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Edexcel S3 2023 January Q3
9 marks Moderate -0.8
3 A mobile phone company offers an insurance policy to its customers when they purchase a mobile phone. The company conducted a survey on the age of the customers and whether or not claims were made. A random sample of 1200 customers from this company was investigated for 2020 and the results are shown in the table below.
Claim made in 2020No claim made in 2020Total
\multirow{3}{*}{Age}17-20 years24176200
21-50 years48652700
51 years and over14286300
Total8611141200
The data are to be used to determine whether or not making a claim is independent of age.
  1. Calculate the expected frequencies for the age group 51 years and over that
    1. made a claim in 2020
    2. did not make a claim in 2020 The 4 classes of customers aged between 17 and 50 give a value of \(\sum \frac { ( O - E ) ^ { 2 } } { E } = 7.123\) correct to 3 decimal places.
  2. Test, at the \(1 \%\) level of significance, whether or not making a claim is independent of age. Show your working clearly, stating your hypotheses, the degrees of freedom, the test statistic and the critical value used.
Edexcel S3 2023 January Q4
14 marks Standard +0.3
4 A research student is investigating the number of children who are girls in families with 4 children. The table below shows her results for 200 such families.
Number of girls01234
Frequency1568693810
The research student suggests that a binomial distribution with \(p = \frac { 1 } { 2 }\) could be a suitable model for the number of children who are girls in a family of 4 children.
  1. Using her results and a \(5 \%\) significance level, test the research student's claim. You should state your hypotheses, expected frequencies, test statistic and the critical value used. The research student decides to refine the model and retains the idea of using a binomial distribution but does not specify the probability that the child is a girl.
  2. Use the data in the table to show that the probability that a child is a girl is 0.45 The research student uses the probability from part (b) to calculate a new set of expected frequencies, none of which are less than 5
    The statistic \(\sum \frac { ( O - E ) ^ { 2 } } { E }\) is evaluated and found to be 2.47
  3. Test, at the \(5 \%\) significance level, whether using a binomial distribution is suitable to model the number of children who are girls in a family of 4 children. You should state your hypotheses and the critical value used.
Edexcel S3 2023 January Q5
10 marks Standard +0.3
5 Claire grows strawberries on her farm. She wants to compare two brands of fertiliser, brand \(A\) and brand \(B\). She grows two sets of plants of the same variety of strawberries under the same conditions, fertilising one set with brand \(A\) and the other with brand \(B\). The yields per plant, in grams, from each set of plants are summarised below.
MeanStandard deviationNumber of plants
Fertiliser A137717.850
Fertiliser B136818.440
  1. Stating your hypotheses clearly, carry out a suitable test to assess whether the mean yield from plants using fertiliser \(A\) is greater than the mean yield from plants using fertiliser \(B\).
    Use a 1\% level of significance and state your test statistic and critical value. The total cost of fertiliser \(A\) for Claire's 50 plants was \(\pounds 75\) The total cost of fertiliser \(B\) for Claire's 40 plants was \(\pounds 50\) Claire sells all her strawberries at \(\pounds 3\) per kilogram.
  2. Use this information, together with your answer in part (a), to advise Claire on which of the two brands of fertiliser she should use next year in order to maximise her expected profit per plant, giving a reason for your answer.
Edexcel S3 2023 January Q6
10 marks Moderate -0.3
6 A garden centre sells bags of stones and large bags of gravel.
The weight, \(X\) kilograms, of stones in a bag can be modelled by a normal distribution with unknown mean \(\mu\) and known standard deviation 0.4 The stones in each of a random sample of 36 bags from a large batch is weighed. The total weight of stones in these 36 bags is found to be 806.4 kg
  1. Find a 98\% confidence interval for the mean weight of stones in the batch.
  2. Explain why the use of the Central Limit theorem is not required to answer part (a) The manufacturer of these bags of stones claims that bags in this batch have a mean weight of 22.5 kg
  3. Using your answer to part (a), comment on the claim made by the manufacturer. The weight, \(Y\) kilograms, of gravel in a large bag can be modelled by a normal distribution with mean 850 kg and standard deviation 5 kg A builder purchases 10 large bags of gravel.
