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mathematics
cambridge international as & a level mathematics probability & statistics

Questions and Answers of Cambridge International AS & A Level Mathematics Probability & Statistics

The ages, x years, of 150 cars are summarised by Σx = 645 and Σx2 = 8287.5. Find (x – x̅)2, where x denotes the mean of x.
A set of data values is 152,164, 177,191, 207, 250 and 258.Compare the proportional change in the standard deviation with the proportional change in the interquartile range when the value 250 in the
A shop has in its stock 80 rectangular celebrity posters. All of these posters have a width to height ratio of 1: √2, and their mean perimeter is 231.8cm.Given that the sum of the squares of the
At a village fair, visitors were asked to guess how many sweets are in a glass jar. The best six guesses were 180, 211, 230,199, 214 and 166.a. Show that the mean of these guesses is 200, and use SD
The number of women in senior management positions at a number of companies was investigated. The number of women at each of the 25 service companies and at each of the 16 industrial companies are
The ages, a years, of the five members of the boy-band AlphaArise are such that Σ(a − 21)2 = 11.46 and Σ(a − 21) = −6.The ages, b years, of the seven members of the boy-band BetaBeat are
A total of 112 candidates took a multiple-choice test that had 40 questions. The numbers of correct answers given by the candidates are shown in the following table.a. State which class contains the
Two players, A and B, both played seven matches to reach the final of a tennis tournament. The number of games that each of them won in these matches are given in the following back-to-back
At a factory, 50-metre lengths of cotton thread are wound onto bobbins. Due to fraying, it is common for a length, l cm, of cotton to be removed after it has been wound onto a bobbin. The following
People applying to a Computing college are given an aptitude test. Those who are accepted take a progress test 3 months after the course has begun. The following table gives the aptitude test scores,
The growth of 200 tomato plants, half of which were treated with a growth hormone, was monitored over a 5-day period and is summarised in the following graphs.Use the graphs to describe two
A survey of a random sample of 23 people recorded the number of unwanted emails they received in a particular week. The results are given belowa. Represent the data in a stem-and-leaf diagram.b.
The volumes of water, x ×106 litres, needed to fill six Olympic-sized pools are 2.82, 2.50, 2.75, 3.14, 3.66 and 3.07.a. Find the value of Σ(x − 2) and of Σ(x − 2)2.b. Use your answers to
The speeds of 72 coaches at a certain point on their journeys between two cities were recorded. The results are given in the following table.a. State the number of coaches whose speeds were between
The following table shows the masses, m grams, of 100 unsealed bags of plain potato crisps.a. Show that the heights of the columns in a histogram illustrating these data must be in the ratio 2 : 3 :
The masses, in carats, of a sample of 200 pearls are summarised in the following cumulative frequency graph. One carat is equivalent to 200 milligrams.a. Use the graph to estimate, in carats:i. The
The amounts spent, S dollars, by six customers at a hairdressing salon yesterday were as follows.12.50, 15.75, 41.30, 34.20, 10.80, 40.85.Each of the customers paid with a $50 note and each received
Find the range and the interquartile range of the following sets of data.a. 5, 8, 13, 17, 22, 25, 30b. 7, 13, 21, 2, 37, 28, 17, 11, 2c. 42, 47, 39, 51, 73, 18, 83, 29, 41, 64d. 113, 97, 36, 81, 49,
Find the mean and the standard deviation for these sets of numbers.a. 27, 43, 29, 34, 53, 37,19 and 58.b. 6.2, − 8.5, 7.7, − 4.3,13.5 and −11.9.
The following table shows the number of pets owned by each of 35 families.Find the mean and variance of the number of pets. Ollies (f) No. per 1 2 4 am ilies (f) 6 12 3 4,
Given that:a. Σv2 = 5480, Σv = 288 and n = 64, find the variance of v.b. Σw2 = 4000, w̅ = 5.2 and n = 36, find the standard deviation of w.c. Σx2f = 6120, Σf = 40 and the standard
Two years ago, the standard deviation of the masses of a group of men and a group of women were 8kg and 6kg, respectively. Today, all the men are 5kg heavier and all the women are 3kg lighter. Find
The range of prices of the newspapers sold at a kiosk is $0.80. After 6p.m. all prices are reduced by 20%. Find the range of the prices after 6p.m.
