5.04b Linear combinations: of normal distributions

7 At a particular supermarket, the times taken to serve each customer in a queue at a standard checkout may be modelled by a normal distribution with mean 240 seconds and standard deviation 20 seconds. There is a queue of 3 customers at a standard checkout.
Making a reasonable assumption about the times taken to serve these customers,

1. find the probability that the total time taken to serve the 3 customers will be less than 11 minutes.
2. State the assumption you have made in part (a) In the supermarket there is also an express checkout, which is reserved for customers buying 10 or fewer items. The time taken to serve a customer at this express checkout may be modelled by a normal distribution with mean 100 seconds and standard deviation 8 seconds. On a particular day Jiang has 8 items to pay for and has to choose whether to join a queue of 3 customers waiting at a standard checkout or a queue of 7 customers waiting at the express checkout. Using a similar assumption to that made in part (a),
3. find the probability that the total time taken to serve the 3 customers at the standard checkout will exceed the total time taken to serve the 7 customers at the express checkout.

1. Small containers and large containers are independently filled with fruit juice.

The amounts of fruit juice in small containers are normally distributed with mean 180 ml and standard deviation 4.5 ml The amounts of fruit juice in large containers are normally distributed with mean 330 ml and standard deviation 6.7 ml The random variable \(W\) represents the total amount of fruit juice in a random sample of 2 small containers minus the amount of fruit juice in 1 randomly selected large container. \(W \sim \mathrm {~N} ( a , b )\) where \(a\) and \(b\) are positive constants.

1. Find the value of \(a\) and the value of \(b\)
2. Find the probability that a randomly chosen large container of fruit juice contains more than 1.8 times the amount of fruit juice in a randomly chosen small container. A random sample of 3 small containers of fruit juice is taken.
3. Find the probability that the first container of fruit juice in this sample contains at least 5 ml more than the mean amount of fruit juice in all 3 small containers.

7. The random variable \(X\) is defined as $$X = 4 Y - 3 W$$ where \(Y \sim \mathrm {~N} \left( 40,3 ^ { 2 } \right) , W \sim \mathrm {~N} \left( 50,2 ^ { 2 } \right)\) and \(Y\) and \(W\) are independent.

1. Find \(\mathrm { P } ( X > 25 )\) The random variables \(Y _ { 1 } , Y _ { 2 }\) and \(Y _ { 3 }\) are independent and each has the same distribution as \(Y\). The random variable \(A\) is defined as $$A = \sum _ { i = 1 } ^ { 3 } Y _ { i }$$ The random variable \(C\) is such that \(C \sim \mathrm {~N} \left( 115 , \sigma ^ { 2 } \right)\) Given that \(\mathrm { P } ( A - C < 0 ) = 0.2\) and that \(A\) and \(C\) are independent,
2. find the variance of \(C\).

6. The random variable \(W\) is defined as $$W = 3 X - 4 Y$$ where \(X \sim \mathrm {~N} \left( 21,2 ^ { 2 } \right)\) and \(Y \sim \mathrm {~N} \left( 8.5 , \sigma ^ { 2 } \right)\) and \(X\) and \(Y\) are independent.
Given that \(\mathrm { P } ( W < 44 ) = 0.9\)

1. find the value of \(\sigma\), giving your answer to 2 decimal places. The random variables \(A _ { 1 } , A _ { 2 }\) and \(A _ { 3 }\) each have the same distribution as \(A\), where \(A \sim \mathrm {~N} \left( 28,5 ^ { 2 } \right)\) The random variable \(B\) is defined as $$B = 2 X + \sum _ { i = 1 } ^ { 3 } A _ { i }$$ where \(X , A _ { 1 } , A _ { 2 }\) and \(A _ { 3 }\) are independent.
2. Find \(\mathrm { P } ( B \leqslant 145 \mid B > 120 )\)

7. The independent random variables \(X\) and \(Y\) are such that $$X \sim \mathrm {~N} \left( 30,4.5 ^ { 2 } \right) \text { and } Y \sim \mathrm {~N} \left( 20,3.5 ^ { 2 } \right)$$ The random variables \(X _ { 1 } , X _ { 2 }\) and \(X _ { 3 }\) are independent and each has the same distribution as \(X\). The random variables \(Y _ { 1 }\) and \(Y _ { 2 }\) are independent and each has the same distribution as \(Y\). Given that the random variable \(A\) is defined as $$A = \frac { X _ { 1 } + X _ { 2 } + X _ { 3 } + Y _ { 1 } + Y _ { 2 } } { 5 }$$