  4. Find the probability that the mean weight of gravel in the 10 large bags is less than 848 kg
Edexcel S3 2023 January Q7
8 marks Standard +0.3
7 At a particular supermarket, the times taken to serve each customer in a queue at a standard checkout may be modelled by a normal distribution with mean 240 seconds and standard deviation 20 seconds. There is a queue of 3 customers at a standard checkout.
Making a reasonable assumption about the times taken to serve these customers,
  1. find the probability that the total time taken to serve the 3 customers will be less than 11 minutes.
  2. State the assumption you have made in part (a) In the supermarket there is also an express checkout, which is reserved for customers buying 10 or fewer items. The time taken to serve a customer at this express checkout may be modelled by a normal distribution with mean 100 seconds and standard deviation 8 seconds. On a particular day Jiang has 8 items to pay for and has to choose whether to join a queue of 3 customers waiting at a standard checkout or a queue of 7 customers waiting at the express checkout. Using a similar assumption to that made in part (a),
  3. find the probability that the total time taken to serve the 3 customers at the standard checkout will exceed the total time taken to serve the 7 customers at the express checkout.
Edexcel S3 2024 January Q1
8 marks Standard +0.3
  1. Chen is treating vines to prevent fungus appearing. One month after the treatment, Chen monitors the vines to see if fungus is present.
The contingency table shows information about the type of treatment for a sample of 150 vines and whether or not fungus is present.
\multirow{2}{*}{}Type of treatment
NoneSulphurCopper sulphate
No fungus present205548
Fungus present1089
Test, at the \(5 \%\) level of significance, whether or not there is any association between the type of treatment and the presence of fungus.
Show your working clearly, stating your hypotheses, expected frequencies, test statistic and critical value.
Edexcel S3 2024 January Q2
7 marks Easy -1.8
  1. A company has 800 employees.
The manager of the company is going to take a sample of 80 employees.
  1. Explain how this sample can be taken using systematic sampling. The company has offices in London, Edinburgh and Cardiff. The table shows the number of employees in each city.
    CityLondonEdinburghCardiff
    Number of employees430250120
    The president of the company is going to take a sample of 100 employees to determine the average time employees spend in front of a computer each week.
  2. Explain how this sample can be taken using stratified sampling.
  3. Explain an advantage of using stratified sampling rather than simple random sampling.
Edexcel S3 2024 January Q3
12 marks Standard +0.3
  1. The table shows the annual tea consumption, \(t\) (kg/person), and population, \(p\) (millions), for a random sample of 7 European countries.
CountryABCDEFG
Annual tea consumption, \(\boldsymbol { t }\) (kg/person)0.270.150.420.061.940.780.44
Population, \(\boldsymbol { p }\) (millions)5.45.8910.267.917.18.7
$$\text { (You may use } \mathrm { S } _ { t t } = 2.486 \quad \mathrm {~S} _ { p p } = 3026.234 \quad \mathrm {~S} _ { p t } = 83.634 \text { ) }$$ Angela suggests using the product moment correlation coefficient to calculate the correlation between annual tea consumption and population.
  1. Use Angela's suggestion to test, at the \(5 \%\) level of significance, whether or not there is evidence of any correlation between annual tea consumption and population. State your hypotheses clearly and the critical value used. Johan suggests using Spearman's rank correlation coefficient to calculate the correlation between the rank of annual tea consumption and the rank of population.
  2. Calculate Spearman's rank correlation coefficient between the rank of annual tea consumption and the rank of population.
  3. Use Johan's suggestion to test, at the \(5 \%\) level of significance, whether or not there is evidence of a positive correlation between annual tea consumption and population.
    State your hypotheses clearly and the critical value used.
Edexcel S3 2024 January Q4
10 marks Standard +0.3
  1. The number of jobs sent to a printer per hour in a small office is recorded for 120 hours. The results are summarised in the following table.
