2.04e Normal distribution: as model N(mu, sigma^2)

6 The battery in Sue's phone runs out at random moments. Over a long period, she has found that the battery runs out, on average, 3.3 times in a 30-day period.

1. Find the probability that the battery runs out fewer than 3 times in a 25-day period.
2. (a) Use an approximating distribution to find the probability that the battery runs out more than 50 times in a year ( 365 days).
   (b) Justify the approximating distribution used in part (ii)(a).
3. Independently of her phone battery, Sue's computer battery also runs out at random moments. On average, it runs out twice in a 15-day period. Find the probability that the total number of times that her phone battery and her computer battery run out in a 10-day period is at least 4 .

3 The length, in centimetres, of a certain type of snake is modelled by the random variable \(X\) with mean 52 and standard deviation 6.1. A random sample of 75 snakes is selected, and the sample mean, \(\bar { X }\), is found.

1. Find \(\mathrm { P } ( 51 < \bar { X } < 53 )\).
2. Explain why it was necessary to use the Central Limit theorem in the solution to part (i).

6 The masses, in kilograms, of cartons of sugar and cartons of flour have the distributions \(\mathrm { N } \left( 78.8,12.6 ^ { 2 } \right)\) and \(\mathrm { N } \left( 62.0,10.0 ^ { 2 } \right)\) respectively.

1. The standard load for a certain crane is 8 cartons of sugar and 3 cartons of flour. The maximum load that can be carried safely by the crane is 900 kg . Stating a necessary assumption, find the percentage of standard loads that will exceed the maximum safe load.
2. Find the probability that a randomly chosen carton of sugar has a smaller mass than a randomly chosen carton of flour.

4 Each year a transport firm uses \(X\) litres of gasoline and \(Y\) litres of diesel fuel, where \(X\) and \(Y\) have the independent distributions \(X \sim \mathrm {~N} ( 10700,950 ) ^ { 2 }\) and \(Y \sim \mathrm {~N} \left( 13400,1210 ^ { 2 } \right)\).

1. Find the probability that in a randomly chosen year the firm uses more gasoline than diesel fuel.
   The costs per litre of gasoline and diesel fuel are \\(0.80 and \\)0.85 respectively.
2. Find the probability that the total cost of gasoline and diesel fuel in a randomly chosen year is between \(\\) 20000\( and \)\\( 22000\).

7 The volume of liquid in cans of cola is normally distributed with mean 330 millilitres and standard deviation 5.2 millilitres. The volume of liquid in bottles of tonic water is normally distributed with mean 500 millilitres and standard deviation 7.1 millilitres.

1. Find the probability that 3 randomly chosen cans of cola contain less liquid than 2 randomly chosen bottles of tonic water.
2. A new drink is made by mixing the contents of 2 cans of cola with half a bottle of tonic water. Find the probability that the volume of the new drink is more than 900 millilitres.

2 A computer user finds that unwanted emails arrive randomly at a uniform average rate of 1.27 per hour.

1. Find the probability that more than 1 unwanted email arrives in a period of 5 hours.
2. Find the probability that more than 850 unwanted emails arrive in a period of 700 hours.

3 An airline knows that some people who have bought tickets may not arrive for the flight. The airline therefore sells more tickets than the number of seats that are available. For one flight there are 210 seats available and 213 people have bought tickets. The probability of any person who has bought a ticket not arriving for the flight is \(\frac { 1 } { 50 }\).

1. By considering the number of people who do not arrive for the flight, use a suitable approximation to calculate the probability that more people will arrive than there are seats available. Independently, on another flight for which 135 people have bought tickets, the probability of any person not arriving is \(\frac { 1 } { 75 }\).
2. Calculate the probability that, for both these flights, the total number of people who do not arrive is 5 .

7

1. Random variables \(Y\) and \(X\) are related by \(Y = a + b X\), where \(a\) and \(b\) are constants and \(b > 0\). The standard deviation of \(Y\) is twice the standard deviation of \(X\). The mean of \(Y\) is 7.92 and is 0.8 more than the mean of \(X\). Find the values of \(a\) and \(b\).
2. Random variables \(R\) and \(S\) are such that \(R \sim \mathrm {~N} \left( \mu , 2 ^ { 2 } \right)\) and \(S \sim \mathrm {~N} \left( 2 \mu , 3 ^ { 2 } \right)\). It is given that \(\mathrm { P } ( R + S > 1 ) = 0.9\).
   1. Find \(\mu\).
   2. Hence find \(\mathrm { P } ( S > R )\).

1 A random variable has the distribution \(\mathrm { Po } ( 31 )\). Name an appropriate approximating distribution and state the mean and standard deviation of this approximating distribution.

