Statistics and probability

A machine fills bottles of sparkling water. The volume, V ml, in a bottle is normally distributed with mean 500 ml and unknown standard deviation σ ml. Quality checks show that 3% of bottles contain less than 490 ml.

Each working day Priya travels to work either by bicycle (C) or by bus (B). If she cycles one day, the probability that she cycles the next day is 0.7. If she takes the bus one day, the probability that she cycles the next day is 0.4.

A machine fills bags of rice. The label states that each bag contains 500 g. The masses of the bags filled are normally distributed with a known standard deviation of 4.8 g. A consumer group weighs a random sample of 40 bags and finds a sample mean of 502.3 g.

A helpline receives calls at random at an average rate of 3.2 calls per 10-minute period. Let X be the number of calls in a 10-minute period.

A bus journey has two stages. The time for stage 1, A minutes, is normally distributed with mean 12 and standard deviation 1.5. The time for stage 2, B minutes, is normally distributed with mean 8 and standard deviation 2. The two times are independent. Let T = A + B.

During a meteor shower, meteors are seen by an observer at random at an average rate of 6 per hour. Assume the number of meteors seen in any period follows a Poisson distribution.

The peak running speeds of 16 randomly chosen players in a youth football league are measured. The sample mean is 31.4 km h⁻¹ and the sample standard deviation is 1.8 km h⁻¹. The speeds may be assumed to be normally distributed.

Each day Noor travels to the sports centre either by train or by bus. If she takes the train one day, the probability she takes the train the next day is 0.7. If she takes the bus one day, the probability she takes the train the next day is 0.4. She takes the train on day 1.

The area N cm² covered by algae in a pond sample, t days after it was first measured, is modelled by N = 120 × 1.18^t. For this model R² = 0.994.

Swarms leave an apiary at random times during May. The number of swarms per week is modelled by a Poisson distribution with mean 2.4.

A beekeeper fills jars with honey. The mass of honey in a jar is normally distributed with mean 454 g and standard deviation 5 g. The mass of an empty jar is normally distributed with mean 310 g and standard deviation 4 g. The two masses are independent. Let \(M\) be the mass of a full jar.

Each Grand Prix weekend race day is classed as Dry (D) or Wet (W) at a certain circuit. Weather on successive race days is modelled by a Markov chain with \(\mathbf{s}_{n+1}=\mathbf{T}\mathbf{s}_n\), in which each column of \(\mathbf{T}\) gives the probabilities of tomorrow's weather. In the order D, W:
\(\mathbf{T}=\begin{pmatrix}0.7&a\\0.3&0.6\end{pmatrix}\).

The waiting times, in minutes, of a random sample of 20 patients at a clinic have mean 41.6 and a sample standard deviation (dividing by \(n\)) of 7.8. The waiting times are assumed to be normally distributed.

Explosive eruptions at Mount Thessa occur independently and at random, at a constant mean rate of 2.4 per year. The number of eruptions in a given period is modelled by a Poisson distribution. Give all probabilities to 3 significant figures.

A volcanologist records the mass of sulfur dioxide released by the Orlo volcano, in tonnes, on 12 randomly chosen days:
412, 389, 455, 431, 398, 467, 420, 441, 405, 436, 473, 394.
The daily mass may be assumed to be normally distributed.

A theatre sells season subscriptions. At the end of each season, a fraction a of subscribers do not renew, and 15% of non-subscribers take out a subscription. Each person is either a subscriber or a non-subscriber, and the changes are modelled by a Markov chain with transition matrix T, with the first row and column representing subscribers and the second non-subscribers; columns give the probabilities of moving from each state.

At an archaeological dig, the number of bronze coins found by a team in one day is modelled by a Poisson distribution with mean 3.2. Daily counts are independent. Give answers to 3 significant figures.

In a cycling time trial, a rider's time for stage A, A minutes, has mean 42 and standard deviation 3.5. Her time for stage B, B minutes, has mean 55 and standard deviation 4.2. The two stage times are independent. Let \(T=A+B\) be her total time.