Number of jobs012345
Frequency2434282185
  1. Show that the mean number of jobs sent to the printer per hour for these data is 1.75 The office manager believes that the number of jobs sent to the printer per hour can be modelled using a Poisson distribution. The office manager uses the mean given in part (a) to calculate the expected frequencies for this model. Some of the results are given in the following table.
    Number of jobs012345 or more
    Expected frequency20.8536.4931.93\(r\)\(s\)3.95
  2. Show that the value of \(s\) is 8.15 to 2 decimal places.
  3. Find the value of \(r\) to 2 decimal places. The value of \(\sum \frac { \left( O _ { i } - E _ { i } \right) ^ { 2 } } { E _ { i } }\) for the first four frequencies in the table is 1.43
  4. Test, at the \(5 \%\) level of significance, whether or not the number of jobs sent to the printer per hour can be modelled using a Poisson distribution. Show your working clearly, stating your hypotheses, test statistic and critical value.
Edexcel S3 2024 January Q5
9 marks Standard +0.3
  1. A professor claims that undergraduates studying History have a typing speed of more than 15 words per minute faster than undergraduates studying Maths.
A sample is taken of 38 undergraduates studying History and 45 undergraduates studying Maths. The typing speed, \(x\) words per minute, of each undergraduate is recorded. The results are summarised in the table below.
\(n\)\(\bar { x }\)\(s ^ { 2 }\)
Undergraduates studying History3856.327.2
Undergraduates studying Maths4539.818.5
  1. Use a suitable test, at the \(5 \%\) level of significance, to investigate the professor's claim.
    State clearly your hypotheses, test statistic and critical value.
  2. State two assumptions you have made in carrying out the test in part (a).
Edexcel S3 2024 January Q6
15 marks Standard +0.3
  1. A random sample of 8 three-month-old golden retriever dogs is taken.
The heights of the golden retrievers are recorded.
Using this sample, a 95\% confidence interval for the mean height, in cm, of three-month-old golden retrievers is found to be \(( 45.72,53.88 )\)
  1. Find a 99\% confidence interval for the mean height. You may assume that the heights are normally distributed with known population standard deviation. Some summary statistics for the weights, \(x \mathrm {~kg}\), of this sample are given below. $$\sum x = 91.2 \quad \sum x ^ { 2 } = 1145.16 \quad n = 8$$
  2. Calculate unbiased estimates of the mean and the variance of the weights of three-month-old golden retrievers. A further random sample of 24 three-month-old golden retrievers is taken. The unbiased estimates of the mean and the variance of the weights, in kg , from this sample are found to be 10.8 and 17.64 respectively.
  3. Estimate the standard error of the mean weight for the combined sample of 32 three-month-old golden retrievers.
Edexcel S3 2024 January Q7
14 marks Standard +0.8
  1. Small containers and large containers are independently filled with fruit juice.
The amounts of fruit juice in small containers are normally distributed with mean 180 ml and standard deviation 4.5 ml The amounts of fruit juice in large containers are normally distributed with mean 330 ml and standard deviation 6.7 ml The random variable \(W\) represents the total amount of fruit juice in a random sample of 2 small containers minus the amount of fruit juice in 1 randomly selected large container. \(W \sim \mathrm {~N} ( a , b )\) where \(a\) and \(b\) are positive constants.
  1. Find the value of \(a\) and the value of \(b\)
  2. Find the probability that a randomly chosen large container of fruit juice contains more than 1.8 times the amount of fruit juice in a randomly chosen small container. A random sample of 3 small containers of fruit juice is taken.
  3. Find the probability that the first container of fruit juice in this sample contains at least 5 ml more than the mean amount of fruit juice in all 3 small containers.
Edexcel S3 2014 June Q1
8 marks Easy -1.8
  1. A tennis club's committee wishes to select a sample of 50 members to fill in a questionnaire about the club's facilities. The 300 members, of whom 180 are males, are listed in alphabetical order and numbered \(1 - 300\) in the club’s membership book.
The club's committee decides to use a random number table to obtain its sample.
The first three lines of the random number table used are given below.