1. find \(\mathrm { P } ( A < 24 )\) The random variable \(W\) is such that \(W \sim \mathrm {~N} \left( \mu , 2.8 ^ { 2 } \right)\) Given that \(\mathrm { P } ( W - X < 4 ) = 0.1\) and that \(W\) and \(X\) are independent,
2. find the value of \(\mu\), giving your answer to 3 significant figures.

7.(i)As part of a recruitment exercise candidates are required to complete three separate tasks.The times taken,\(A , B\) and \(C\) ,in minutes,for candidates to complete the three tasks are such that $$A \sim \mathrm {~N} \left( 21,2 ^ { 2 } \right) , B \sim \mathrm {~N} \left( 32,7 ^ { 2 } \right) \text { and } C \sim \mathrm {~N} \left( 45,9 ^ { 2 } \right)$$ The time taken by an individual candidate to complete each task is assumed to be independent of the time taken to complete each of the other tasks. A candidate is selected at random.

1. Find the probability that the candidate takes a total time of more than 90 minutes to complete all three tasks.
2. Find \(\mathrm { P } ( A > B )\) (ii)A simple random sample,\(X _ { 1 } , X _ { 2 } , X _ { 3 } , X _ { 4 }\) ,is taken from a normal population with mean \(\mu\) and standard deviation \(\sigma\) Given that $$\bar { X } = \frac { X _ { 1 } + X _ { 2 } + X _ { 3 } + X _ { 4 } } { 4 }$$ and that $$\mathrm { P } \left( X _ { 1 } > \bar { X } + k \sigma \right) = 0.1$$ where \(k\) is a constant,
   find the value of \(k\) ,giving your answer correct to 3 significant figures.

   END

1. A baker produces bread buns and bread rolls. The weights of buns, \(B\) grams, and the weights of rolls, \(R\) grams, are such that \(B \sim \mathrm {~N} \left( 55,1.3 ^ { 2 } \right)\) and \(R \sim \mathrm {~N} \left( 51,1.2 ^ { 2 } \right)\)

A bun and a roll are selected at random.

1. Find the probability that the bun weighs less than \(110 \%\) of the weight of the roll. Two buns are chosen at random.
2. Find the probability that their weights differ by more than 1 gram. The baker sells bread in bags. Each bag contains either 10 buns or 11 rolls. The weight of an empty bag, \(S\) grams, is such that \(S \sim \mathrm {~N} \left( 3,0.2 ^ { 2 } \right)\)
3. Find the probability that a bag of buns weighs less than a bag of rolls.

6 A particular lift has a maximum load capacity of 700 kg .
The weights of men are normally distributed with mean 80 kg and standard deviation 10 kg . The weights of women are normally distributed with mean 69 kg and standard deviation 5 kg . You may assume that weights of people are independent.

1. Find the probability that when 6 men and 3 women are in the lift, the load exceeds 700 kg . A sign in the lift states: "Maximum number of people in the lift is \(c\) "
2. Find the value of \(c\) such that the probability of the load exceeding 700 kg is less than \(2.5 \%\) no matter the gender of the occupants.

1. The continuous random variable \(X\) is normally distributed with

$$X \sim \mathrm {~N} \left( \mu , 5 ^ { 2 } \right)$$ A random sample of 10 observations of \(X\) is taken and \(\bar { X }\) denotes the sample mean.

1. Show that a \(90 \%\) confidence interval for \(\mu\), in terms of \(\bar { x }\), is given by $$( \bar { x } - 2.60 , \bar { x } + 2.60 )$$ The continuous random variable \(Y\) is normally distributed with $$Y \sim \mathrm {~N} \left( \mu , 3 ^ { 2 } \right)$$ A random sample of 20 observations of \(Y\) are taken and \(\bar { Y }\) denotes the sample mean.
2. Find a 95\% confidence interval for \(\mu\), in terms of \(\bar { y }\)
3. Given that \(X\) and \(Y\) are independent,
   1. find the distribution of \(\bar { X } - \bar { Y }\)
   2. calculate the probability that the two confidence intervals from part (a) and part (b) do not overlap.