4 The masses, in milligrams, of three minerals found in 1 tonne of a certain kind of rock are modelled by three independent random variables \(P , Q\) and \(R\), where \(P \sim \mathrm {~N} \left( 46,19 ^ { 2 } \right) , Q \sim \mathrm {~N} \left( 53,23 ^ { 2 } \right)\) and \(R \sim \mathrm {~N} \left( 25,10 ^ { 2 } \right)\). The total value of the minerals found in 1 tonne of rock is modelled by the random variable \(V\), where \(V = P + Q + 2 R\). Use the model to find the probability of finding minerals with a value of at least 93 in a randomly chosen tonne of rock.

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.

7 Kieran and Andreas are long-jumpers. They model the lengths, in metres, that they jump by the independent random variables \(K \sim \mathrm {~N} ( 5.64,0.0576 )\) and \(A \sim \mathrm {~N} ( 4.97,0.0441 )\) respectively. They each make a jump and measure the length. Find the probability that

1. the sum of the lengths of their jumps is less than 11 m ,
2. Kieran jumps more than 1.2 times as far as Andreas.

6 The lifetimes, in hours, of Longlive light bulbs and Enerlow light bulbs have the independent distributions \(\mathrm { N } \left( 1020,45 ^ { 2 } \right)\) and \(\mathrm { N } \left( 2800,52 ^ { 2 } \right)\) respectively.

1. Find the probability that the total of the lifetimes of 5 randomly chosen Longlive bulbs is less than 5200 hours.
2. Find the probability that the lifetime of a randomly chosen Enerlow bulb is at least 3 times that of a randomly chosen Longlive bulb.

1 The masses, in grams, of potatoes of types \(A\) and \(B\) have the distributions \(\mathrm { N } \left( 175,60 ^ { 2 } \right)\) and \(\mathrm { N } \left( 105,28 ^ { 2 } \right)\) respectively. Find the probability that a randomly chosen potato of type \(A\) has a mass that is at least twice the mass of a randomly chosen potato of type \(B\).

1 The random variable \(H\) has the distribution \(\mathrm { N } \left( \mu , 5 ^ { 2 } \right)\). It is given that \(\mathrm { P } ( H < 22 ) = 0.242\). Find the value of \(\mu\).

2 A continuous random variable has a normal distribution with mean 25.0 and standard deviation \(\sigma\). The probability that any one observation of the random variable is greater than 20,0 is 0.75 . Find the value of \(\sigma\).

3

1. The random variable \(X\) has a \(\mathrm { B } ( 60,0.02 )\) distribution. Use an appropriate approximation to find \(\mathrm { P } ( X \leqslant 2 )\).
2. The random variable \(Y\) has a \(\operatorname { Po } ( 30 )\) distribution. Use an appropriate approximation to find \(\mathrm { P } ( Y \leqslant 38 )\).

3 The continuous random variable \(T\) has mean \(\mu\) and standard deviation \(\sigma\). It is known that \(\mathrm { P } ( T < 140 ) = 0.01\) and \(\mathrm { P } ( T < 300 ) = 0.8\).

1. Assuming that \(T\) is normally distributed, calculate the values of \(\mu\) and \(\sigma\). In fact, \(T\) represents the time, in minutes, taken by a randomly chosen runner in a public marathon, in which about \(10 \%\) of runners took longer than 400 minutes.
2. State with a reason whether the mean of \(T\) would be higher than, equal to, or lower than the value calculated in part (i).

3 The random variable \(G\) has the distribution \(\mathbf { N } \left( \mu , \boldsymbol { \sigma } ^ { 2 } \right)\). One hundred observations of \(G\) are taken. The results are summarised in the following table.

1. By considering \(\mathrm { P } ( G < 40.0 )\), write down an equation involving \(\mu\) and \(\sigma\). [2]
2. Find a second equation involving \(\mu\) and \(\sigma\). Hence calculate values for \(\mu\) and \(\sigma\). [4]
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3. Explain why your answers are only estimates. [1]

4 The random variable \(G\) has mean 20.0 and standard deviation \(\sigma\). It is given that \(\mathrm { P } ( G > 15.0 ) = 0.6\). Assume that \(G\) is normally distributed.

1. (a) Find the value of \(\sigma\).
   (b) Given that \(\mathrm { P } ( G > g ) = 0.4\), find the value of \(\mathrm { P } ( G > 2 g )\).
2. It is known that no values of \(G\) are ever negative. State with a reason what this tells you about the assumption that \(G\) is normally distributed.

2 The drug EPO (erythropoetin) is taken by some athletes to improve their performance. This drug is in fact banned and blood samples taken from athletes are tested to measure their 'hematocrit level'. If the level is over 50 it is considered that the athlete is likely to have taken EPO and the result is described as 'positive'. The measured hematocrit level of each athlete varies over time, even if EPO has not been taken.

