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

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.
   

5. Rosie keeps bees. The amount of honey, in kg, produced by a hive of Rosie's bees in a season, is modelled by a normal distribution with a mean of 22 kg and a standard deviation of 10 kg .

1. Find the probability that a hive of Rosie's bees produces less than 18 kg of honey in a season. The local bee keepers' club awards a certificate to every hive that produces more than 39 kg of honey in a season, and a medal to every hive that produces more than 50 kg in a season. Given that one of Rosie's bee hives is awarded a certificate
2. find the probability that this hive is also awarded a medal.
   (5) Sam also keeps bees. The amount of honey, in kg, produced by a hive of Sam's bees in a season, is modelled by a normal distribution with mean \(\mu \mathrm { kg }\) and standard deviation \(\sigma \mathrm { kg }\). The probability that a hive of Sam's bees produces less than 28 kg of honey in a season is 0.8413 Only 20\% of Sam's bee hives produce less than 18 kg of honey in a season.
3. Find the value of \(\mu\) and the value of \(\sigma\). Give your answers to 2 decimal places.
   (6)
   
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1. In a factory, a machine is used to fill bags of rice. The weights of bags of rice are modelled using a normal distribution with mean 250 g .

Only \(1 \%\) of the bags of rice weigh more than 256 g .

1. Write down the percentage of bags of rice with weights between 244 g and 256 g .
2. Find the standard deviation of the weights of the bags of rice. An inspection consists of selecting a bag of rice at random and checking if its weight is within 4 g of the mean. If the weight is more than 4 g away from the mean, then a second bag of rice is selected at random and checked. If the weight of each of the 2 bags of rice is more than 4 g away from the mean, then the machine is shut down.
3. Find the probability that the machine is shut down following an inspection.

7. The weights, \(G\), of a particular breed of gorilla are normally distributed with mean 180 kg and standard deviation 15 kg .

1. Find the proportion of these gorillas whose weights exceed 174 kg .
2. Find, to 1 decimal place, the value of \(k\) such that \(\mathrm { P } ( k < G < 174 ) = 0.3196\) The weights, \(B\), of a particular breed of buffalo are normally distributed with mean 216 kg and standard deviation 30 kg . Given that \(\mathrm { P } ( G > w ) = \mathrm { P } ( B < w ) = p\)
3. 1. find the value of \(w\)
   2. find the value of \(p\) and standard deviation 15 kg .
      1. Find the proportion of these gorillas whose weights exceed 174 kg .
      2. Find, to 1 decimal place, the value of \(k\) such that \(\mathrm { P } ( k < G < 174 ) = 0.3196\)

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The weights of women boxers in a tournament are normally distributed with mean 64 kg and standard deviation 8 kg .

1. Find the probability that a randomly chosen woman boxer in the tournament weighs less than 51 kg . In the tournament, women boxers who weigh less than 51 kg are classified as lightweight. Ren weighs 49 kg and she has a match against another randomly selected, lightweight woman boxer.
2. Find the probability that Ren weighs less than the other boxer. In the tournament, women boxers who weigh more than \(H \mathrm {~kg}\) are classified as heavyweight. Given that \(10 \%\) of the women boxers in the tournament are classified as heavyweight,
3. find the value of \(H\).
   

3. The weights of packages that arrive at a factory are normally distributed with a mean of 18 kg and a standard deviation of 5.4 kg

1. Find the probability that a randomly selected package weighs less than 10 kg The heaviest 15\% of packages are moved around the factory by Jemima using a forklift truck.
2. Find the weight, in kg , of the lightest of these packages that Jemima will move. One of the packages not moved by Jemima is selected at random.
3. Find the probability that it weighs more than 18 kg A delivery of 4 packages is made to the factory. The weights of the packages are independent.
4. Find the probability that exactly 2 of them will be moved by Jemima.

The lengths, \(L \mathrm {~mm}\), of housefly wings are normally distributed with \(L \sim \mathrm {~N} \left( 4.5,0.4 ^ { 2 } \right)\)

1. Find the probability that a randomly selected housefly has a wing length of less than 3.86 mm .
2. Find
   1. the upper quartile ( \(Q _ { 3 }\) ) of \(L\)
   2. the lower quartile ( \(Q _ { 1 }\) ) of \(L\) A value that is greater than \(Q _ { 3 } + 1.5 \times \left( Q _ { 3 } - Q _ { 1 } \right)\) or smaller than \(Q _ { 1 } - 1.5 \times \left( Q _ { 3 } - Q _ { 1 } \right)\) is defined as an outlier.
3. Find these two outlier limits. A housefly is selected at random.
4. Using standardisation, show that the probability that this housefly is not an outlier is 0.993 to 3 decimal places. Given that this housefly is not an outlier,
5. showing your working, find the probability that the wing length of this housefly is greater than 5 mm .