Each night the stage lighting rig in a theatre is either working (W) or faulty (F). If it is working one night, the probability that it is working the next night is 0.9. If it is faulty one night, the probability that it is repaired and working the next night is 0.6.

A mountain rescue team receives call-outs at random, independently, at a constant average rate of 3.2 per day during the summer season. Let X be the number of call-outs received on a randomly chosen summer day.

A school canteen offers two lunch options each day: vegetarian (V) or standard (S). Each pupil chooses one option per day. Pupils' choices follow a Markov chain with transition matrix T, in which each column gives the probabilities of the next day's choice, as shown:
\(T=\begin{pmatrix}0.92&0.15\\0.08&0.85\end{pmatrix}\) where the first row and column correspond to V and the second to S. On day 0, 40% of pupils choose V and 60% choose S.

A food scientist compares the protein content, in g per 100 g, of loaves made from two flours. Six loaves are made with flour A and seven with flour B. The results are:
Flour A: 11.8, 12.4, 12.1, 11.6, 12.9, 12.3
Flour B: 12.9, 13.4, 12.7, 13.8, 13.1, 12.6, 13.5
Assume that both populations are normally distributed with equal variances. A two-tailed pooled two-sample t-test is carried out at the 5% significance level.

A sauce is left to cool in a kitchen. The temperature excess, y °C, of the sauce above room temperature is recorded at time t minutes:
t: 0, 5, 10, 15, 20, 25
y: 92, 71, 56, 43, 33, 26
The data are modelled by \(y=ab^t\).

Light aircraft land at a small grass airstrip at random times, at a constant average rate of 3.2 landings per hour. Let \(X\) be the number of landings in a one-hour period.

Each week, listeners of an online radio station choose either the Classical channel (C) or the Jazz channel (J). The transition matrix is \(\mathbf{T} = \begin{pmatrix} 0.8 & p \\ 0.2 & 1-p \end{pmatrix}\), where each column gives the probabilities of moving from that channel (first column from C, second from J) and the first row is the probability of moving to C. In the long run, 60% of listeners choose Classical.

A flying club records the take-off mass \(m\) (in tonnes) of seven cargo aircraft and the ground run \(x\) (in metres) each needs on the same runway.
\(m\): 42, 48, 55, 61, 66, 72, 79
\(x\): 620, 710, 830, 910, 1010, 1100, 1230

The mass of a hen's egg from a large farm is modelled by a normal distribution with mean \(\mu\) g and standard deviation \(\sigma\) g. 10.6% of eggs have a mass below 52 g and 4.01% of eggs have a mass above 64 g.

Small explosions at a volcanic crater occur randomly and independently, at a constant mean rate of 2.5 per day. Let X be the number of explosions in a given day, and assume X ~ Po(2.5). Give all answers to 3 significant figures.

A new musical opens at a theatre. The average nightly attendance y in each of the first eight weeks (x = 1, 2, …, 8) is shown below.

The annual container throughput T (thousands of containers) of a port in year t, where t = 0 is 2015, is modelled by T = 412 × 1.058^t. The model was fitted by exponential regression to data for t = 0 to 7, and has R² = 0.962.

A theatre has 45 seats for a small studio performance but accepts 48 bookings. Each booked person independently fails to attend with probability 0.07. Let X be the number of people who fail to attend.

Bird strikes on aircraft at a small regional airport are recorded. The number of strikes in one week, X, is modelled by a Poisson distribution with mean 2.4.

A cargo aircraft is loaded with 8 freight boxes. The mass of each box, in kg, is modelled as N(24, 3.5²), and box masses are independent. Let T be the total mass of the 8 boxes.

On a farm, each year a field is planted with either wheat (W) or barley (B). If the field was planted with wheat one year, the probability that it is planted with wheat the next year is 0.3. If it was planted with barley, the probability that it is planted with wheat the next year is 0.5. In year 1 the field is planted with wheat. Use a transition matrix T whose columns give the probabilities of moving from each crop, in the order W, B.

