5.04a Linear combinations: E(aX+bY), Var(aX+bY)

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)

3. Given the random variables \(X \sim \mathrm {~N} ( 20,5 )\) and \(Y \sim \mathrm {~N} ( 10,4 )\) where \(X\) and \(Y\) are independent, find

1. \(\mathrm { E } ( X - Y )\),
2. \(\operatorname { Var } ( X - Y )\),
3. \(\mathrm { P } ( 13 < X - Y < 16 )\).

7. The random variable \(D\) is defined as $$D = A - 3 B + 4 C$$ where \(A \sim \mathrm {~N} \left( 5,2 ^ { 2 } \right) , B \sim \mathrm {~N} \left( 7,3 ^ { 2 } \right)\) and \(C \sim \mathrm {~N} \left( 9,4 ^ { 2 } \right)\), and \(A , B\) and \(C\) are independent.

1. Find \(\mathrm { P } ( \mathrm { D } < 44 )\). The random variables \(B _ { 1 } , B _ { 2 }\) and \(B _ { 3 }\) are independent and each has the same distribution as \(B\). The random variable \(X\) is defined as $$X = A - \sum _ { i = 1 } ^ { 3 } B _ { i } + 4 C .$$
2. Find \(\mathrm { P } ( X > 0 )\). \section*{END}

7. A set of scaffolding poles come in two sizes, long and short. The length \(L\) of a long pole has the normal distribution \(\mathrm { N } \left( 19.7,0.5 ^ { 2 } \right)\). The length \(S\) of a short pole has the normal distribution \(\mathrm { N } \left( 4.9,0.2 ^ { 2 } \right)\). The random variables \(L\) and \(S\) are independent. A long pole and a short pole are selected at random.

1. Find the probability that the length of the long pole is more than 4 times the length of the short pole. Four short poles are selected at random and placed end to end in a row. The random variable \(T\) represents the length of the row.
2. Find the distribution of \(T\).
3. Find \(\mathrm { P } ( | L - T | < 0.1 )\).

The weights of adult men are normally distributed with a mean of 84 kg and a standard deviation of 11 kg .

1. Find the probability that the total weight of 4 randomly chosen adult men is less than 350 kg . The weights of adult women are normally distributed with a mean of 62 kg and a standard deviation of 10 kg .
2. Find the probability that the weight of a randomly chosen adult man is less than one and a half times the weight of a randomly chosen adult woman.

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.
   

2. The weights of pears in an orchard are assumed to have unknown mean \(\mu\) and unknown standard deviation \(\sigma\). A random sample of 20 pears is taken and their weights recorded.
The sample is represented by \(X _ { 1 } , X _ { 2 } , \ldots , X _ { 20 }\). State whether or not the following are statistics. Give reasons for your answers.

1. 1. \(\frac { X _ { 1 } + 3 X _ { 20 } } { 2 }\)
   2. \(\sum _ { i = 1 } ^ { 20 } \left( X _ { i } - \mu \right)\)
   3. \(\sum _ { i = 1 } ^ { 20 } \left( \frac { X _ { i } - \mu } { \sigma } \right)\)
2. Find the mean and variance of \(\frac { 3 X _ { 1 } - X _ { 20 } } { 2 }\)

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).

6

1. The length of one-metre galvanised-steel straps used in house building may be modelled by a normal distribution with a mean of 1005 mm and a standard deviation of 15 mm . The straps are supplied to house builders in packs of 12, and the straps in a pack may be assumed to be a random sample. Determine the probability that the mean length of straps in a pack is less than one metre.
2. Tania, a purchasing officer for a nationwide house builder, measures the thickness, \(x\) millimetres, of each of a random sample of 24 galvanised-steel straps supplied by a manufacturer. She then calculates correctly that the value of \(\bar { x }\) is 4.65 mm .
   1. Assuming that the thickness, \(X \mathrm {~mm}\), of such a strap may be modelled by the distribution \(\mathrm { N } \left( \mu , 0.15 ^ { 2 } \right)\), construct a \(99 \%\) confidence interval for \(\mu\).
   2. Hence comment on the manufacturer's specification that the mean thickness of such straps is greater than 4.5 mm .

5 When a particular make of tennis ball is dropped from a vertical distance of 250 cm on to concrete, the height, \(X\) centimetres, to which it first bounces may be assumed to be normally distributed with a mean of 140 and a standard deviation of 2.5.

1. Determine:
   1. \(\mathrm { P } ( X < 145 )\);
   2. \(\mathrm { P } ( 138 < X < 142 )\).
2. Determine, to one decimal place, the maximum height exceeded by \(85 \%\) of first bounces.
3. Determine the probability that, for a random sample of 4 first bounces, the mean height is greater than 139 cm .

3 Each day, Margot completes the crossword in her local morning newspaper. Her completion times, \(X\) minutes, can be modelled by a normal random variable with a mean of 65 and a standard deviation of 20 .

1. Determine:
   1. \(\mathrm { P } ( X < 90 )\);
   2. \(\mathrm { P } ( X > 60 )\).
2. Given that Margot's completion times are independent from day to day, determine the probability that, during a particular period of 6 days:
   1. she completes one of the six crosswords in exactly 60 minutes;
   2. she completes each crossword in less than 60 minutes;
   3. her mean completion time is less than 60 minutes.
      
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2 The diameter, \(D\) millimetres, of an American pool ball may be modelled by a normal random variable with mean 57.15 and standard deviation 0.04 .

