomly chosen peeled orange weighs less than 250 grams. I randomly choose 5 oranges to buy.
2. Find the probability that the total weight of the 5 unpeeled oranges is at least 1800 grams.
3. I peel three of the oranges and leave the remaining two unpeeled. Find the probability that the total weight of the two unpeeled oranges is greater than the total weight of the three peeled ones.