Three boys and seven girls are asked how much money they have in their pockets. The boys have $2.50 each and the mean amount that the 10 children have is $3.90.a. Show that the girls have a total of
a. Find the range and the interquartile range of the dataset represented in the following box plot.b. What type of skewness would you expect this set of data to have? 2.
Last term Abraham sat three tests in each of his science subjects. His raw percentage marks for the tests, in the order they were completed, are listed.a. Calculate the variance of Abraham’s marks
A building is occupied by n companies. The number of people employed by these companies is denoted by x. Find the mean number of employees, given that Σx2 = 8900, Σx = 220 and that the standard
Twenty readings of y are summarised by the totals Σ(y − 5)2 = 890 and Σ(y − 5) = 130. Find the standard deviation of y.
Find the standard deviation of x, given that 4x2 = 14600, Σ2x = 420 and n = 20.
Jean Luc was asked to record the times of 20 athletes in a long distance race. He started his stopwatch when the race began and then went to sit in the shade, where he fell asleep. On waking, he
Twenty-five values of p are such that Σp2 = 6006 and Σp = 388, and 25 values of q are such that Σq2 = 6114 and Σq = 387. Calculate the variance of the 50 values of p and q together.
The amounts of rainfall, rmm, at a certain location were recorded on 365 consecutive days and are summarised by Σ(r − 3)2 = 9950 and Σ(r − 3) = 1795.8. Calculate the mean daily rainfall and
The values of x given in the table on the left have a standard deviation of 0.88. Find the standard deviation of the values of y.The temperatures, T° Celsius, at seven locations in the Central
The quiz marks of nine students are written down in ascending order and it is found that the range and interquartile range are equal. Find the greatest possible number of distinct marks that were
The numbers of fouls made in eight hockey matches and in eight football matches played at the weekend are shown in the following back-to-back stem-and-leaf diagram.a. Is it true to say that the
The numbers of cobs produced by 360 maize plants are shown in the following table.a. Calculate the mean and the standard deviation.b. Find the interquartile range and give an example of what it tells
In a class of 30 students, the mean mass of the 14 boys is 63.5kg and the mean mass of the girls is 57.3kg. Calculate the mean and standard deviation of the masses of all the students together, given
Exactly 20 years ago, the mean age of a group of boys was 15.7 years and the sum of the squares of their ages was 16000. If the sum of the squares of their ages has increased by 8224 in this 20-year
The temperatures, T° Celsius, at seven locations in the Central Kalahari Game Reserve were recorded at 4p.m. one January afternoon. The values of T, correct to 1 decimal place, were:32.1, 31.7,
The times spent, in minutes, by 30 girls and by 40 boys on an assignment are detailed in the following table.a. For the boys and for the girls, calculate estimates of the mean and standard
Two days before a skiing competition, the depths of snow, x metres, at 32 points on the course were measured and it was discovered that the numerical values of Σx and Σx2 were equal.a. Given that
Rishi and Daisy take the same seven tests in Mathematics, and both students’ marks improve on successive tests. Their percentage marks are as follows.Rishi’s marks
The following table shows the front tyre pressure, in psi, of five 4-wheeled vehicles, A to E.a. Show that the variance of the pressure in all of these front tyres is 7.65psi2.b. Rear tyre pressures
Readings from a device, denoted by y, are such that Σ(y − 3)2 = 2775, Σy = 105 and the standard deviation of y is 13. Find the number of readings that were taken.
Building plots are offered for sale at $315 per square metre. The seller has to pay a lawyer’s fee of $500 from the money received. Salome’s plot is 240 square metres larger than Nadia’s plot.
The following stem-and-leaf diagram shows the marks out of 50 obtained by 15 students in a Science test.a. Find the range and interquartile range of the marks.b. Illustrate the data in a
The following box plots summarise the percentage scores of a class of students in the three Mathematics tests they took this term.a. Describe the progress made by the class in Mathematics tests this
The following table shows the maximum speeds, skm/h, of some vintage cars.a. On the same sheet of graph paper, construct a cumulative frequency polygon and a box-and whisker diagram to illustrate the
The lengths, correct to the nearest centimetre, of 50 rods are given in the following table.Calculate an estimate of the standard deviation of the lengths. Length (cm) 15-17 18-24 25-29 30-37 No.