319952241343278811394165008413063179749
722962334461267114806992414837837657339
470684554127067459142920144575311605412
Starting with the top left-hand corner (319) and working across, the committee selects 50 random numbers. The first 2 suitable numbers are 241 and 278. Numbers greater than 300 are ignored.
  1. Find the next two suitable numbers. When the club's committee looks at the members corresponding to their random numbers they find that only 1 female has been selected.
    The committee does not want to be accused of being biased towards males so considers using a systematic sample instead.
    1. Explain clearly how the committee could take a systematic sample.
    2. Explain why a systematic sample may not give a sample that represents the proportion of males and females in the club. The committee decides to use a stratified sample instead.
  3. Describe how to choose members for the stratified sample.
  4. Explain an advantage of using a stratified sample rather than a quota sample.
Edexcel S3 2014 June Q2
7 marks Standard +0.3
2. The random variable \(X\) follows a continuous uniform distribution over the interval \([ \alpha - 3,2 \alpha + 3 ]\) where \(\alpha\) is a constant.
The mean of a random sample of size \(n\) is denoted by \(\bar { X }\)
  1. Show that \(\bar { X }\) is a biased estimator of \(\alpha\), and state the bias. Given that \(Y = k \bar { X }\) is an unbiased estimator for \(\alpha\)
  2. find the value of \(k\). A random sample of 10 values of \(X\) is taken and the results are as follows $$\begin{array} { l l l l l l l l l l } 3 & 5 & 8 & 12 & 4 & 13 & 10 & 8 & 5 & 12 \end{array}$$
  3. Hence estimate the maximum value of \(X\)
Edexcel S3 2014 June Q3
7 marks Standard +0.3
3. A grocer believes that the average weight of a grapefruit from farm \(A\) is greater than the average weight of a grapefruit from farm \(B\). The weights, in grams, of 80 grapefruit selected at random from farm \(A\) have a mean value of 532 g and a standard deviation, \(s _ { A }\), of 35 g . A random sample of 100 grapefruit from farm \(B\) have a mean weight of 520 g and a standard deviation, \(s _ { B }\), of 28 g . Stating your hypotheses clearly and using a 1\% level of significance, test whether or not the grocer's belief is supported by the data.
Edexcel S3 2014 June Q4
12 marks Standard +0.3
4. In a survey 10 randomly selected men had their systolic blood pressure, \(x\), and weight, \(w\), measured. Their results are as follows
Man\(\boldsymbol { A }\)\(\boldsymbol { B }\)\(\boldsymbol { C }\)\(\boldsymbol { D }\)\(\boldsymbol { E }\)\(\boldsymbol { F }\)\(\boldsymbol { G }\)\(\boldsymbol { H }\)\(\boldsymbol { I }\)\(\boldsymbol { J }\)
\(x\)123128137143149153154159162168
\(w\)78938583759888879599
  1. Calculate the value of Spearman's rank correlation coefficient between \(x\) and \(w\).
  2. Stating your hypotheses clearly, test at the \(5 \%\) level of significance, whether or not there is evidence of a positive correlation between systolic blood pressure and weight. The product moment correlation coefficient for these data is 0.5114
  3. Use the value of the product moment correlation coefficient to test, at the \(5 \%\) level of significance, whether or not there is evidence of a positive correlation between systolic blood pressure and weight.
  4. Using your conclusions to part (b) and part (c), describe the relationship between systolic blood pressure and weight.
Edexcel S3 2014 June Q5
12 marks Standard +0.3
  1. A random sample of 200 people were asked which hot drink they preferred from tea, coffee and hot chocolate. The results are given below.
\cline { 3 - 6 } \multicolumn{2}{|c|}{}
\multirow{2}{*}{Total}
\cline { 3 - 5 } \multicolumn{2}{|c|}{}TeaCoffeeHot Chocolate
\multirow{2}{*}{Gender}Males57261194
\cline { 2 - 6 }Females424717106
Total997328200
  1. Test, at the \(5 \%\) significance level, whether or not there is an association between type of drink preferred and gender. State your hypotheses and show your working clearly. You should state your expected frequencies to 2 decimal places.
