Expectation and variance with context application

2 An examination consists of a written paper and a practical test. The written paper marks ( \(M\) ) have mean 54.8 and standard deviation 16.0. The practical test marks ( \(P\) ) are independent of the written paper marks and have mean 82.4 and standard deviation 4.8. The final mark is found by adding \(75 \%\) of \(M\) to \(25 \%\) of \(P\). Find the mean and standard deviation of the final marks for the examination. [3]

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$.

4 The masses, in kilograms, of chemicals \(A\) and \(B\) produced per day by a factory are modelled by the independent random variables \(X\) and \(Y\) respectively, where \(X \sim \mathrm {~N} ( 10.3,5.76 )\) and \(Y \sim \mathrm {~N} ( 11.4,9.61 )\). The income generated by the chemicals is \(
) 2.50\( per kilogram for \)A\( and \)\\( 3.25\) per kilogram for \(B\).

1. Find the mean and variance of the daily income generated by chemical \(A\).
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2. Find the probability that, on a randomly chosen day, the income generated by chemical \(A\) is greater than the income generated by chemical \(B\).

4 The masses, in kilograms, of chemicals \(A\) and \(B\) produced per day by a factory are modelled by the independent random variables \(X\) and \(Y\) respectively, where \(X \sim \mathrm {~N} ( 10.3,5.76 )\) and \(Y \sim \mathrm {~N} ( 11.4,9.61 )\). The income generated by the chemicals is \(
) 2.50\( per kilogram for \)A\( and \)\\( 3.25\) per kilogram for \(B\).

1. Find the mean and variance of the daily income generated by chemical \(A\).
2. Find the probability that, on a randomly chosen day, the income generated by chemical \(A\) is greater than the income generated by chemical \(B\).

2 A mathematics module is assessed by an examination and by coursework. The examination makes up \(75 \%\) of the total assessment and the coursework makes up \(25 \%\). Examination marks, \(X\), are distributed with mean 53.2 and standard deviation 9.3. Coursework marks, \(Y\), are distributed with mean 78.0 and standard deviation 5.1. Examination marks and coursework marks are independent. Find the mean and standard deviation of the combined mark \(0.75 X + 0.25 Y\).

1 At an internet café, the charge for using a computer is 5 cents per minute. The number of minutes for which people use a computer has mean 23 and standard deviation 8.

1. Find, in cents, the mean and standard deviation of the amount people pay when using a computer.
2. Each day, 15 people use computers independently. Find, in cents, the mean and standard deviation of the total amount paid by 15 people.

6 The numbers of barrels of oil, in millions, extracted per day in two oil fields \(A\) and \(B\) are modelled by the independent random variables \(X\) and \(Y\) respectively, where \(X \sim \mathrm {~N} \left( 3.2,0.4 ^ { 2 } \right)\) and \(Y \sim \mathrm {~N} \left( 4.3,0.6 ^ { 2 } \right)\). The income generated by the oil from the two fields is \(
) 90\( per barrel for \)A\( and \)\\( 95\) per barrel for \(B\).

1. Find the mean and variance of the daily income, in millions of dollars, generated by field \(A\). [3]
2. Find the probability that the total income produced by the two fields in a day is at least \(
   ) 670$ million.

4 The cost of electricity for a month in a certain town under scheme \(A\) consists of a fixed charge of 600 cents together with a charge of 5.52 cents per unit of electricity used. Stella uses scheme \(A\). The number of units she uses in a month is normally distributed with mean 500 and variance 50.41.

1. Find the mean and variance of the total cost of Stella's electricity in a randomly chosen month. Under scheme \(B\) there is no fixed charge and the cost in cents for a month is normally distributed with mean 6600 and variance 421. Derek uses scheme \(B\).
2. Find the probability that, in a randomly chosen month, Derek spends more than twice as much as Stella spends.

5 The marks of candidates in Mathematics and English in 2009 were represented by the independent random variables \(X\) and \(Y\) with distributions \(\mathrm { N } \left( 28,5.6 ^ { 2 } \right)\) and \(\mathrm { N } \left( 52,12.4 ^ { 2 } \right)\) respectively. Each candidate's marks were combined to give a final mark \(F\), where \(F = X + \frac { 1 } { 2 } Y\).

