Questions — Edexcel S3 (332 questions)

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Edexcel S3 Q6
12 marks Moderate -0.8
As part of her statistics project, Deepa decided to estimate the amount of time A-level students at her school spend on private study each week. She took a random sample of students from those studying Arts subjects, Science subjects and a mixture of Arts and Science subjects. Each student kept a record of the time they spent on private study during the third week of term.
  1. Write down the name of the sampling method used by Deepa. [1]
  2. Give a reason for using this method and give one advantage this method has over simple random sampling. [2]
The results Deepa obtained are summarised in the table below.
Type of studentSample sizeMean number of hours
Arts1212.6
Science1214.1
Mixture810.2
  1. Show that an estimate of the mean time spent on private study by A level students at Deepa's school, based on these 32 students is 12.56, to 2 decimal places. [3]
The standard deviation of the time spent on private study by students at the school was 2.48 hours.
  1. Assuming that the number of hours spent on private study is normally distributed, find a 95% confidence interval for the mean time spent on private study by A level students at Deepa's school. [4]
A member of staff at the school suggested that A level students should spend on average 12 hours each week on private study.
  1. Comment on this suggestion in the light of your interval. [2]
Edexcel S3 Q7
16 marks Standard +0.3
For one of the activities at a gymnastics competition, 8 gymnasts were awarded marks out of 10 for each of artistic performance and technical ability. The results were as follows.
Gymnast\(A\)\(B\)\(C\)\(D\)\(E\)\(F\)\(G\)\(H\)
Technical ability8.58.69.57.56.89.19.49.2
Artistic performance6.27.58.26.76.07.28.09.1
The value of the product moment correlation coefficient for these data is 0.774.
  1. Stating your hypotheses clearly and using a 1% level of significance, interpret this value. [5]
  2. Calculate the value of the rank correlation coefficient for these data. [6]
  3. Stating your hypotheses clearly and using a 1% level of significance, interpret this coefficient. [3]
  4. Explain why the rank correlation coefficient might be the better one to use with these data. [2]
Edexcel S3 Specimen Q1
4 marks Easy -1.8
The 240 members of a bowling club are listed alphabetically in the club's membership book. The committee wishes to select a sample of 30 members to fill in a questionnaire about the facilities the club offers.
  1. Explain how the committee could use a table of random numbers to take a systematic sample. [3]
  2. Give one advantage of this method over taking a simple random sample. [1]
Edexcel S3 Specimen Q2
5 marks Moderate -0.8
The weights of pears, \(P\) grams, are normally distributed with a mean of 110 and a standard deviation of 8. Geoff buys a bag of 16 pears.
  1. Write down the distribution of \(\overline{P}\), the mean weight of the 16 pears. [2]
  2. Find P\((110 < \overline{P} < 113)\). [3]
Edexcel S3 Specimen Q3
10 marks Standard +0.3
The three tasks most frequently carried out in a garage are \(A\), \(B\) and \(C\). For each of the tasks the times, in minutes, taken by the garage mechanics are assumed to be normally distributed with means and standard deviations given in the following table.
TaskMeanStandard deviation
\(A\)22538
\(B\)16523
\(C\)18527
Assuming that the times for the three tasks are independent, calculate the probability that
  1. the total time taken by a single randomly chosen mechanic to carry out all three tasks lies between 533 and 655 minutes, [5]
  2. a randomly chosen mechanic takes longer to carry out task \(B\) than task \(C\). [5]
Edexcel S3 Specimen Q4
11 marks Standard +0.3
At the end of a season a league of eight ice hockey clubs produced the following table showing the position of each club in the league and the average attendances (in hundreds) at home matches.
Club\(A\)\(B\)\(C\)\(D\)\(E\)\(F\)\(G\)\(H\)
Position12345678
Average3738192734262232
  1. Calculate the Spearman rank correlation coefficient between position in the league and average home attendance. [5]
  2. Stating clearly your hypotheses and using a 5\% two-tailed test, interpret your rank correlation coefficient. [4]
Many sets of data include tied ranks.