1. Determine:
   1. \(\mathrm { P } ( D < 57.2 )\);
   2. \(\mathrm { P } ( 57.1 < D < 57.2 )\).
2. A box contains 16 of these pool balls. Given that the balls may be regarded as a random sample, determine the probability that:
   1. all 16 balls have diameters less than 57.2 mm ;
   2. the mean diameter of the 16 balls is greater than 57.16 mm .

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 .

6 The weight, \(X\) kilograms, of sand in a bag can be modelled by a normal random variable with unknown mean \(\mu\) and known standard deviation 0.4 .

1. The sand in each of a random sample of 25 bags from a large batch is weighed. The total weight of sand in these 25 bags is found to be 497.5 kg .
   1. Construct a 98\% confidence interval for the mean weight of sand in bags in the batch.
   2. Hence comment on the claim that bags in the batch contain an average of 20 kg of sand.
   3. State why use of the Central Limit Theorem is not required in answering part (a)(i).
2. The weight, \(Y\) kilograms, of cement in a bag can be modelled by a normal random variable with mean 25.25 and standard deviation 0.35. A firm purchases 10 such bags. These bags may be considered to be a random sample.
   1. Determine the probability that the mean weight of cement in the 10 bags is less than 25 kg .
   2. Calculate the probability that the weight of cement in each of the 10 bags is more than 25 kg .
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6. The distributions of two independent discrete random variables \(X\) and \(Y\) are given in the tables: The random variable \(Z\) is defined to be the sum of one observation from \(X\) and one from \(Y\).

1. Tabulate the probability distribution for \(Z\).
2. Calculate \(\mathrm { E } ( Z )\).
3. Calculate (i) \(\mathrm { E } \left( Z ^ { 2 } \right)\), (ii) \(\operatorname { Var } ( Z )\).
4. Calculate Var (3Z-4).

3. A study was made of the heights of boys of different ages in Lancashire. The study concluded that the heights of 13 year-old boys are normally distributed with a mean of 156 cm and a variance of \(73 \mathrm {~cm} ^ { 2 }\). Find the probability that a 13 year-old boy chosen at random will be

1. more than 165 cm tall,
2. between 156 and 165 cm tall. The study also concluded that the heights of 14 year-old boys are normally distributed with a mean of 160 cm and a variance of \(79 \mathrm {~cm} ^ { 2 }\). One 13 year-old and one 14 year-old boy are chosen at random.
3. Find the probability that both boys are more than 165 cm tall.
4. State, with a reason, whether the probability that the combined height of the two boys is more than 330 cm is more or less than your answer to part (c).
   (2 marks)

4. A company offering a bicycle courier service within London collected data on the delivery times for a sample of jobs completed by staff at each of its two offices. The times, \(t\) minutes, for 20 deliveries handled by the company's Hammersmith office were summarised by $$\Sigma t = 427 , \text { and } \Sigma t ^ { 2 } = 11077$$

1. Find the mean and variance of the delivery times in this sample. The company's Holborn office handles more business, so the delivery times for a sample of 30 jobs handled by this office was taken. The mean and standard deviation of this sample were 18.5 minutes and 8.2 minutes respectively.
2. Find the mean and variance of the delivery times of the combined sample of 50 deliveries.

6. In a game two spinners are used. The score on the first spinner is given by the random variable \(A\), which has the following probability distribution:

1. State the name of this distribution.
2. Write down \(\mathrm { E } ( A )\). The score on the second spinner is given by the random variable \(B\), which has the following probability distribution:

   \(b\)	1	2	3
   \(\mathrm { P } ( B = b )\)	\(\frac { 1 } { 2 }\)	\(\frac { 1 } { 4 }\)	\(\frac { 1 } { 4 }\)

3. Find \(\mathrm { E } ( B )\). On each player's turn in the game, both spinners are used and the scores on the two spinners are added together. The total score on the two spinners is given by the random variable \(C\).
4. Show that \(\mathrm { P } ( C = 2 ) = \frac { 1 } { 6 }\).
5. Find the probability distribution of \(C\).
6. Show that \(\mathrm { E } ( C ) = \mathrm { E } ( A ) + \mathrm { E } ( B )\).

4

1. A red biased tetrahedral die is rolled. The number, \(X\), on the face on which it lands has the probability distribution given by

   \(\boldsymbol { x }\)	1	2	3	4
   \(\mathbf { P } ( \boldsymbol { X } = \boldsymbol { x } )\)	0.2	0.1	0.4	0.3

   1. Calculate \(\mathrm { E } ( X )\) and \(\operatorname { Var } ( X )\).
   2. The red die is now rolled three times. The random variable \(S\) is the sum of the three numbers obtained. Find \(\mathrm { E } ( S )\) and \(\operatorname { Var } ( S )\).
2. A blue biased tetrahedral die is rolled. The number, \(Y\), on the face on which it lands has the probability distribution given by $$\mathrm { P } ( Y = y ) = \begin{cases} \frac { y } { 20 } & y = 1,2 \text { and } 3 \\ \frac { 7 } { 10 } & y = 4 \end{cases}$$ The random variable \(T\) is the value obtained when the number on the face on which it lands is multiplied by 3 . Calculate \(\mathrm { E } ( T )\) and \(\operatorname { Var } ( T )\).
3. Calculate:
   1. \(\mathrm { P } ( X > 1 )\);
   2. \(\mathrm { P } ( X + T \leqslant 9\) and \(X > 1 )\);
   3. \(\mathrm { P } ( X + T \leqslant 9 \mid X > 1 )\).