1. Find \(\mathrm { E } ( F )\) and \(\operatorname { Var } ( F )\).
2. The final marks of a random sample of 10 candidates from Grinford in 2009 had a mean of 49. Test at the 5\% significance level whether this result suggests that the mean final mark of all candidates from Grinford in 2009 was lower than elsewhere.

3 The cost of hiring a bicycle consists of a fixed charge of 500 cents together with a charge of 3 cents per minute. The number of minutes for which people hire a bicycle has mean 142 and standard deviation 35.

1. Find the mean and standard deviation of the amount people pay when hiring a bicycle.
2. 6 people hire bicycles independently. Find the mean and standard deviation of the total amount paid by all 6 people.

4 Each week a farmer sells \(X\) litres of milk and \(Y \mathrm {~kg}\) of cheese, where \(X\) and \(Y\) have the independent distributions \(\mathrm { N } \left( 1520,53 ^ { 2 } \right)\) and \(\mathrm { N } \left( 175,12 ^ { 2 } \right)\) respectively.

1. Find the mean and standard deviation of the total amount of milk that the farmer sells in 4 randomly chosen weeks. During a year when milk prices are low, the farmer makes a loss of 2 cents per litre on milk and makes a profit of 21 cents per kg on cheese, so the farmer's overall weekly profit is \(( 21 Y - 2 X )\) cents.
2. Find the probability that, in a randomly chosen week, the farmer's overall profit is positive.

2 The operator of a section of motorway toll road records its weekly takings according to the types of vehicles using the motorway. For purposes of charging, there are three types of vehicle: cars, coaches, lorries. The weekly takings (in thousands of pounds) for each type are assumed to be Normally distributed. These distributions are independent of each other and are summarised in the table.

1. Find the probability that the weekly takings for coaches are less than \(\pounds 40000\).
2. Find the probability that the weekly takings for lorries exceed the weekly takings for cars.
3. Find the probability that over a 4 -week period the total takings for cars exceed \(\pounds 225000\). What assumption must be made about the four weeks?
4. Each week the operator allocates part of the takings for repairs. This is determined for each type of vehicle according to estimates of the long-term damage caused. It is calculated as follows: \(5 \%\) of takings for cars, \(10 \%\) for coaches and \(20 \%\) for lorries. Find the probability that in any given week the total amount allocated for repairs will exceed \(\pounds 20000\).

1

1. Let \(X\) be a random variable with variance \(\sigma ^ { 2 }\). The independent random variables \(X _ { 1 }\) and \(X _ { 2 }\) are both distributed as \(X\). Write down the variances of \(X _ { 1 } + X _ { 2 }\) and \(2 X\); explain why they are different. A large company has produced an aptitude test which consists of three parts. The parts are called mathematical ability, spatial awareness and communication. The scores obtained by candidates in the three parts are continuous random variables \(X , Y\) and \(W\) which have been found to have independent Normal distributions with means and standard deviations as shown in the table.

   	Mean	Standard deviation
   Mathematical ability, \(X\)	30.1	5.1
   Spatial awareness, \(Y\)	25.4	4.2
   Communication, \(W\)	28.2	3.9

2. Find the probability that a randomly selected candidate obtains a score of less than 22 in the mathematical ability part of the test.
3. Find the probability that a randomly selected candidate obtains a total score of at least 100 in the whole test.
4. For a particular role in the company, the score \(2 X + Y\) is calculated. Find the score that is exceeded by only \(2 \%\) of candidates.
5. For a different role, a candidate must achieve a score in communication which is at least \(60 \%\) of the score obtained in mathematical ability. What proportion of candidates do not achieve this?

5 In the manufacture of desk drawer fronts, a machine cuts sheets of veneered chipboard into rectangular pieces of width \(W\) millimetres and height \(H\) millimetres. The 4 edges of each of these pieces are then covered with matching veneered tape. The distributions of \(W\) and \(H\) are such that $$\mathrm { E } ( W ) = 350 \quad \operatorname { Var } ( W ) = 5 \quad \mathrm { E } ( H ) = 210 \quad \operatorname { Var } ( H ) = 4 \quad \rho _ { W H } = 0.75$$

1. Calculate the mean and the variance of the length of tape, \(T = 2 W + 2 H\), needed for the edges of a drawer front.
2. A desk has 4 such drawers whose sizes may be assumed to be independent. Given that \(T\) may be assumed to be normally distributed, determine the probability that the total length of tape needed for the edges of the desk's 4 drawer fronts does not exceed 4.5 metres.
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