  1. Explain briefly how tied ranks can be dealt with. [2]
Edexcel S3 Specimen Q5
11 marks Moderate -0.3
For a six-sided die it is assumed that each of the sides has an equal chance of landing uppermost when the die is rolled.
  1. Write down the probability function for the random variable \(X\), the number showing on the uppermost side after the die has been rolled. [2]
  2. State the name of the distribution. [1]
A student wishing to check the above assumption rolled the die 300 times and for the sides 1 to 6, obtained the frequencies 41, 49, 52, 58, 37 and 63 respectively.
  1. Analyse these data and comment on whether or not the assumption is valid for this die. Use a 5\% level of significance and state your hypotheses clearly. [8]
Edexcel S3 Specimen Q6
11 marks Standard +0.3
A sociologist was studying the smoking habits of adults. A random sample of 300 adult smokers from a low income group and an independent random sample of 400 adult smokers from a high income group were asked what their weekly expenditure on tobacco was. The results are summarised below.
\(N\)means.d.
Low income group300£6.40£6.69
High income group400£7.42£8.13
  1. Using a 5\% significance level, test whether or not the two groups differ in the mean amounts spent on tobacco. [9]
  2. Explain briefly the importance of the central limit theorem in this example. [2]
Edexcel S3 Specimen Q7
11 marks Moderate -0.3
A survey in a college was commissioned to investigate whether or not there was any association between gender and passing a driving test. A group of 50 male and 50 female students were asked whether they passed or failed their driving test at the first attempt. All the students asked had taken the test. The results were as follows.
PassFail
Male2327
Female3218
Stating your hypotheses clearly test, at the 10\% level, whether or not there is any evidence of an association between gender and passing a driving test at the first attempt. [11]
Edexcel S3 Specimen Q8
12 marks Moderate -0.3
Observations have been made over many years of \(T\), the noon temperature in °C, on 21st March at Sunnymere. The records for a random sample of 12 years are given below. 5.2, 3.1, 10.6, 12.4, 4.6, 8.7, 2.5, 15.3, \(-1.5\), 1.8, 13.2, 9.3.
  1. Find unbiased estimates of the mean and variance of \(T\). [5]
Over the years, the standard deviation of \(T\) has been found to be 5.1.
  1. Assuming a normal distribution find a 90\% confidence interval for the mean of \(T\). [5]
A meteorologist claims that the mean temperature at noon in Sunnymere on 21st March is 4 °C.
  1. Use your interval to comment on the meteorologist's claim. [2]
Edexcel S3 Q1
6 marks Easy -1.8
  1. Explain briefly the method of quota sampling. [3]
  2. Give one disadvantage of quota sampling compared with stratified sampling. [1]
  3. Describe a situation in which you would choose to use quota sampling rather than stratified sampling and explain why. [2]
Edexcel S3 Q2
9 marks Standard +0.3
Commentators on a game of cricket say that a certain batsman is "playing shots all round the ground". A sports statistician wishes to analyse this claim and records the direction of shots played by the batsman during the course of his innings. She divides the \(360°\) around the batsman into six sectors, measuring the angle of each shot clockwise from the line between the wickets, and obtains the following results:
Sector\(0° -\)\(45° -\)\(90° -\)\(180° -\)\(270° -\)\(315° - 360°\)
No. of Shots18191520915
Stating your hypotheses clearly and using a 5% level of significance test whether or not these data can be modelled by a continuous uniform distribution. [9]
Edexcel S3 Q3
11 marks Moderate -0.3
A film-buff is interested in how long it takes for the credits to roll at the end of a movie. She takes a random sample of 20 movies from those that she has bought on DVD and finds that the credits on these films last for a total of 46 minutes and 15 seconds
  1. Assuming that the time for the credits to roll follows a Normal distribution with a standard deviation of 23 seconds, use her data to calculate a 90% confidence interval for the mean time taken for the credits to roll. [5]
  2. Find the minimum number of movies she would need to have included in her sample for her confidence interval to have a width of less than 10 seconds. [5]
  3. Explain why her sample might not be representative of all movies. [1]
Edexcel S3 Q4
11 marks Standard +0.3
A hospital administrator is assessing staffing needs for its Accident and Emergency Department at different times of day. The administrator already has data on the number of admissions at different times of day but needs to know if the proportion of the cases that are serious remains constant. Staff are asked to assess whether each person arriving at Accident and Emergency has a "minor" or "serious" problem and the results for three different time periods are shown below.