The distance an athlete can throw a discus is normally distributed with mean 40 m and standard deviation 4 m

1. Using standardisation, show that the probability that this athlete throws the discus less than 38.8 m is 0.3821 This athlete enters a discus competition.
   To qualify for the final, they have 3 attempts to throw the discus a distance of more than 38.8 m
   Once they qualify, they do not use any of their remaining attempts.
   Given that they qualified for the final and that throws are independent,
2. find the probability that this athlete qualified for the final on their second throw with a distance of more than 44 m

1. The random variable \(X\) is normally distributed with mean \(\mu\) and variance 36

Given that $$\mathrm { P } ( \mu - 2 k < X < \mu + 2 k ) = 0.6$$

1. find the value of \(k\) The random variable \(Y\) is normally distributed with mean \(\mu\) and standard deviation \(\sigma\) Given that $$2 \mu = 3 \sigma ^ { 2 } \quad \text { and } \quad \mathrm { P } \left( \mathrm { Y } > \frac { 3 } { 2 } \mu \right) = 0.0668$$
2. find the value of \(\mu\) and the value of \(\sigma\)

4. The random variable \(Y \sim \mathrm {~N} \left( \mu , \sigma ^ { 2 } \right)\) Given that \(\mathrm { P } ( Y < 17 ) = 0.6\) find

1. \(\mathrm { P } ( Y > 17 )\)
2. \(\mathrm { P } ( \mu < Y < 17 )\)
3. \(\mathrm { P } ( Y < \mu \mid Y < 17 )\)

7. One event at Pentor sports day is throwing a tennis ball. The distance a child throws a tennis ball is modelled by a normal distribution with mean 32 m and standard deviation 12 m . Any child who throws the tennis ball more than 50 m is awarded a gold certificate.

1. Show that, to 3 significant figures, 6.68\% of children are awarded a gold certificate. A silver certificate is awarded to any child who throws the tennis ball more than \(d\) metres but less than 50 m . Given that 19.1\% of the children are awarded a silver certificate,
2. find the value of \(d\). Three children are selected at random from those who take part in the throwing a tennis ball event.
3. Find the probability that 1 is awarded a gold certificate and 2 are awarded silver certificates. Give your answer to 2 significant figures.

Police measure the speed of cars passing a particular point on a motorway. The random variable \(X\) is the speed of a car. \(X\) is modelled by a normal distribution with mean 55 mph (miles per hour).

1. Draw a sketch to illustrate the distribution of \(X\). Label the mean on your sketch. The speed limit on the motorway is 70 mph . Car drivers can choose to travel faster than the speed limit but risk being caught by the police. The distribution of \(X\) has a standard deviation of 20 mph .
2. Find the percentage of cars that are travelling faster than the speed limit. The fastest \(1 \%\) of car drivers will be banned from driving.
3. Show that the lowest speed, correct to 3 significant figures, for a car driver to be banned is 102 mph . Show your working clearly. Car drivers will just be given a caution if they are travelling at a speed \(m\) such that $$\mathrm { P } ( 70 < X < m ) = 0.1315$$
4. Find the value of \(m\). Show your working clearly.

2. The random variable \(X\) is normally distributed with mean \(\mu\) and variance \(\sigma ^ { 2 }\).

1. Write down 3 properties of the distribution of \(X\). Given that \(\mu = 27\) and \(\sigma = 10\)
2. find \(\mathrm { P } ( 26 < X < 28 )\).

2 A random variable \(C\) has the distribution \(\mathrm { N } \left( \mu , \sigma ^ { 2 } \right)\). A random sample of 10 observations of \(C\) is obtained, and the results are summarised as $$n = 10 , \Sigma c = 380 , \Sigma c ^ { 2 } = 14602 .$$

1. Calculate unbiased estimates of \(\mu\) and \(\sigma ^ { 2 }\).
2. Hence calculate an estimate of the probability that \(C > 40\).