The totals Σx2 = 7931, Σx = 397 and Σy = 499 are given by 29 values of x and n values of y. All the values of x and y together have a variance of 52.a. Express y2 in terms of n.b. Find the
Mei measured the heights of her classmates and, after correctly analysing her data, she found the mean and standard deviation to be 163.8cm and 7.6cm. Decide whether or not these measures are valid,
Temperatures in degrees Celsius (°C) can be converted to temperatures in degrees Fahrenheit (°F) using the formula F = 1.8C + 32.a. The temperatures yesterday had a range of 15°C. Express this
The following table shows the mean and standard deviation of the lengths of 75 adult puff adders (Bitis arietans), which are found in Africa and on the Arabian peninsula.a. Find the mean length of
Twenty adults are selected at random, and each is asked to state the number of trips abroad that they have made. The results are shown in the following back-to-back stem-and-leaf diagram.a. i. Draw
For the dataset denoted by x in the following table, k is a constant.Find the value of k and calculate the variance of x, given that x = 17. 15 16 17 18 19 20 2k k+5 k - 3 10 8.
The five values in a dataset have a sum of 250 and standard deviation of 15. A sixth value is added to the dataset, such that the mean is now 40. Find the variance of the six values in the dataset.
A transport company runs 21 coaches between two cities every week. In the past, the mean and variance of the journey times were 4 hours 35 minutes and 53.29minutes2.What would be the mean and
Ten items were selected from each of four sections at a supermarket. Details of the prices of those items, in dollars, on 1st April and on 1st June are shown in the following table.For which
The scores obtained by 11 people throwing three darts each at a dartboard are 54, 46, 43, 52,180, 50, 41, 56, 52, 49 and 54.a. Find the range, the interquartile range and the standard deviation of
The resistances, in ohms (Ω), of 100 conductors are represented in the following graph.Find, to an appropriate degree of accuracy, an estimate of:a. The interquartile range b. The 90th
The following table illustrates the heights, in centimetres, of 150 children.Height (cm) ...................... No. children ( f )140 up to 144 .................................... a144
A group of 10 friends played a mini-golf competition. Eight of the friends tied for second place, each with a score of 34, and the other two friends tied for first place. Find the winning score,
During a sale, a boy bought six pairs of jeans, each with leg length xcm. He also bought four pairs of pants, each with leg length (x – 2)cm. The boy is quite short, so his father removed 4cm from
The lengths of 45 ropes used at an outdoor recreational centre can be extended by 30% when stretched. The sum of the squares of their stretched lengths is 0.0507km2 and their natural lengths, x
The heights, xcm, of a group of 28 people were measured. The mean height was found to be 172.6cm and the standard deviation was found to be 4.58cm. A person whose height was 161.8cm left the group.i.
The areas, in cm2, of some circuit boards are represented in the following graph.a. State the greatest possible range of the data.b. Construct a box-and-whisker diagram to illustrate the data.c. Find
Kristina plans to raise money for charity. Her plan is to walk 217km in 7 days so that she walks k + 2n – n2 km on the nth day. Find the standard deviation of the daily distances she plans to walk,
The numbers of goals, x, scored by a team in each of its previous 25 games are summarised by the totalsΣ(x −1)2 = 30 and Σ(x −1) = 12.a. Find the mean number of goals that the team scored
The lengths of some insects of the same type from two countries, X and Y, were measured. The stem and-leaf diagram shows the results.i. Find the median and interquartile range of the lengths of the
A teacher randomly selects one student from a group of 12 boys and 24 girls. Find the probability that the teacher selects:a. A particular boy b. A girl.
Find the probability that the number rolled with an ordinary fair die is:a. A prime number or a 4 b. A square number or a multiple of 3c. More than 3 or a factor of 8.
Using a tree diagram, find the probability that exactly one head is obtained when two fair coins are tossed.
Y and Z are independent events. P(Y) = 0.7 and P(Z) = 0.9. Find P(Y ∩ Z).
