Questions — Edexcel S3 (332 questions)

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Edexcel S3 2015 June Q5
17 marks Standard +0.8
    1. The volume, \(B \mathrm { ml }\), in a bottle of Burxton's water has a normal distribution \(B \sim \mathrm {~N} \left( 325,6 ^ { 2 } \right)\) and the volume, \(H \mathrm { ml }\), in a bottle of Hargate's water has a normal distribution \(H \sim \mathrm {~N} \left( 330,4 ^ { 2 } \right)\).
      Rebecca buys 5 bottles of Burxton's water and one bottle of Hargate's water.
      Find the probability that the total volume in the 5 bottles of Burxton's water is more than 5 times the volume in the bottle of Hargate's water.
      (5)
    2. Two independent random samples \(X _ { 1 } , X _ { 2 } , X _ { 3 } , X _ { 4 } , X _ { 5 }\) and \(Y _ { 1 } , Y _ { 2 } , Y _ { 3 } , