Questions — Edexcel S3 (313 questions)

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Edexcel S3 2018 June Q2
  1. Merchandise is sold at concerts. The manager of a concert claims that the mean value of merchandise sold to premium ticket holders is more than \(\pounds 6\) greater than the mean value of merchandise sold to standard ticket holders.
    1. Given that all the tickets for the next concert have been sold, describe how a stratified sample should be taken at the concert.
    The mean value of merchandise sold to a random sample of 60 standard ticket holders at the concert is \(\pounds 15\) with a standard deviation of \(\pounds 10\). The mean value of merchandise sold to a random sample of 55 premium ticket holders at the concert is \(\pounds 23\) with a standard deviation of \(\pounds 8\).
  2. Test the manager's claim at the \(5 \%\) level of significance. State your hypotheses clearly.
  3. For the test in part (b), state whether or not it is necessary to assume that values of merchandise sold have normal distributions. Give a reason for your answer.
    REA
Edexcel S3 2018 June Q3
  1. A random sample of repair times, in hours, was taken for an electronic component. The 4 observed times are shown below.
    1.3
    1.7
    1.4
    1.8
    1. Calculate unbiased estimates of the mean and the variance of the population of repair times for this electronic component.
    The population standard deviation of the repair times for this electronic component is known to be 0.5 hours. An estimate of the population mean is required to be within 0.1 hours of its true value with a probability of at least 0.99
  2. Find the minimum sample size required.
Edexcel S3 2018 June Q4
  1. The waiting times, in minutes, of patients at a doctor's surgery follows a normal distribution with unknown mean \(\mu\) and known standard deviation \(\sigma\)
A random sample of 120 patients was taken.
  1. Find, in the form \(k \sigma\), the width of a \(99 \%\) confidence interval for \(\mu\) based on this sample. Give the value of \(k\) to 2 decimal places. A further random sample of 100 patients from the surgery gave a \(90 \%\) confidence interval for \(\mu\) of \(( 5.14,6.25 )\)
  2. Use this confidence interval to determine whether or not it provides evidence that \(\mu = 6\) State the hypotheses being tested here and write down the significance level being used. You do not need to carry out any further calculations.
  3. Find the value of \(\sigma\)
Edexcel S3 2018 June Q5
5. The weights, in kg , of cars may be assumed to follow the normal distribution \(\mathrm { N } \left( 1000,250 ^ { 2 } \right)\). The weights, in kg , of lorries may be assumed to follow the normal distribution \(\mathrm { N } \left( 2800,650 ^ { 2 } \right)\). A lorry and a car are chosen at random.
  1. Find the probability that the lorry weighs more than 3 times the weight of the car. A ferry carries vehicles across a river. The ferry is designed to carry a maximum weight of 20000 kg .
  2. One morning, 8 cars and 3 lorries drive on to the ferry. Find the probability that their total weight will exceed the recommended maximum weight of 20000 kg .
  3. State a necessary assumption needed for the calculation in part (b).
Edexcel S3 2018 June Q6
  1. David carries out an experiment with 4 identical dice, each with faces numbered 1 to 6 . He rolls the 4 dice and counts the number of dice showing an even number on the uppermost face. He repeats this 150 times. The results are summarised in the table below.
No. of dice showing an even number01234
Frequency1245363918
David defines the random variable \(C\) as the number of dice showing an even number on the uppermost face when the four dice are thrown. David claims that \(C \sim \mathrm {~B} ( 4,0.5 )\)
  1. Stating your hypotheses clearly and using a \(1 \%\) level of significance, test David's claim. Show your working clearly. John claims that \(C \sim \mathrm {~B} ( 4 , p )\)
  2. Calculate an estimate of the value of \(p\) from the summary of the results of David's experiment. Show your working clearly. John decides to test his claim. He calculates expected frequencies using the results of David's experiment and obtains the following table.
    No. of dice showing an even number01234
    Expected frequency8.6536.00\(d\)39.00\(e\)
  3. Calculate, to 2 decimal places, the value of \(d\) and the value of \(e\)
  4. State suitable hypotheses to test John’s claim. John obtained a test statistic of 16.9 and carries out a test at the \(1 \%\) level of significance.
  5. State what conclusion John should make about his claim.
    END
Edexcel S3 Q1
  1. A random sample \(X _ { 1 } , X _ { 2 } , \ldots , X _ { 10 }\) is taken from a normal population with mean 100 and standard deviation 14.
    1. Write down the distribution of \(\bar { X }\), the mean of this sample.
    2. Find \(\mathrm { P } ( | \bar { X } - 100 | > 5 )\).
    3. A random sample of the invoices, for books purchased by the customers of a large bookshop, was classified by book cover (hardback, paperback) and type of book (novel, textbook, general interest). As part of the analysis of these invoices, an approximate \(\chi ^ { 2 }\) statistic was calculated and found to be 11.09 .
    Assuming that there was no need to amalgamate any of the classifications, carry out an appropriate test to determine whether or not there was any association between book cover and type of book. State your hypotheses clearly and use a \(5 \%\) level of significance.
    (6 marks)
Edexcel S3 Q3
3. As part of a research project into the role played by cholesterol in the development of heart disease a random sample of 100 patients was put on a special fish-based diet. A different random sample of 80 patients was kept on a standard high-protein low-fat diet. After several weeks their blood cholesterol was measured and the results summarised in the table below.
GroupSample size
Mean drop in
cholesterol (mg/dl)
Standard deviation
Special diet1007522
Standard diet806431
  1. Stating your hypotheses clearly and using a \(5 \%\) level of significance, test whether or not the special diet is more effective in reducing blood cholesterol levels than the standard diet.
  2. Explain briefly any assumptions you made in order to carry out this test.
Edexcel S3 Q4
4. Breakdowns on a certain stretch of motorway were recorded each day for 80 consecutive days. The results are summarised in the table below.
Number of
breakdowns
012\(> 2\)
Frequency3832100
It is suggested that the number of breakdowns per day can be modelled by a Poisson distribution. Using a \(5 \%\) level of significance, test whether or not the Poisson distribution is a suitable model for these data. State your hypotheses clearly.
(13 marks)
Edexcel S3 Q5
5. The random variable \(R\) is defined as \(R = X + 4 Y\) where \(X \sim \mathrm {~N} \left( 8,2 ^ { 2 } \right) , Y \sim \mathrm {~N} \left( 14,3 ^ { 2 } \right)\) and \(X\) and \(Y\) are independent. Find
  1. \(\mathrm { E } ( R )\),
  2. \(\operatorname { Var } ( R )\),
  3. \(\mathrm { P } ( R < 41 )\) The random variables \(Y _ { 1 } , Y _ { 2 }\) and \(Y _ { 3 }\) are independent and each has the same distribution as \(Y\). The random variable \(S\) is defined as $$S = \sum _ { i = 1 } ^ { 3 } Y _ { i } - \frac { 1 } { 2 } X$$
  4. Find Var (S).
Edexcel S3 Q6
6. 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.
  2. Give a reason for using this method and give one advantage this method has over simple random sampling. The results Deepa obtained are summarised in the table below.
    Type of studentSample size
    Mean number of
    hours
    Arts1212.6
    Science1214.1
    Mixture810.2
  3. 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 marks) The standard deviation of the time spent on private study by students at the school was 2.48 hours.
  4. 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. A member of staff at the school suggested that A level students should spend on average 12 hours each week on private study.
  5. Comment on this suggestion in the light of your interval.
Edexcel S3 Q7
7. 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.
  2. Calculate the value of the rank correlation coefficient for these data.
  3. Stating your hypotheses clearly and using a \(1 \%\) level of significance, interpret this coefficient.
  4. Explain why the rank correlation coefficient might be the better one to use with these data. END
Edexcel S3 Specimen Q1
  1. 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.
    2. Give one advantage of this method over taking a simple random sample.
    3. 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.
    4. Write down the distribution of \(\bar { P }\), the mean weight of the 16 pears.
    5. Find \(\mathrm { P } ( 110 < \bar { P } < 113 )\).
    6. 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
  2. the total time taken by a single randomly chosen mechanic to carry out all three tasks lies between 533 and 655 minutes,
  3. a randomly chosen mechanic takes longer to carry out task \(B\) than task \(C\).
Edexcel S3 Specimen Q4
4. 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.
  2. Stating clearly your hypotheses and using a \(5 \%\) two-tailed test, interpret your rank correlation coefficient. Many sets of data include tied ranks.
  3. Explain briefly how tied ranks can be dealt with.
Edexcel S3 Specimen Q5
5. 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. State the name of the distribution. 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.
  3. 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
6. 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.
\(\boldsymbol { N }\)means.d.
Low income group300\(\pounds 6.40\)\(\pounds 6.69\)
High income group400\(\pounds 7.42\)\(\pounds 8.13\)
  1. Using a \(5 \%\) significance level, test whether or not the two groups differ in the mean amounts spent on tobacco.
  2. Explain briefly the importance of the central limit theorem in this example.
    (2)
Edexcel S3 Specimen Q7
7. 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.
Edexcel S3 Specimen Q8
8. Observations have been made over many years of \(T\), the noon temperature in \({ } ^ { \circ } \mathrm { C }\), on 21 st 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 \(T\). Over the years, the standard deviation of \(T\) has been found to be 5.1.
  2. 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 \(4 ^ { \circ } \mathrm { C }\).
  3. Use your interval to comment on the meteorologist's claim.
Edexcel S3 Q1
  1. A museum is open to the public for six hours a day from Monday to Friday every week. The number of visitors, \(V\), to the museum on ten randomly chosen days were as follows:
$$\begin{array} { l l l l l l l l l l } 182 & 172 & 113 & 99 & 168 & 183 & 135 & 129 & 150 & 108 \end{array}$$
  1. Calculate an unbiased estimate of the mean of \(V\). Assuming that \(V\) is normally distributed with a variance of 130 ,
  2. find a 95\% confidence interval for the mean of \(V\).
Edexcel S3 Q2
2. (a) Explain what is meant by a simple random sample.
(b) Explain briefly how you could use a table of random numbers to select a simple random sample of size 12 from a list of the 70 junior members of a tennis club.
(c) Give an example of a situation in which you might choose to take a stratified sample and explain why.
Edexcel S3 Q3
3. The time that a school pupil spends on French homework each week is normally distributed with a mean of 55 minutes and a standard deviation of 10 minutes. The time that this pupil spends on English homework each week is normally distributed with a mean of 1 hour 30 minutes and a standard deviation of 18 minutes. Find the probability that in a randomly chosen week
  1. the pupil spends more than 2 hours in total doing French and English homework,
  2. the pupil spends more than twice as long doing English homework as he spends doing French homework.
    (6 marks)
Edexcel S3 Q4
4. A group of 40 males and 40 females were asked which of three "Reality TV" shows they liked most - Watched, Stranded or One-2-Win. The results were as follows:
\cline { 2 - 4 } \multicolumn{1}{c|}{}WatchedStrandedOne-2-Win
Males21613
Females151015
Stating your hypotheses clearly, test at the \(10 \%\) level whether or not there is a significant difference in the preferences of males and females.
Edexcel S3 Q5
5. A marathon runner believes that she is more likely to win a medal at her national championships the higher the temperature is on the day of the race. She records the temperature at the start of each of eight races against fields of a similar standard and her finishing position in each race. Her results are shown in the table below.
Temperature \(\left( { } ^ { \circ } \mathrm { C } \right)\)1691157211215
Finishing position215519104611
  1. Calculate Spearman's rank correlation coefficient for these data.
  2. Using a 5\% level of significance and stating your hypotheses clearly, interpret your result. Another runner suggests that she should use her time in each race instead of her finishing position and calculate the product moment correlation coefficient for the data.
  3. Comment on this suggestion.
Edexcel S3 Q6
6. The weight of a particular electrical component is normally distributed with a mean of 46.7 grams and a variance of 1.8 grams \(^ { 2 }\). The component is sold in boxes of 12 .
  1. State the distribution of the mean weight of the components in one box.
  2. Find the probability that the mean weight of the components in a randomly chosen box is more than 47 grams.
    (3 marks)
    After a break in production the component manufacturer wishes to find out if the mean weight of the components has changed. A random sample of 30 components is found to have a mean weight of 46.5 grams.
  3. Assuming that the variance of the weight of the components is unchanged, test at the \(5 \%\) level of significance if there has been any change in the mean weight of the components.
    (7 marks)
Edexcel S3 Q7
7. A student collects data on whether competitors in local tennis tournaments are right, or left-handed. The table below shows the number of left-handed players who reached the last 16 for fifty tournaments.
No. of Left-handed Players01234\(\geq 5\)
No. of Tournaments412181150
The student believes that a binomial distribution with \(n = 16\) and \(p = 0.1\) could be a suitable model for these data.
  1. Stating your hypotheses clearly test the student's model at the \(5 \%\) level of significance.
    (13 marks)
    To improve the model the student decides to estimate \(p\) using the data in the table. Using this value of \(p\) to calculate expected frequencies the student had 5 classes after combining and calculated that \(\sum \frac { ( O - E ) ^ { 2 } } { E } = 2.127\)
  2. Test at the \(5 \%\) level of significance whether or not the binomial distribution is a suitable model for the number of left-handed players who reach the last 16 in local tennis tournaments. \section*{END}
Edexcel S3 Q1
  1. (a) Explain briefly the method of quota sampling.
    (b) Give one disadvantage of quota sampling compared with stratified sampling.
    (c) Describe a situation in which you would choose to use quota sampling rather than stratified sampling and explain why.
    (2 marks)
  2. 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 \(360 ^ { \circ }\) 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\(0 ^ { \circ } -\)\(45 ^ { \circ } -\)\(90 ^ { \circ } -\)\(180 ^ { \circ } -\)\(270 ^ { \circ } -\)\(315 ^ { \circ } - 360 ^ { \circ }\)
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 marks)