One letter is randomly selected from the six letters in the word BANANA. Find the probability that:a. An N is selected, given that an A is not selectedb. An A is selected, given that an N is not
Two ties are taken at random from a bag of three plain and five striped ties. By use of a tree diagram, or otherwise, find the probability that both ties are:a. Plain b. Striped.
Four quality-control officers were asked to test 1214 randomly selected electronic components from a company’s production line, and to report the proportions that they found to be defective. The
United’s manager estimates that the team has a 65% chance of winning any particular game and an 85% chance of not drawing any particular game.a. What are the manager’s estimates most likely to be
A group of 40 students took a test in Economics. The following Venn diagram shows that 19 boys (B) took the test and that seven students failed the test (F).a. Describe the 21 students who are
Two ordinary fair dice are rolled. Using a possibility diagram, find the probability of obtaining:a. Two 6s b. Two even numbers c. Two numbers whose product is 6.
Two independent events are M and N. Given that P(M) = 0.75 and P(M ∩ N) = 0.21, find P(N).
One hundred children were each asked whether they have brothers (B) and whether they have sisters (S). Their responses are given in the Venn diagram opposite.Find the probability that a randomly
There are four toffee sweets and seven nutty sweets in a girl’s pocket. Find the probability that two sweets, selected at random, one after the other, are not the same type.
Three referees are needed at an international tournament and there are 12 to choose from: three from Bosnia, four from Chad and five from Denmark. If the referees are selected at random, find the
Katya randomly picks one of the 10 cards shown.If she repeats this 40 times, how many times is Katya expected to pick a card that is not blue and does not have a letter B on it? A B A B.
The following table gives information about all the animals on a farm.a. Find the probability that a randomly selected animal is:i. Male or a goat ii. A sheep or female.b. Find a different way
It is known that 8% of all new FunX cars develop a mechanical fault within a year and that 15% independently develop an electrical fault within a year. Find the probability that within a year a new
Independent events S and T are such that P(S) = 0.4 and P(T') = 0.2. Find:a. P(S ∩ T)b. P(S' ∩ T ).
Two photographs are randomly selected from a pack of 12 colour and eight black and white photographs. Find the probability that the second photograph selected is colour, given that the first is:a.
On a library shelf there are seven novels, three dictionaries and two atlases. Two books are randomly selected without replacement from these. Find the probability that the selected books are:a. Both
The diagram opposite gives details about a company’s 115 employees. For example, it employs four unqualified, part-time females. Two employees are selected at random. Find the probability that:a.
A numbered wheel is divided into eight sectors of equal size, as shown. The wheel is spun until it stops with the arrow pointing at one of the numbers. Axel decides to spin the wheel 400 times.a.
When a coin is randomly selected from a savings box, each coin has a 98% chance of not being selected. How many coins are in the savings box?
A fair six-sided spinner, P, has edges marked 0, 1, 2, 2, 3 and 4.A fair four-sided spinner, Q, has edges marked 0, –1, –1 and –2.Each spinner is spun once and the numbers on which they come to
Three friends, Rick, Brenda and Ali, go to a football match but forget to say which entrance to the ground they will meet at. There are four entrances, A, B,C and D. Each friend chooses an entrance
Two ordinary fair dice are rolled and three events are:X: the sum of the two numbers rolled is 6.Y: the difference between the two numbers rolled is zero.Z: both of the numbers rolled are even.a.
A certain horse has a 70% chance of winning any particular race. Find the probability that it wins exactly one of its next two races.
A, B and C are independent events, and it is given that P(A ∩ B) = 0.35, 5, P(B ∩ C) = 0.56 and P(A ∩ C) = 0.4.a. Express P(A) in terms of:i. P(B)ii. P(C).b. Use your answers to part a to
The Venn diagram opposite shows the responses of 40 girls who were asked if they have an interest in a career in nursing (N), dentistry (D) or human rights (H).a. Find the probability that a randomly
A woman travels to work by bicycle 70% of the time and by scooter 30% of the time. If she uses her bicycle she is late 3% of the time but if she uses her scooter she is late only 2% of the time.a.
The numbers of books read in the past 3 months by the members of a reading club are shown in the following table.Find the probability that three randomly selected members have all read fewer than

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