  2. State what difference using a \(0.5 \%\) significance level would make to your conclusion. Give a reason for your answer.
Edexcel S3 2014 June Q6
14 marks Standard +0.3
6. Eight tasks were given to each of 125 randomly selected job applicants. The number of tasks failed by each applicant is recorded. The results are as follows
Number of tasks failed by an applicant0123456 or more
Frequency22145421230
  1. Show that the probability of a randomly selected task, from this sample, being failed is 0.3 An employer believes that a binomial distribution might provide a good model for the number of tasks, out of 8, that an applicant fails. He uses a binomial distribution, with the estimated probability 0.3 of a task being failed. The calculated expected frequencies are as follows
    Number of tasks failed by an applicant0123456 or more
    Expected frequency7.2124.7137.06\(r\)17.025.83\(s\)
  2. Find the value of \(r\) and the value of \(s\) giving your answers to 2 decimal places.
  3. Test, at the \(5 \%\) level of significance, whether or not a binomial distribution is a suitable model for these data. State your hypotheses and show your working clearly. The employer believes that all applicants have the same probability of failing each task.
  4. Use your result from part(c) to comment on this belief.
Edexcel S3 2014 June Q7
15 marks Standard +0.3
7. The random variable \(X\) is defined as $$X = 4 Y - 3 W$$ where \(Y \sim \mathrm {~N} \left( 40,3 ^ { 2 } \right) , W \sim \mathrm {~N} \left( 50,2 ^ { 2 } \right)\) and \(Y\) and \(W\) are independent.
  1. Find \(\mathrm { P } ( X > 25 )\) The random variables \(Y _ { 1 } , Y _ { 2 }\) and \(Y _ { 3 }\) are independent and each has the same distribution as \(Y\). The random variable \(A\) is defined as $$A = \sum _ { i = 1 } ^ { 3 } Y _ { i }$$ The random variable \(C\) is such that \(C \sim \mathrm {~N} \left( 115 , \sigma ^ { 2 } \right)\) Given that \(\mathrm { P } ( A - C < 0 ) = 0.2\) and that \(A\) and \(C\) are independent,
  2. find the variance of \(C\).
Edexcel S3 2015 June Q1
5 marks Standard +0.3
  1. The names of the 720 members of a swimming club are listed alphabetically in the club's membership book. The chairman of the swimming club wishes to select a systematic sample of 40 names. The names are numbered from 001 to 720 and a number between 001 and \(w\) is selected at random. The corresponding name and every \(x\) th name thereafter are included in the sample.
    1. Find the value of \(w\).
    2. Find the value of \(x\).
    3. Write down the probability that the sample includes both the first name and the second name in the club's membership book.
    4. State one advantage and one disadvantage of systematic sampling in this case.
    5. Nine dancers, Adilzhan \(( A )\), Bianca \(( B )\), Chantelle \(( C )\), Lee \(( L )\), Nikki \(( N )\), Ranjit \(( R )\), Sergei \(( S )\), Thuy \(( T )\) and Yana \(( Y )\), perform in a dancing competition.
    Two judges rank each dancer according to how well they perform. The table below shows the rankings of each judge starting from the dancer with the strongest performance.
    Rank123456789
    Judge 1\(S\)\(N\)\(B\)\(C\)\(T\)\(A\)\(Y\)\(R\)\(L\)
    Judge 2\(S\)\(T\)\(N\)\(B\)\(C\)\(Y\)\(L\)\(A\)\(R\)
  2. Calculate Spearman's rank correlation coefficient for these data.
  3. Stating your hypotheses clearly, test at the \(1 \%\) level of significance, whether or not the two judges are generally in agreement.
Edexcel S3 2015 June Q3
11 marks Standard +0.3
  1. The number of accidents on a particular stretch of motorway was recorded each day for 200 consecutive days. The results are summarised in the following table.
Number of accidents012345
Frequency4757463596
  1. Show that the mean number of accidents per day for these data is 1.6 A motorway supervisor believes that the number of accidents per day on this stretch of motorway can be modelled by a Poisson distribution. She uses the mean found in part (a) to calculate the expected frequencies for this model. Her results are given in the following table.