1. The random variable \(X\) is defined as

$$X = 4 A - 3 B$$ where \(A\) and \(B\) are independent and $$A \sim \mathrm {~N} \left( 15,5 ^ { 2 } \right) \quad B \sim \mathrm {~N} \left( 10,4 ^ { 2 } \right)$$

1. Find \(\mathrm { P } ( X < 40 )\) The random variable \(C\) is such that \(C \sim \mathrm {~N} \left( 20 , \sigma ^ { 2 } \right)\) The random variables \(C _ { 1 } , C _ { 2 }\) and \(C _ { 3 }\) are independent and each has the same distribution as \(C\) The random variable \(D\) is defined as $$D = \sum _ { i = 1 } ^ { 3 } C _ { i }$$ Given that \(\mathrm { P } ( A + B + D < 76 ) = 0.2420\) and that \(A , B\) and \(D\) are independent,
2. showing your working clearly, find the standard deviation of \(C\)

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 \(\mathrm { P } ( G < 2 T + 20 )\)

7. A company makes cricket balls and tennis balls. The weights of cricket balls, \(C\) grams, follow a normal distribution $$C \sim \mathrm {~N} \left( 160,1.25 ^ { 2 } \right)$$ Three cricket balls are selected at random.

1. Find the probability that their total weight is more than 475.8 grams. The weights of tennis balls, \(T\) grams, follow a normal distribution $$T \sim \mathrm {~N} \left( 60,2 ^ { 2 } \right)$$ Five tennis balls and two cricket balls are selected at random.
2. Find the probability that the total weight of the five tennis balls and the two cricket balls is more than 625 grams. A random sample of \(n\) tennis balls \(T _ { 1 } , T _ { 2 } , \ldots , T _ { n }\) is taken.
   The random variable \(Y = ( n - 1 ) T _ { 1 } - \sum _ { r = 2 } ^ { n } T _ { r }\) Given that \(\mathrm { P } ( Y > 40 ) = 0.0838\) correct to 4 decimal places,
3. find \(n\).
   

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   Q7

1. A company produces bricks.

The weight of a brick, \(B \mathrm {~kg}\), is such that \(B \sim \mathrm {~N} \left( 1.96 , \sqrt { 0.003 } ^ { 2 } \right)\) Two bricks are chosen at random.

1. Find the probability that the difference in weight of the 2 bricks is greater than 0.1 kg A random sample of \(n\) bricks is to be taken.
2. Find the minimum sample size such that the probability of the sample mean being greater than 2 is less than 1\% The bricks are randomly selected and stacked on pallets.
   The weight of an empty pallet, \(E \mathrm {~kg}\), is such that \(E \sim \mathrm {~N} \left( 21.8 , \sqrt { 0.6 } ^ { 2 } \right)\) The random variable \(M\) represents the total weight of a pallet stacked with 500 bricks. The random variable \(T\) represents the total weight of a container of cement.
   Given that \(T\) is independent of \(M\) and that \(T \sim \mathrm {~N} \left( 774 , \sqrt { 1.8 } ^ { 2 } \right)\)
3. calculate \(\mathrm { P } ( 4 T > 100 + 3 M )\)

4. A farm produces potatoes. The potatoes are packed into sacks. The weight of a sack of potatoes is modelled by a normal distribution with mean 25.6 kg and standard deviation 0.24 kg

1. Find the probability that two randomly chosen sacks of potatoes differ in weight by more than 0.5 kg Sacks of potatoes are randomly selected and packed onto pallets. The weight of an empty pallet is modelled by a normal distribution with mean 20.0 kg and standard deviation 0.32 kg Each full pallet of potatoes holds 30 sacks of potatoes.
2. Find the probability that the total weight of a randomly chosen full pallet of potatoes is greater than 785 kg
   
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2. Philip and James are racing car drivers. Philip's lap times, in seconds, are normally distributed with mean 90 and variance 9. James' lap times, in seconds, are normally distributed with mean 91 and variance 12. The lap times of Philip and James are independent. Before a race, they each take a qualifying lap.