A grower compares two fertilisers, A and B, on tomato plants. The yield per plant (kg) is normally distributed with the same variance for both fertilisers. Summary data are:
Fertiliser A: n = 10, mean 6.42, standard deviation 0.58.
Fertiliser B: n = 12, mean 5.97, standard deviation 0.71.
(The standard deviations are unbiased sample estimates.) The grower tests, at the 5% significance level, whether fertiliser A gives a greater mean yield than B.

The controller of a warehouse robot arm logs software faults. The number of faults in one week may be modelled by a Poisson distribution with mean 2.4. Faults in different weeks are independent.

A fashion studio cuts fabric for dresses. The length of fabric used for one dress is normally distributed with mean 2.4 m and standard deviation 0.15 m. The lengths used for different dresses are independent. Let T be the total length used for three dresses.

A fashion company tests a new fabric treatment. The number of wash cycles survived before visible fading is recorded for a random sample of fabric pieces. Both populations are assumed to be normally distributed with equal variances.

A fisheries officer classifies each day at a harbour as Good (G) or Poor (P) for catches. The day-to-day changes are modelled by a Markov chain with transition matrix

The number of radio dropouts, \(X\), between a Formula 1 car and its pit wall during a race follows a Poisson distribution with mean \(\lambda\). Data from many races show that \(P(X=0)=0.2\).

In a hospital outpatient clinic, the time a patient spends at triage is \(X\) minutes, with \(E(X)=12\) and \(\mathrm{Var}(X)=16\). The time in consultation is \(Y\) minutes, with \(E(Y)=20\) and \(\mathrm{Var}(Y)=25\). \(X\) and \(Y\) are independent. The clinic estimates the cost of a visit, in euros, as \(C=2X+3Y\).

A beekeeper models the number of bees \(y\) (in thousands) in a colony against hive volume \(x\) litres by \(y=1.8x^{0.65}\), for \(10\le x\le 60\). For this model, \(R^2=0.97\).

At a regional clinic, patients presenting with a rare tropical infection arrive independently at a constant mean rate of 3.2 per day. The number of such patients in one day, X, is modelled by a Poisson distribution.

A retailer packs a jar of honey and a gift box together for a parcel. The mass of a jar, J, is normally distributed with mean 450 g and standard deviation 6 g. The mass of a box, B, is normally distributed with mean 85 g and standard deviation 4 g. J and B are independent. Let T = J + B be the total mass of a parcel.

A supermarket tracks whether shoppers buy its own-brand pasta (O) or a competitor's brand (C) each week. The transition matrix is T = (0.85, 0.30; 0.15, 0.70), written row by row, where the first column gives the probabilities of moving from O to O and from O to C, and the second column gives those for C. Each column sums to 1. At the start, 40% of shoppers buy own-brand and 60% buy the competitor's.

A café manager records the number of customers, y, in the café during the x-th hour after opening on one day.

Hour x: 1, 2, 3, 4, 5, 6, 7, 8
Customers y: 42, 63, 80, 88, 91, 85, 74, 58
The manager models the data by a quadratic function y = ax² + bx + c.

Faults on a stretch of railway track occur randomly and independently, at a constant average rate of 2.4 faults per 10 km.

The state of a commuter train on any morning is either on time (O) or delayed (D). The transition matrix is
\(\mathbf{T}=\begin{pmatrix}0.9 & 0.35\\ 0.1 & 0.65\end{pmatrix}\), where the columns give the current state (O, D) and the rows give the next day's state (O, D), so that \(\mathbf{s}_{n+1}=\mathbf{T}\mathbf{s}_n\).

The number of daily passengers, P thousand, on a rail line is modelled by P = 62e^0.018t, where t is the number of months after 1 January 2024. The model was obtained by fitting an exponential curve to monthly data, and its coefficient of determination is R² = 0.964.