    Number of accidents012345 or more
    Frequency40.3864.61\(r\)27.5711.03\(s\)
  2. Find the value of \(r\) and the value of \(s\), giving your answers to 2 decimal places.
  3. Stating your hypotheses clearly, use a \(10 \%\) level of significance to test the motorway supervisor's belief. Show your working clearly.
Edexcel S3 2015 June Q4
11 marks Challenging +1.2
  1. A farm produces potatoes. The potatoes are packed into sacks.
The weight of a sack of potatoes is modelled by a normal distribution with mean 25.6 kg and standard deviation 0.24 kg
  1. Find the probability that two randomly chosen sacks of potatoes differ in weight by more than 0.5 kg
    (6) Sacks of potatoes are randomly selected and packed onto pallets.
    The weight of an empty pallet is modelled by a normal distribution with mean 20.0 kg and standard deviation 0.32 kg Each full pallet of potatoes holds 30 sacks of potatoes.
  2. Find the probability that the total weight of a randomly chosen full pallet of potatoes is greater than 785 kg
Edexcel S3 2015 June Q5
12 marks Standard +0.3
  1. A Head of Department at a large university believes that gender is independent of the grade obtained by students on a Business Foundation course. A random sample was taken of 200 male students and 160 female students who had studied the course.
The results are summarised below.
\cline { 3 - 4 } \multicolumn{2}{c|}{}MaleFemale
\multirow{3}{*}{Grade}Distinction\(18.5 \%\)\(27.5 \%\)
\cline { 2 - 4 }Merit\(63.5 \%\)\(60.0 \%\)
\cline { 2 - 4 }Unsatisfactory\(18.0 \%\)\(12.5 \%\)
Stating your hypotheses clearly, test the Head of Department's belief using a 5\% level of significance. Show your working clearly.
Edexcel S3 2015 June Q6
13 marks Standard +0.3
  1. As part of an investigation, a random sample was taken of 50 footballers who had completed an obstacle course in the early morning. The time taken by each of these footballers to complete the obstacle course, \(x\) minutes, was recorded and the results are summarised by
$$\sum x = 1570 \quad \text { and } \quad \sum x ^ { 2 } = 49467.58$$
  1. Find unbiased estimates for the mean and variance of the time taken by footballers to complete the obstacle course in the early morning. An independent random sample was taken of 50 footballers who had completed the same obstacle course in the late afternoon. The time taken by each of these footballers to complete the obstacle course, \(y\) minutes, was recorded and the results are summarised as $$\bar { y } = 30.9 \quad \text { and } \quad s _ { y } ^ { 2 } = 3.03$$
  2. Test, at the \(5 \%\) level of significance, whether or not the mean time taken by footballers to complete the obstacle course in the early morning, is greater than the mean time taken by footballers to complete the obstacle course in the late afternoon. State your hypotheses clearly.
  3. Explain the relevance of the Central Limit Theorem to the test in part (b).
  4. State an assumption you have made in carrying out the test in part (b).
Edexcel S3 2015 June Q7
5 marks Standard +0.3
  1. A fair six-sided die is labelled with the numbers \(1,2,3,4,5\) and 6 The die is rolled 40 times and the score, \(S\), for each roll is recorded.
    1. Find the mean and the variance of \(S\).
    2. Find an approximation for the probability that the mean of the 40 scores is less than 3 (3)
    3. A factory produces steel sheets whose weights \(X \mathrm {~kg}\), are such that \(X \sim \mathrm {~N} \left( \mu , \sigma ^ { 2 } \right)\)
    A random sample of these sheets is taken and a \(95 \%\) confidence interval for \(\mu\) is found to be (29.74, 31.86)
  2. Find, to 2 decimal places, the standard error of the mean.
  3. Hence, or otherwise, find a \(90 \%\) confidence interval for \(\mu\) based on the same sample of sheets. Using four different random samples, four \(90 \%\) confidence intervals for \(\mu\) are to be found.
  4. Calculate the probability that at least 3 of these intervals will contain \(\mu\).