1. Find the probability that James' time for the qualifying lap is less than Philip's. The race is made up of 60 laps. Assuming that they both start from the same starting line and lap times are independent,
2. find the probability that Philip beats James in the race by more than 2 minutes.

2. A workshop makes two types of electrical resistor. The resistance, \(X\) ohms, of resistors of Type A is such that \(X \sim \mathrm {~N} ( 20,4 )\).
The resistance, \(Y\) ohms, of resistors of Type B is such that \(Y \sim \mathrm {~N} ( 10,0.84 )\).
When a resistor of each type is connected into a circuit, the resistance \(R\) ohms of the circuit is given by \(R = X + Y\) where \(X\) and \(Y\) are independent. Find

1. \(\mathrm { E } ( R )\),
2. \(\operatorname { Var } ( R )\),
3. \(\mathrm { P } ( 28.9 < R < 32.64 )\) (6)

7. The heights, in cm, of the male employees in a large company follow a normal distribution with mean 177 and standard deviation 5 The heights, in cm, of the female employees follow a normal distribution with mean 163 and standard deviation 4 A male employee and a female employee are chosen at random.

1. Find the probability that the male employee is taller than the female employee. Six male employees and four female employees are chosen at random.
2. Find the probability that their total height is less than 17 m .

8. A farmer supplies both duck eggs and chicken eggs. The weights of duck eggs, \(D\) grams, and chicken eggs, \(C\) grams, are such that $$D \sim \mathrm {~N} \left( 54,1.2 ^ { 2 } \right) \text { and } C \sim \mathrm {~N} \left( 44,0.8 ^ { 2 } \right)$$

1. Find the probability that the weights of 2 randomly selected duck eggs will differ by more than 3 g .
2. Find the probability that the weight of a randomly selected chicken egg is less than \(\frac { 4 } { 5 }\) of the weight of a randomly selected duck egg. Eggs are packed in boxes which contain either 6 randomly selected duck eggs or 6 randomly selected chicken eggs. The weight of an empty box has distribution \(\mathrm { N } \left( 28 , \sqrt { 5 } ^ { 2 } \right)\).
3. Find the probability that a full box of duck eggs weighs at least 50 g more than a full box of chicken eggs.

1. Blumen is a perfume sold in bottles. The amount of perfume in each bottle is normally distributed. The amount of perfume in a large bottle has mean 50 ml and standard deviation 5 ml . The amount of perfume in a small bottle has mean 15 ml and standard deviation 3 ml .

One large and 3 small bottles of Blumen are chosen at random.

1. Find the probability that the amount in the large bottle is less than the total amount in the 3 small bottles. A large bottle and a small bottle of Blumen are chosen at random.
2. Find the probability that the large bottle contains more than 3 times the amount in the small bottle.
   

A company produces two types of milk powder, 'Semi-Skimmed' and 'Full Cream'. In tests, each type of milk powder is used to make a large number of cups of coffee. The mass, \(S\) grams, of 'Semi-Skimmed' milk powder used in one cup of coffee is modelled by \(S \sim \mathrm {~N} \left( 4.9,0.8 ^ { 2 } \right)\). The mass, \(C\) grams, of 'Full Cream' milk powder used in one cup of coffee is modelled by \(C \sim \mathrm {~N} \left( 2.5,0.4 ^ { 2 } \right)\)

1. Two cups of coffee, one with each type of milk powder, are to be selected at random. Find the probability that the mass of 'Semi-Skimmed' milk powder used will be at least double that of the 'Full Cream' milk powder used.
2. 'Semi-Skimmed' milk powder is sold in 500 g packs. Find the probability that one pack will be sufficient for 100 cups of coffee.
   

1. The random variable \(A\) is defined as

$$A = B + 4 C - 3 D$$ where \(B\), \(C\) and \(D\) are independent random variables with $$B \sim \mathrm {~N} \left( 6,2 ^ { 2 } \right) \quad C \sim \mathrm {~N} \left( 7,3 ^ { 2 } \right) \quad D \sim \mathrm {~N} \left( 4,1.5 ^ { 2 } \right)$$ Find \(\mathrm { P } ( A < 45 )\)

