Questions S3 (597 questions)

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Edexcel S3 2017 June Q1
  1. A company director decides to survey staff about changes to the company calendar. The company has staff in 4 different job roles
72 managers, 108 drivers, 180 administrators and 360 warehouse staff.
The director decides to take a stratified sample.
  1. Write down one advantage of using a stratified sample rather than a simple random sample for this survey.
  2. Find the number of staff in each job role that will be included in a stratified sample of 40 staff.
  3. Describe how to choose managers for the stratified sample.
Edexcel S3 2017 June Q2
2. \begin{figure}[h]
\includegraphics[alt={},max width=\textwidth]{585de4b0-906e-40c4-9045-966d68505eff-04_430_438_260_753} \captionsetup{labelformat=empty} \caption{Figure 1}
\end{figure} The pointer shown in Figure 1 is spun so that it comes to rest between 0 and 360 degrees.
Linda claims that it is equally likely to come to rest at any point between 0 and 360 degrees. She spins the pointer 100 times and her results are summarised in the table below. She calculates expected frequencies for some of the possible outcomes and these are also given in the table below.
Angle (degrees)\(0 - 45\)\(45 - 90\)\(90 - 180\)\(180 - 315\)\(315 - 360\)
Frequency1816182919
Expected frequency12.5\(a\)\(b\)\(c\)12.5
  1. Find the values of the missing expected frequencies \(a , b\) and \(c\).
  2. Stating your hypotheses clearly and using a \(5 \%\) level of significance, test whether or not Linda's claim is supported by these data.
Edexcel S3 2017 June Q3
  1. A junior judge is being trained by a senior judge to learn how to assess ice skaters. After the training, the judges each assess 6 ice skaters \(A , B , C , D , E\) and \(F\). They each list them in order of preference with the best ice skater first. The results are shown in the table below.
Rank123456
Senior Judge\(A\)\(B\)\(D\)\(C\)\(F\)\(E\)
Junior Judge\(B\)\(D\)\(A\)\(F\)\(C\)\(E\)
  1. Calculate Spearman's rank correlation coefficient for these data.
  2. Test, at the \(5 \%\) level of significance, whether or not there is evidence of a positive correlation between the rankings of the junior judge and the senior judge. State your hypotheses clearly.
  3. Comment on the effectiveness of the training delivered by the senior judge.
Edexcel S3 2017 June Q4
4. A psychologist carries out a survey of the perceived body weight of 150 randomly chosen people. He asks them if they think they are underweight, about right or overweight. His results are summarised in the table below.
\cline { 2 - 4 } \multicolumn{1}{c|}{}UnderweightAbout rightOverweight
Male202230
Female162834
The psychologist calculates two of the expected frequencies, to 2 decimal places, for a test of independence between perceived body weight and gender. These results are shown in the table below.
\cline { 2 - 4 } \multicolumn{1}{c|}{}UnderweightAbout rightOverweight
Male17.28
Female18.72
  1. Complete the table of expected frequencies shown above.
  2. Test, at the \(10 \%\) level of significance, whether or not perceived body weight is independent of gender. State your hypotheses clearly. The psychologist now combines the male and female data to test whether or not body weight types are chosen equally.
  3. Find the smallest significance level, from the tables in the formula booklet, for which there is evidence of a preference.
Edexcel S3 2017 June Q5
5. Paul takes the company bus to work. According to the bus timetable he should arrive at work at 0831. Paul believes the bus is not reliable and often arrives late. Paul decides to test the arrival time of the bus and carries out a survey. He records the values of the random variable $$X = \text { number of minutes after } 0831 \text { when the bus arrives. }$$ His results are summarised below. $$n = 15 \quad \sum x = 60 \quad \sum x ^ { 2 } = 1946$$
  1. Calculate unbiased estimates of the mean, \(\mu\), and the variance of \(X\). Using the mean of Paul's sample and given \(X \sim \mathrm {~N} \left( \mu , 10 ^ { 2 } \right)\)
    1. calculate a 95\% confidence interval for the mean arrival time at work for this company bus.
    2. State an assumption you made about the values in the sample obtained by Paul.
  3. Comment on Paul's belief. Justify your answer.
Edexcel S3 2017 June Q6
6. An engineer has developed a new battery. She claims that the new battery will last more than 8 hours longer, on average, than the old battery. To test the claim, the engineer randomly selects a sample of 50 new batteries and 40 old batteries. She records how long each battery lasts, \(x\) hours for the new batteries and \(y\) hours for the old batteries. The results are summarised in the table below.
\cline { 2 - 4 } \multicolumn{1}{c|}{}\(n\)Sample mean\(s ^ { 2 }\)
New battery50\(\bar { x } = 83\)7
Old battery40\(\bar { y } = 74\)6
  1. Test, at the \(5 \%\) level of significance, whether or not there is evidence to support the engineer's claim. State your hypotheses and show your working clearly.
  2. Explain the relevance of the Central Limit Theorem to the test in part (a).
Edexcel S3 2017 June Q7
7. Sugar is packed into medium bags and large bags. The weights of the medium bags of sugar are normally distributed with mean 520 grams and standard deviation 10 grams. The weights of the large bags of sugar are normally distributed with mean 1510 grams and standard deviation 20 grams.
  1. Find the probability that a randomly chosen large bag of sugar weighs at least 15 grams more than the combined weight of 3 randomly chosen medium bags of sugar.
  2. Find the probability that a randomly chosen large bag of sugar weighs less than 3 times the weight of a randomly chosen medium bag of sugar. A random sample of 5 medium bags of sugar is taken.
  3. Find the value of \(d\) so that the probability that all 5 bags of sugar each weigh more than 520 grams is equal to the probability that the mean weight of the 5 bags of sugar is more than \(d\) grams.
Edexcel S3 2018 June Q1
  1. Phil measures the concentration of a radioactive element, \(c\), and the amount of dissolved solids, \(a\), of 8 random samples of groundwater. His results are shown in the table below.
Sample\(A\)\(B\)\(C\)\(D\)\(E\)\(F\)\(G\)\(H\)
\(c\)625700650645720600825665
\(a\)1.281.301.001.201.551.151.401.45
Given that $$\mathrm { S } _ { c c } = 34787.5 \quad \mathrm {~S} _ { a a } = 0.2172875 \quad \mathrm {~S} _ { c a } = 47.7625$$
  1. calculate, to 3 decimal places, the product moment correlation coefficient between the concentration of the radioactive element and the amount of dissolved solids for these groundwater samples.
  2. Use your value of the product moment correlation coefficient to test whether or not there is evidence of a positive correlation between the concentration of this radioactive element and the amount of dissolved solids in groundwater. Use a \(5 \%\) significance level. State your hypotheses clearly.
  3. Calculate, to 3 decimal places, Spearman's rank correlation coefficient between the concentration of the radioactive element and the amount of dissolved solids.
  4. Use your value of Spearman's rank correlation coefficient to test for evidence of a positive correlation between the concentration of the radioactive element and the amount of dissolved solids. Use a \(5 \%\) significance level. State your hypotheses clearly.
  5. Using your conclusions in part (b) and part (d), comment on the possible relationship between these variables.
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.