MinorSerious
8 a.m. – 6 p.m.4511
6 p.m. – 2 a.m.4922
2 a.m. – 8 a.m.147
Stating your hypotheses clearly, test at the 5% level of significance whether or not there is evidence of the proportion of serious injuries being different at different times of day. [11]
Edexcel S3 Q5
12 marks Standard +0.3
In a competition, a wine-enthusiast has to rank ten bottles of wine, \(A\) to \(J\), in order starting with the one he thinks is the most expensive. The table below shows his rankings and the actual order according to price.
Rank12345678910
Enthusiast\(D\)\(C\)\(J\)\(A\)\(G\)\(F\)\(B\)\(E\)\(I\)\(H\)
Price\(A\)\(C\)\(D\)\(H\)\(J\)\(B\)\(F\)\(I\)\(G\)\(E\)
  1. Calculate Spearman's rank correlation coefficient for these data. [6]
  2. Stating your hypotheses clearly, test at the 5% level of significance whether or not there is evidence of positive correlation. [4]
  3. Explain briefly how you would have been able to carry out the test if bottles \(B\) and \(F\) had the same price. [2]
Edexcel S3 Q6
13 marks Standard +0.3
A researcher collects data on the height of boys aged between nine and nine and-a-half years and their diet. The data on the height, \(V\) cm, of the 80 boys who had always eaten a vegetarian diet is summarised by $$\Sigma V = 10\,367, \quad \Sigma V^2 = 1\,350\,314.$$
  1. Calculate unbiased estimates of the mean and variance of \(V\). [5]
The researcher calculates unbiased estimates of the mean and variance of the height of boys whose diet has included meat from a sample of size 280, giving values of 130.5 cm and 96.24 cm\(^2\) respectively.
  1. Stating your hypotheses clearly, test at the 1% level whether or not there is a significant difference in the heights of boys of this age according to whether or not they have a vegetarian diet. [8]
Edexcel S3 Q7
13 marks Standard +0.8
An examiner believes that once she has marked the first 20 papers the time it takes her to mark one paper for a particular exam follows a Normal distribution. Having already marked more than 20 papers for each of the \(P1\), \(M1\) and \(S1\) modules set one summer, the mean and standard deviation, in seconds, of the time it takes her to mark a paper for each module are as shown in the table below.
MeanStandard Deviation
\(P1\)25217
\(M1\)31442
\(S1\)28429
  1. Find the probability that the difference in the time it takes her to mark two randomly chosen \(P1\) papers is less than 5 seconds. [6]
  2. Find the probability that it takes her less than 10 hours to mark 45 \(M1\) and 80 \(S1\) papers. [7]
Edexcel S3 Q1
5 marks Easy -1.8
A researcher wishes to take a sample of size 9, without replacement, from a list of 72 people involved in the trial of a new computer keyboard. She numbers the people from 01 to 72 and uses the table of random numbers given in the formula book. She starts with the left-hand side of the sixth row of the table and works across the row. The first two numbers she writes down are 56 and 32.
  1. Find the other six numbers in the sample. [3 marks]
  2. Give one advantage and one disadvantage of using random numbers when taking a sample. [2 marks]
Edexcel S3 Q2
6 marks Moderate -0.8
The length of time that registered customers spend on each visit to a supermarket's website is normally distributed with a mean of 28.5 minutes and a standard deviation of 7.2 minutes. Eight visitors to the site are selected at random and the length of time, \(T\) minutes, that each stays is recorded.
