2.04f Find normal probabilities: Z transformation

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.

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 \(\mu\).

1 The random variable \(T\) is normally distributed with mean \(\mu\) and standard deviation \(\sigma\). It is given that \(\mathrm { P } ( T > 80 ) = 0.05\) and \(\mathrm { P } ( T > 50 ) = 0.75\). Find the values of \(\mu\) and \(\sigma\).

4 The random variable \(Y\) has the distribution \(\mathrm { N } \left( \mu , \sigma ^ { 2 } \right)\). The results of 40 independent observations of \(Y\) are summarised by $$\Sigma y = 3296.0 , \quad \Sigma y ^ { 2 } = 286800.40$$

1. Calculate unbiased estimates of \(\mu\) and \(\sigma ^ { 2 }\).
2. Use your answers to part (i) to estimate the probability that a single random observation of \(Y\) will be less than 60.0.
3. Explain whether it is necessary to know that \(Y\) is normally distributed in answering part (i) of this question.

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.

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 .

1 The random variable \(X\) represents the time taken in minutes for a haircut at a barber's shop. \(X\) is Normally distributed with mean 11 and standard deviation 3 .

1. Find \(\mathrm { P } ( X < 10 )\).
2. Find the probability that exactly 3 out of 8 randomly selected haircuts take less than 10 minutes.
3. Use a suitable approximating distribution to find the probability that at least 50 out of 100 randomly selected haircuts take less than 10 minutes. A new hairdresser joins the shop. The shop manager suspects that she takes longer on average than the other staff to do a haircut. In order to test this, the manager records the time taken for 25 randomly selected cuts by the new hairdresser. The mean time for these cuts is 12.34 minutes. You should assume that the time taken by the new hairdresser is Normally distributed with standard deviation 3 minutes.
4. Write down suitable null and alternative hypotheses for the test.
5. Carry out the test at the \(5 \%\) level.

3 The number of calls received at an office per 5 minutes is modelled by a Poisson distribution with mean 3.2.

1. Find the probability of
   (A) exactly one call in a 5 -minute period,
   (B) at least 6 calls in a 5 -minute period.
2. Find the probability of
   (A) exactly one call in a 1 -minute period,
   (B) exactly one call in each of five successive 1-minute periods.
3. Use a suitable approximating distribution to find the probability of at most 45 calls in a period of 1 hour. Two assumptions required for a Poisson distribution to be a suitable model are that calls arrive

4 The sexes and ages of a random sample of 300 runners taking part in marathons are classified as follows.

1. Carry out a test at the \(5 \%\) significance level to examine whether there is any association between age group and sex. State carefully your null and alternative hypotheses. Your working should include a table showing the contributions of each cell to the test statistic.
2. Does your analysis support the suggestion that women are less likely than men to enter marathons as they get older? Justify your answer. For marathons in general, on average \(3 \%\) of runners are 'Female, 50 and over'. The random variable \(X\) represents the number of 'Female, 50 and over' runners in a random sample of size 300.
3. Use a suitable approximating distribution to find \(\mathrm { P } ( X \geqslant 12 )\).

2 A public water supply contains bacteria. Each day an analyst checks the water quality by counting the number of bacteria in a random sample of 5 ml of water. Throughout this question, you should assume that the bacteria occur randomly at a mean rate of 0.37 bacteria per 5 ml of water.

1. Use a Poisson distribution to
   (A) find the probability that a 5 ml sample contains exactly 2 bacteria,
   (B) show that the probability that a 5 ml sample contains more than 2 bacteria is 0.0064 .
2. The month of September has 30 days. Find the probability that during September there is at most one day when a 5 ml sample contains more than 2 bacteria. The daily 5 ml sample is the first stage of the quality control process. The remainder of the process is as follows.

3 A company has a fleet of identical vans. Company policy is to replace all of the tyres on a van as soon as any one of them is worn out. The random variable \(X\) represents the number of miles driven before the tyres on a van are replaced. \(X\) is Normally distributed with mean 27500 and standard deviation 4000.

1. Find \(\mathrm { P } ( X > 25000 )\).
2. 10 vans in the fleet are selected at random. Find the probability that the tyres on exactly 7 of them last for more than 25000 miles.
3. The tyres of \(99 \%\) of vans last for more than \(k\) miles. Find the value of \(k\). A tyre supplier claims that a different type of tyre will have a greater mean lifetime. A random sample of 15 vans is fitted with these tyres. For each van, the number of miles driven before the tyres are replaced is recorded. A hypothesis test is carried out to investigate the claim. You may assume that these lifetimes are also Normally distributed with standard deviation 4000.
4. Write down suitable null and alternative hypotheses for the test.
5. For the 15 vans, it is found that the mean lifetime of the tyres is 28630 miles. Carry out the test at the \(5 \%\) level.

3 Bill and Ben run their own gardening company. At regular intervals throughout the summer they come to work on my garden, mowing the lawns, hoeing the flower beds and pruning the bushes. From past experience it is known that the times, in minutes, spent on these tasks can be modelled by independent Normally distributed random variables as follows.

1. Find the probability that, on a randomly chosen visit, it takes less than 50 minutes to mow the lawns.
2. Find the probability that, on a randomly chosen visit, the total time for hoeing and pruning is less than 50 minutes.
3. If Bill mows the lawns while Ben does the hoeing and pruning, find the probability that, on a randomly chosen visit, Ben finishes first. Bill and Ben do my gardening twice a month and send me an invoice at the end of the month.
4. Write down the mean and variance of the total time (in minutes) they spend on mowing, hoeing and pruning per month.
5. The company charges for the total time spent at 15 pence per minute. There is also a fixed charge of \(\pounds 10\) per month. Find the probability that the total charge for a month does not exceed \(\pounds 40\).