The time, T seconds, taken by swimmers in a club to swim 100 m freestyle is normally distributed with mean 58.4 and standard deviation 1.9.

A wind farm records unplanned turbine shutdowns. They occur independently and at random, at a constant mean rate of 0.8 per week. Let X be the number of unplanned shutdowns in a period of 3 weeks.

At a leisure centre, each swimmer chooses the Fast lane (F) or the Steady lane (S) for each session. If a swimmer used F last session, the probability they choose F again is 0.8. If a swimmer used S last session, the probability they choose F is p. The transition matrix, with columns giving the current state in the order F, S, is T = [[0.8, p], [0.2, 1 − p]].

The wind speed W (m s⁻¹) at a proposed turbine site is modelled by W ~ N(9.5, 3.2²). A turbine only generates power when 3.5 ≤ W ≤ 25.

A library records the daily rainfall x (mm) and the number of visitors y on 8 days.
Rainfall x: 0, 2, 5, 8, 11, 14, 18, 22
Visitors y: 214, 205, 188, 181, 160, 157, 131, 118

Dolphin sightings from a research boat are modelled by a Poisson distribution with a mean of 2.4 sightings per hour. Let \(X\) be the number of sightings in one hour.

Seals use two haul-out sites, A and B. Each day a seal is at exactly one site. If a seal is at A today, the probability that it is at A tomorrow is 0.75. If it is at B today, the probability that it is at A tomorrow is 0.40. The transition matrix is \(\mathbf{T}=\begin{pmatrix}0.75&0.40\\0.25&0.60\end{pmatrix}\), where each column gives the probabilities for a seal at the site it heads (first column: from A; second column: from B).

A courier company records whether 300 deliveries arrived on time or late, for three delivery methods.

A new delivery app records its weekly orders. The table shows the number of orders, \(y\) hundred, in week \(x\) after launch.

Eruptions of a volcano occur at random and independently, at a constant mean rate of 2.4 per year. Let X be the number of eruptions in one year.

The running time of Act 1 of a play is a random variable A ~ N(52, 4²) minutes and that of Act 2 is B ~ N(47, 5²) minutes. The two running times are independent. The interval between the acts lasts exactly 20 minutes. Let T = A + B.

A ferry company compares the crossing times, in minutes, of two operators on the same route. Six sailings by operator A took 52, 55, 49, 58, 54 and 51 minutes. Seven sailings by operator B took 57, 60, 55, 62, 58, 56 and 59 minutes. Assume that both sets of times come from normal populations with the same variance. A test is carried out at the 5 % significance level to see whether the mean crossing times differ.

The alert status of a volcano is recorded each week as Calm (C) or Alert (A). If the status is Calm one week, then it is Calm the next week with probability 0.85. If the status is Alert, then it is Calm the next week with probability 0.40. The status is Calm this week.

The height of the ash plume produced by eruptions of a volcano is modelled by a normal distribution with mean 8.4 km and standard deviation 1.6 km.

A theatre sells 40 tickets for a matinee. Independently, each ticket-holder fails to attend with probability 0.08. Let X be the number of ticket-holders who fail to attend.

At a theatre, the number of mobile phone interruptions in a performance is modelled by a Poisson distribution with mean 2.4. Let X be the number of interruptions in a performance.

A cargo ship carries two types of container. The mass of a type A container is normally distributed with mean 18.5 tonnes and standard deviation 2.1 tonnes. The mass of a type B container is normally distributed with mean 12.4 tonnes and standard deviation 1.5 tonnes. The masses of all containers are independent. A crane lifts one type A container and two different type B containers together. Let L tonnes be the total mass lifted.

A camera on a mountain-top observatory records meteors from a shower. The number of meteors, X, recorded in a 10-minute interval is modelled by a Poisson distribution with mean 3.5. Assume the meteors arrive independently of each other.

A commuter train makes a journey in two legs. The time for the first leg, A minutes, is normally distributed with mean 42 and standard deviation 3.2. The time for the second leg, B minutes, is normally distributed with mean 28 and standard deviation 2.4. The two times are independent. Let T = A + B be the total journey time.

