Same variable, two observations

5

1. Two random variables \(X\) and \(Y\) have the independent distributions \(\mathrm { N } ( 7,3 )\) and \(\mathrm { N } ( 6,2 )\) respectively. A random value of each variable is taken. Find the probability that the two values differ by more than 2 .
2. Each candidate's overall score in a science test is calculated as follows. The mark for theory is denoted by \(T\), the mark for practical is denoted by \(P\), and the overall score is given by \(T + 1.5 P\). The variables \(T\) and \(P\) are assumed to be independent with distributions \(\mathrm { N } ( 62,158 )\) and \(\mathrm { N } ( 42,108 )\) respectively. You should assume that no continuity corrections are needed when using these distributions.
   1. A pass is awarded to candidates whose overall score is at least 90 . Find the proportion of candidates who pass.
   2. Comment on the assumption that the variables \(T\) and \(P\) are independent.

5 The heights of buildings in a large city are normally distributed with mean 18.3 m and standard deviation 2.5 m .

1. Find the probability that the total height of 5 randomly chosen buildings in the city is more than 95 m .
2. Find the probability that the difference between the heights of two randomly chosen buildings in the city is less than 1 m .

5 The thickness of books in a large library is normally distributed with mean 2.4 cm and standard deviation 0.3 cm .

1. Find the probability that the total thickness of 6 randomly chosen books is more than 16 cm .
2. Find the probability that the thickness of a book chosen at random is less than 1.1 times the thickness of a second book chosen at random.

5 Cans of drink are packed in boxes, each containing 4 cans. The weights of these cans are normally distributed with mean 510 g and standard deviation 14 g . The weights of the boxes, when empty, are independently normally distributed with mean 200 g and standard deviation 8 g .

1. Find the probability that the total weight of a full box of cans is between 2200 g and 2300 g .
2. Two cans of drink are chosen at random. Find the probability that they differ in weight by more than 20 g .

4 The masses of a certain variety of potato are normally distributed with mean 180 g and variance \(1550 \mathrm {~g} ^ { 2 }\). Two potatoes of this variety are chosen at random. Find the probability that the mass of one of these potatoes is at least twice the mass of the other.

1

1. Sarah travels home from work each evening by bus; there is a bus every 20 minutes. The time at which Sarah arrives at the bus stop varies randomly in such a way that the probability density function of \(X\), the length of time in minutes she has to wait for the next bus, is given by $$\mathrm { f } ( x ) = k ( 20 - x ) \text { for } 0 \leqslant x \leqslant 20 \text {, where } k \text { is a constant. }$$
   1. Find \(k\). Sketch the graph of \(\mathrm { f } ( x )\) and use its shape to explain what can be deduced about how long Sarah has to wait.
   2. Find the cumulative distribution function of \(X\) and hence, or otherwise, find the probability that Sarah has to wait more than 10 minutes for the bus.
   3. Find the median length of time that Sarah has to wait.
2. 1. Define the term 'simple random sample'.
   2. Explain briefly how to carry out cluster sampling.
   3. A researcher wishes to investigate the attitudes of secondary school pupils to pollution. Explain why he might prefer to collect his data using a cluster sample rather than a simple random sample. An electronics company purchases two types of resistor from a manufacturer. The resistances of the resistors (in ohms) are known to be Normally distributed. Type A have a mean of 100 ohms and standard deviation of 1.9 ohms. Type B have a mean of 50 ohms and standard deviation of 1.3 ohms.
   4. Find the probability that the resistance of a randomly chosen resistor of type A is less than 103 ohms.
   5. Three resistors of type A are chosen at random. Find the probability that their total resistance is more than 306 ohms.
   6. One resistor of type A and one resistor of type B are chosen at random. Find the probability that their total resistance is more than 147 ohms.
   7. Find the probability that the total resistance of two randomly chosen type B resistors is within 3 ohms of one randomly chosen type A resistor.
   8. The manufacturer now offers type C resistors which are specified as having a mean resistance of 300 ohms. The resistances of a random sample of 100 resistors from the first batch supplied have sample mean 302.3 ohms and sample standard deviation 3.7 ohms. Find a \(95 \%\) confidence interval for the true mean resistance of the resistors in the batch. Hence explain whether the batch appears to be as specified.

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 \(\mathrm { P } ( 4 T > 100 + 3 M )\)

7. A manufacturer produces two flavours of soft drink, cola and lemonade. The weights, \(C\) and \(L\), in grams, of randomly selected cola and lemonade cans are such that \(C \sim \mathrm {~N} ( 350,8 )\) and \(L \sim \mathrm {~N} ( 345,17 )\).

1. Find the probability that the weights of two randomly selected cans of cola will differ by more than 6 g . One can of each flavour is selected at random.
2. Find the probability that the can of cola weighs more than the can of lemonade. Cans are delivered to shops in boxes of 24 cans. The weights of empty boxes are normally distributed with mean 100 g and standard deviation 2 g .
3. Find the probability that a full box of cola cans weighs between 8.51 kg and 8.52 kg .
4. State an assumption you made in your calculation in part (c).

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.

4. The weights of eggs are normally distributed with mean 60 g and standard deviation 5 g Sairah chooses 2 eggs at random.

1. Find the probability that the difference in weight of these 2 eggs is more than 2 g
   (5) Sairah is packing eggs into cartons. The weight of an empty egg carton is normally distributed with mean 40 g and standard deviation 1.5 g
2. Find the distribution of the total weight of a carton filled with 12 randomly chosen eggs.
3. Find the probability that a randomly chosen carton, filled with 12 randomly chosen eggs, weighs more than 800 g