1. The volume, \(B \mathrm { ml }\), in a bottle of Burxton's water has a normal distribution \(B \sim \mathrm {~N} \left( 325,6 ^ { 2 } \right)\) and the volume, \(H \mathrm { ml }\), in a bottle of Hargate's water has a normal distribution \(H \sim \mathrm {~N} \left( 330,4 ^ { 2 } \right)\).
   Rebecca buys 5 bottles of Burxton's water and one bottle of Hargate's water.
   Find the probability that the total volume in the 5 bottles of Burxton's water is more than 5 times the volume in the bottle of Hargate's water.
   (5)
2. Two independent random samples \(X _ { 1 } , X _ { 2 } , X _ { 3 } , X _ { 4 } , X _ { 5 }\) and \(Y _ { 1 } , Y _ { 2 } , Y _ { 3 } , Y _ { 4 } , Y _ { 5 }\) are each taken from a normal population with mean \(\mu\) and standard deviation \(\sigma\).

1. Find the distribution of the random variable \(D = Y _ { 1 } - \bar { X }\)
2. Hence show that \(\mathrm { P } \left( Y _ { 1 } > \bar { X } + \sigma \right) = 0.181\) correct to 3 decimal places. Ankit believes that \(\mathrm { P } \left( U _ { 1 } > \bar { U } + \sigma \right) = 0.181\) correct to 3 decimal places, for any random sample \(U _ { 1 } , U _ { 2 } , U _ { 3 } , U _ { 4 } , U _ { 5 }\) taken from a normal population with mean \(\mu\) and standard deviation \(\sigma\).
3. Explain briefly why the result from part (b) should not be used to confirm Ankit's belief.
4. Find, correct to 3 decimal places, the actual value of \(\mathrm { P } \left( U _ { 1 } > \bar { U } + \sigma \right)\).

7. Sugar is packed into medium bags and large bags. The weights of the medium bags of sugar are normally distributed with mean 520 grams and standard deviation 10 grams. The weights of the large bags of sugar are normally distributed with mean 1510 grams and standard deviation 20 grams.

1. Find the probability that a randomly chosen large bag of sugar weighs at least 15 grams more than the combined weight of 3 randomly chosen medium bags of sugar.
2. Find the probability that a randomly chosen large bag of sugar weighs less than 3 times the weight of a randomly chosen medium bag of sugar. A random sample of 5 medium bags of sugar is taken.
3. Find the value of \(d\) so that the probability that all 5 bags of sugar each weigh more than 520 grams is equal to the probability that the mean weight of the 5 bags of sugar is more than \(d\) grams.

5. The weights, in kg , of cars may be assumed to follow the normal distribution \(\mathrm { N } \left( 1000,250 ^ { 2 } \right)\). The weights, in kg , of lorries may be assumed to follow the normal distribution \(\mathrm { N } \left( 2800,650 ^ { 2 } \right)\). A lorry and a car are chosen at random.

1. Find the probability that the lorry weighs more than 3 times the weight of the car. A ferry carries vehicles across a river. The ferry is designed to carry a maximum weight of 20000 kg .
2. One morning, 8 cars and 3 lorries drive on to the ferry. Find the probability that their total weight will exceed the recommended maximum weight of 20000 kg .
3. State a necessary assumption needed for the calculation in part (b).

5 A general store sells lawn fertiliser in 2.5 kg bags, 5 kg bags and 10 kg bags.

1. The actual weight, \(W\) kilograms, of fertiliser in a 2.5 kg bag may be modelled by a normal random variable with mean 2.75 and standard deviation 0.15 . Determine the probability that the weight of fertiliser in a 2.5 kg bag is:
   1. less than 2.8 kg ;
   2. more than 2.5 kg .
2. The actual weight, \(X\) kilograms, of fertiliser in a 5 kg bag may be modelled by a normal random variable with mean 5.25 and standard deviation 0.20 .
   1. Show that \(\mathrm { P } ( 5.1 < X < 5.3 ) = 0.372\), correct to three decimal places.
   2. A random sample of four 5 kg bags is selected. Calculate the probability that none of the four bags contains between 5.1 kg and 5.3 kg of fertiliser.
3. The actual weight, \(Y\) kilograms, of fertiliser in a 10 kg bag may be modelled by a normal random variable with mean 10.75 and standard deviation 0.50. A random sample of six 10 kg bags is selected. Calculate the probability that the mean weight of fertiliser in the six bags is less than 10.5 kg .