A sports scientist records the weekly training time (hours) of seven club runners and their finishing positions in a race (1 = first).
Runner: P, Q, R, S, T, U, V
Training (h): 4, 9, 6, 12, 8, 10, 5
Position: 6, 3, 5, 1, 2, 4, 7

A sports scientist records the running speed, x km h⁻¹, and the heart rate, y beats per minute, of a volunteer on a treadmill.
x: 8, 9, 10, 12, 13, 14, 16
y: 128, 134, 141, 152, 160, 165, 176

A volcano observatory records eruptions of a particular volcano. Eruptions occur independently and at random, at a constant average rate of 3.2 per year. Let X be the number of eruptions in one year, and assume X follows a Poisson distribution.

A theatre sells season subscriptions. Each year, 85% of those who held a subscription the year before renew it, and 40% of those who did not hold one the year before take one out. Let the states be S (holds a subscription) and N (does not). Use the convention that each column of the transition matrix T gives the probabilities of moving from a state, and \(\mathbf{s}_n=\mathbf{T}^n\mathbf{s}_0\).

A theatre compares what patrons spend (£) on refreshments per visit.
Matinee (10 patrons): 7.2, 6.8, 7.9, 6.5, 7.4, 8.1, 7.0, 6.9, 7.6, 7.3
Evening (9 patrons): 6.4, 7.1, 6.2, 6.9, 7.0, 6.5, 6.0, 6.8, 7.2
Assume the populations are normal with equal variances. The manager believes matinee patrons spend more on average and tests this at the 5% significance level.

A café advertises that a large cup of its filter coffee contains a mean of 110 mg of caffeine. A consumer group buys 10 large cups at random and measures the caffeine content, in mg:

92, 105, 98, 110, 101, 95, 99, 107, 103, 96

Assume the caffeine content is normally distributed.

A running coach believes a six-week training programme reduces 5 km race times. Eight runners record their times, in minutes, before and after the programme.

Runner: 1, 2, 3, 4, 5, 6, 7, 8
Before: 24.6, 27.1, 22.8, 30.2, 26.5, 25.9, 28.4, 23.7
After: 24.1, 26.0, 22.9, 29.1, 25.8, 25.2, 27.5, 23.4

The coach carries out a test at the 1% significance level.

A phone manufacturer claims that the mean battery life of its new model is 12.0 hours of continuous video playback. A reviewer suspects the true mean is lower. She tests a random sample of 10 phones and records a sample mean of 11.4 hours and an unbiased estimate of the standard deviation \(s_{n-1} = 0.85\) hours. Battery lives may be assumed to be normally distributed.

She carries out a one-sample t-test at the 5% significance level. The critical value for this test is −1.833.

The number of public electric-vehicle charging points in a city, y hundred, is recorded x years after the start of 2019.

x: 1, 2, 3, 4, 5
y: 3.1, 4.8, 7.6, 11.9, 17.2

Two models are proposed:
Model A: \(y = 3.5x - 1.4\)
Model B: \(y = 0.6x^2 + 0.1x + 2.4\)

A machine fills bags of coffee labelled 250 g. A quality controller takes every 50th bag from the production line until she has 12 bags. She weighs each one and records the mass x grams. The results give \(\sum x = 3018\) and \(\sum x^2 = 759\,160\).

Priya travels to work each weekday either by bicycle (C) or by bus (B). If she cycles on one day, the probability that she cycles the next day is 0.7. If she takes the bus on one day, the probability that she takes the bus the next day is 0.6. Assume her choice depends only on the previous day's choice.

The number of downloads, N, of a new mobile game was recorded at the end of each of its first 9 weeks after launch, where t is the number of weeks. A developer suspects exponential growth, so she calculates ln N for each week. The regression line of ln N on t is
\(\ln N = 0.231t + 5.12\)
with correlation coefficient r = 0.994.