  1. Write down the distribution of \(\overline{T}\), the mean time spent at the site by these eight visitors. [2 marks]
  2. Find \(P(25 < \overline{T} < 30)\). [4 marks]
Edexcel S3 Q3
7 marks Standard +0.3
The discrete random variable \(X\) has the probability distribution given below.
\(x\)247\(k\)
\(P(X = x)\)0.050.150.30.5
  1. Find the mean of \(X\) in terms of \(k\). [2 marks]
  2. Find the bias in using \((2\overline{X} - 5)\) as an estimator of \(k\). [3 marks]
Fifty observations of \(X\) were made giving a sample mean of 8.34 correct to 3 significant figures.
  1. Calculate an unbiased estimate of \(k\). [2 marks]
Edexcel S3 Q4
7 marks Standard +0.8
The mass of waste in filled large dustbin bags is normally distributed with a mean of 6.8 kg and a standard deviation of 1.5 kg. The mass of waste in filled small dustbin bags is normally distributed with a mean of 3.2 kg and a standard deviation of 0.6 kg. One week there are 8 large and 3 small dustbin bags left for collection outside a block of flats. Find the probability that this waste has a total mass of more than 70 kg. [7 marks]
Edexcel S3 Q5
8 marks Standard +0.3
For a project, a student is investigating whether more athletic individuals have better hand-eye coordination. He records the time it takes a number of students to complete a task testing coordination skills and notes whether or not they play for a school sports team. His results are as follows:
Number of StudentsMeanStandard Deviation
In a School Team5032.8 s4.6 s
Not in a Team19035.1 s8.0 s
Stating your hypotheses clearly, test at the 5\% level of significance whether or not there is evidence that those who play in a school team complete the task more quickly on average. [8 marks]
Edexcel S3 Q6
11 marks Standard +0.3
Two schools in the same town advertise at the same time for new heads of English and History departments. The number of applicants for each post are shown in the table below.
EnglishHistory
Highfield School3214
Rowntree School4826
Stating your hypotheses clearly, test at the 10\% level of significance whether or not there is evidence of the proportion of applicants for each job being different in the two schools. [11 marks]
Edexcel S3 Q7
11 marks Standard +0.3
A sports scientist wishes to examine the link between resting pulse and fitness. He records the resting pulse, \(p\), of 20 volunteers and the length of time, \(t\) minutes, that each one can run comfortably at 4 metres per second on a treadmill. The results are summarised by $$\Sigma p = 1176, \quad \Sigma t = 511, \quad \Sigma p^2 = 70932, \quad \Sigma t^2 = 19213, \quad \Sigma pt = 27188.$$
  1. Calculate the product moment correlation coefficient for these data. [5 marks]
  2. Stating your hypotheses clearly, test at the 1\% level of significance whether there is evidence of people with a lower resting pulse having a higher level of fitness as measured by the test. [4 marks]
  3. State an assumption necessary to carry out the test in part (b) and comment on its validity in this case. [2 marks]
Edexcel S3 Q8
20 marks Standard +0.3
A physicist believes that the number of particles emitted by a radioactive source with a long half-life can be modelled by a Poisson distribution. She records the number of particles emitted in 80 successive 5-minute periods and her results are shown in the table below.
No. of Particles012345 or more
No. of Intervals233214830
  1. Comment on the suitability of a Poisson distribution for this situation. [3 marks]
  2. Show that an unbiased estimate of the mean number of particles emitted in a 5-minute period is 1.2 and find an unbiased estimate of the variance. [5 marks]
  3. Explain how your answers to part (b) support the fitting of a Poisson distribution. [1 mark]
  4. Stating your hypotheses clearly and using a 5\% level of significance, test whether or not these data can be modelled by a Poisson distribution. [11 marks]