1. For each athlete in a large population of innocent athletes, the variation in measured hematocrit level is described by the Normal distribution with mean 42.0 and standard deviation 3.0.
   (A) Show that the probability that such an athlete tests positive for EPO in a randomly chosen test is 0.0038 .
   (B) Find the probability that such an athlete tests positive on at least 1 of the 7 occasions during the year when hematocrit level is measured. (These occasions are spread at random through the year and all test results are assumed to be independent.)
   (C) It is standard policy to apply a penalty after testing positive. Comment briefly on this policy in the light of your answer to part (i)(B).
2. Suppose that 1000 tests are carried out on innocent athletes whose variation in measured hematocrit level is as described in part (i). It may be assumed that the probability of a positive result in each test is 0.0038 , independently of all other test results.
   (A) State the exact distribution of the number of positive tests.
   (B) Use a suitable approximating distribution to find the probability that at least 10 tests are positive.
3. Because of genetic factors, a particular innocent athlete has an abnormally high natural hematocrit level. This athlete's measured level is Normally distributed with mean 48.0 and standard deviation 2.0. The usual limit of 50 for a positive test is to be altered for this athlete to a higher value \(h\). Find the value of \(h\) for which this athlete would test positive on average just once in 200 occasions.

3 In a large population, the diastolic blood pressure (DBP) of 5-year-old children is Normally distributed with mean 56 and standard deviation 6.5.

1. Find the probability that the DBP of a randomly selected 5-year-old child is between 52.5 and 57.5. The DBP of young adults is Normally distributed with mean 68 and standard deviation 10.
2. A 5-year-old child and a young adult are selected at random. Find the probability that the DBP of one of them is over 62 and the other is under 62.
3. Sketch both distributions on a single diagram.
4. For another age group, the DBP is Normally distributed with mean 82. The DBP of \(12 \%\) of people in this age group is below 62. Find the standard deviation for this age group.

2 The fuel economy of a car varies from day to day according to weather and driving conditions. Fuel economy is measured in miles per gallon (mpg). The fuel economy of a particular petrol-fuelled type of car is known to be Normally distributed with mean 38.5 mpg and standard deviation 4.0 mpg .

1. Find the probability that on a randomly selected day the fuel economy of a car of this type will be above 45.0 mpg .
2. The manufacturer wishes to quote a fuel economy figure which will be exceeded on \(90 \%\) of days. What figure should be quoted? The daily fuel economy of a similar type of car which is diesel-fuelled is known to be Normally distributed with mean 51.2 mpg and unknown standard deviation \(\sigma \mathrm { mpg }\).
3. Given that on 75\% of days the fuel economy of this type of car is below 55.0 mpg , show that \(\sigma = 5.63\).
4. Draw a sketch to illustrate both distributions on a single diagram.
5. Find the probability that the fuel economy of either the petrol or the diesel model (or both) will be above 45.0 mpg on a randomly selected day. You may assume that the fuel economies of the two models are independent.

1 A low-cost airline charges for breakfasts on its early morning flights. On average, \(10 \%\) of passengers order breakfast.

1. Find the probability that, out of 8 randomly selected passengers, exactly 1 orders breakfast.
2. Use a suitable Poisson approximating distribution to find the probability that the number of breakfasts ordered by 30 randomly selected passengers is
   (A) exactly 6,
   (B) at least 8 .
3. State the conditions under which the use of a Poisson distribution is appropriate as an approximation to a binomial distribution.
4. The aircraft carries 120 passengers and the flight is always full. Find the mean \(\mu\) and variance \(\sigma ^ { 2 }\) of a Normal approximating distribution suitable for modelling the total number of passengers on the flight who order breakfast.
5. Use your Normal approximating distribution to calculate the probability that more than 15 breakfasts are ordered on a particular flight.
6. The airline wishes to be at least \(99 \%\) certain that the plane will have sufficient breakfasts for all passengers who order them. Find the minimum number of breakfasts which should be carried on each flight.

2 The head circumference of 3-year-old boys is known to be Normally distributed with mean 49.7 cm and standard deviation 1.6 cm .

1. Find the probability that the head circumference of a randomly selected 3 -year-old boy will be
   (A) over 51.5 cm ,
   (B) between 48.0 and 51.5 cm .
2. Four 3-year-old boys are selected at random. Find the probability that exactly one of them has head circumference between 48.0 and 51.5 cm .
3. The head circumference of 3-year-old girls is known to be Normally distributed with mean \(\mu\) and standard deviation \(\sigma\). Given that \(60 \%\) of 3-year-old girls have head circumference below 49.0 cm and \(30 \%\) have head circumference below 47.5 cm , find the values of \(\mu\) and \(\sigma\). A nutritionist claims that boys who have been fed on a special organic diet will have a larger mean head circumference than other boys. A random sample of ten 3 -year-old boys who have been fed on this organic diet is selected. It is found that their mean head circumference is 50.45 cm .
4. Using the null and alternative hypotheses \(\mathrm { H } _ { 0 } : \mu = 49.7 \mathrm {~cm} , \mathrm { H } _ { 1 } : \mu > 49.7 \mathrm {~cm}\), carry out a test at the \(10 \%\) significance level to examine the nutritionist's claim. Explain the meaning of \(\mu\) in these hypotheses. You may assume that the standard deviation of the head circumference of organically fed 3 -year-old boys is 1.6 cm .