Faults occur at random along an underground cable at an average rate of 0.8 faults per kilometre.

Customers choose one of three streaming services A, B or C each month. The monthly transition matrix is

\(T=\begin{pmatrix}0.7&0.2&0.1\\0.2&0.6&0.3\\0.1&0.2&0.6\end{pmatrix}\), with columns giving the probabilities of moving from one service (A, B, C).

Initially the market shares are A: 50%, B: 30%, C: 20%.

In a triathlon relay, the swim leg time S and the run leg time R of a team, both in minutes, are independent with S ~ N(24, 2²) and R ~ N(41, 3²). Let T = S + R.

An emergency department receives patients at random, at an average rate of 6 per hour. The number of arrivals in a given period is modelled by a Poisson distribution.

In a Formula 1 qualifying session, the lap times of Mara are independent and normally distributed with mean 92.4 s and standard deviation 0.8 s. Her rival Teo's lap times are independent and normally distributed with mean 92.1 s and standard deviation 0.7 s, independent of Mara's.

A beekeeper records the honey yield, in kg, from a random sample of 12 hives:
28.4, 31.2, 26.9, 33.5, 29.8, 30.6, 27.3, 32.1, 28.8, 34.0, 29.5, 31.7.
Assume the yields are normally distributed.

Ships arrive at the port of Calvera at random and independently, at a mean rate of 5.5 per day. The number of arrivals per day is modelled by a Poisson distribution. Give probabilities to 3 significant figures.

A theatre advertises that its performances of a play last 150 minutes on average. The running times, in minutes, of 10 randomly selected performances were:
152, 156, 149, 158, 154, 151, 160, 147, 155, 157.
Assume running times are normally distributed. A one-sample t-test is carried out at the 5% significance level to decide whether the mean running time differs from the advertised time.

A shipping company cleaned the hulls of 8 ships. Each ship's daily fuel consumption (tonnes) on the same route was recorded before and after cleaning.
Ship 1 to 8, before: 38.2, 41.5, 36.8, 44.0, 39.7, 42.3, 37.5, 40.9
Ship 1 to 8, after: 36.9, 40.8, 37.0, 41.7, 38.1, 41.2, 36.2, 39.3
Differences may be assumed normally distributed. The company carries out a paired t-test at the 5% significance level, with d = before − after.

The alert level of the Thessa volcano is reviewed each day and is Green, Yellow or Red. The day-to-day changes are modelled by a Markov chain with transition matrix T, where the columns give the probabilities of moving from Green, Yellow and Red respectively, and the rows give the probabilities of moving to Green, Yellow and Red respectively:
\(T=\begin{pmatrix}0.80&0.30&0.10\\0.15&0.50&0.30\\0.05&0.20&p\end{pmatrix}\)
Use \(\mathbf{s}_{n}=T^{n}\mathbf{s}_{0}\) for the state vector after n days. Give answers to 3 significant figures where not exact.

A theatre compares two brands of stage-lighting bulb. The lifetimes, in hours, of random samples were:
Brand X (8 bulbs): 1180, 1225, 1310, 1255, 1190, 1270, 1235, 1205
Brand Y (9 bulbs): 1290, 1345, 1260, 1380, 1315, 1330, 1275, 1360, 1300
Lifetimes for both brands are assumed to be normally distributed with the same population variance. A two-sample t-test (pooled) is used at the 5% significance level to test whether the mean lifetimes differ.

A data centre has two independent server halls. In hall A the number of drive failures in a week follows a Poisson distribution with mean 1.4; in hall B it follows a Poisson distribution with mean 0.9. Let T be the total number of drive failures in the data centre in one week. Weekly totals in different weeks are independent.

A web server handles two types of request. The response time of a type 1 request, \(R_1\) milliseconds, is normally distributed with mean 120 and standard deviation 15. The response time of a type 2 request, \(R_2\) milliseconds, is normally distributed with mean 95 and standard deviation 12. The two times are independent.