2 A bus route runs from the centre of town A through the town's urban area to a point B on its boundary and then through the country to a small town C . Because of traffic congestion and general road conditions, delays occur on both the urban and the country sections. All delays may be considered independent. The scheduled time for the journey from A to B is 24 minutes. In fact, journey times over this section are given by the Normally distributed random variable \(X\) with mean 26 minutes and standard deviation 3 minutes. The scheduled time for the journey from B to C is 18 minutes. In fact, journey times over this section are given by the Normally distributed random variable \(Y\) with mean 15 minutes and standard deviation 2 minutes. Journey times on the two sections of route may be considered independent. The timetable published to the public does not show details of times at intermediate points; thus, if a bus is running early, it merely continues on its journey and is not required to wait.

1. Find the probability that a journey from A to B is completed in less than the scheduled time of 24 minutes.
2. Find the probability that a journey from A to C is completed in less than the scheduled time of 42 minutes.
3. It is proposed to introduce a system of bus lanes in the urban area. It is believed that this would mean that the journey time from A to B would be given by the random variable \(0.85 X\). Assuming this to be the case, find the probability that a journey from A to B would be completed in less than the currently scheduled time of 24 minutes.
4. An alternative proposal is to introduce an express service. This would leave out some bus stops on both sections of the route and its overall journey time from A to C would be given by the random variable \(0.9 X + 0.8 Y\). The scheduled time from A to C is to be given as a whole number of minutes. Find the least possible scheduled time such that, with probability 0.75 , buses would complete the journey on time or early.
5. A programme of minor road improvements is undertaken on the country section. After their completion, it is thought that the random variable giving the journey time from B to C is still Normally distributed with standard deviation 2 minutes. A random sample of 15 journeys is found to have a sample mean journey time from B to C of 13.4 minutes. Provide a two-sided \(95 \%\) confidence interval for the population mean journey time from B to C .

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 \(\pounds 20000\).

2 [In this question, you may use the result \(\int _ { 0 } ^ { \infty } u ^ { m } \mathrm { e } ^ { - u } \mathrm {~d} u = m\) ! for any non-negative integer \(m\).]
The random variable \(X\) has probability density function $$\mathrm { f } ( x ) = \begin{cases} \frac { \lambda ^ { k + 1 } x ^ { k } \mathrm { e } ^ { - \lambda x } } { k ! } , & x > 0 \\ 0 , & \text { elsewhere } \end{cases}$$ where \(\lambda > 0\) and \(k\) is a non-negative integer.

1. Show that the moment generating function of \(X\) is \(\left( \frac { \lambda } { \lambda - \theta } \right) ^ { k + 1 }\).
2. The random variable \(Y\) is the sum of \(n\) independent random variables each distributed as \(X\). Find the moment generating function of \(Y\) and hence obtain the mean and variance of \(Y\). [8]
3. State the probability density function of \(Y\).
4. For the case \(\lambda = 1 , k = 2\) and \(n = 5\), it may be shown that the definite integral of the probability density function of \(Y\) between limits 10 and \(\infty\) is 0.9165 . Calculate the corresponding probability that would be given by a Normal approximation and comment briefly.

6. A manufacturer has a machine that fills bags with flour such that the weight of flour in a bag is normally distributed. A label states that each bag should contain 1 kg of flour.

1. The machine is set so that the weight of flour in a bag has mean 1.04 kg and standard deviation 0.17 kg . Find the proportion of bags that weigh less than the stated weight of 1 kg . The manufacturer wants to reduce the number of bags which contain less than the stated weight of 1 kg . At first she decides to adjust the mean but not the standard deviation so that only \(5 \%\) of the bags filled are below the stated weight of 1 kg .
2. Find the adjusted mean. The manufacturer finds that a lot of the bags are overflowing with flour when the mean is adjusted, so decides to adjust the standard deviation instead to make the machine more accurate. The machine is set back to a mean of 1.04 kg . The manufacturer wants \(1 \%\) of bags to be under 1 kg .
3. Find the adjusted standard deviation. Give your answer to 3 significant figures.

1. The resting heart rate, \(h\) beats per minute (bpm), and average length of daily exercise, \(t\) minutes, of a random sample of 8 teachers are shown in the table below.

1. State, with a reason, which variable is the response variable. The equation of the least squares regression line of \(h\) on \(t\) is $$h = 93.5 - 0.43 t$$
2. Give an interpretation of the gradient of this regression line.
3. Find the value of \(\bar { t }\) and the value of \(\bar { h }\)
4. Show that the point \(( \bar { t } , \bar { h } )\) lies on the regression line.
5. Estimate the resting heart rate of a teacher with an average length of daily exercise of 1 hour.
6. Comment, giving a reason, on the reliability of the estimate in part (e). The resting heart rate of teachers is assumed to be normally distributed with mean 73 bpm and standard deviation 8 bpm . The middle \(95 \%\) of resting heart rates of teachers lies between \(a\) and \(b\)
7. Find the value of \(a\) and the value of \(b\).
   

The birth weights, \(W\) grams, of a particular breed of kitten are assumed to be normally distributed with mean 99 g and standard deviation 3.6 g

1. Find \(\mathrm { P } ( W > 92 )\)
2. Find, to one decimal place, the value of \(k\) such that \(\mathrm { P } ( W < k ) = 3 \mathrm { P } ( W > k )\)
3. Write down the name given to the value of \(k\). For a different breed of kitten, the birth weights are assumed to be normally distributed with mean 120 g Given that the 20th percentile for this breed of kitten is 116 g
4. find the standard deviation of the birth weight of this breed of kitten.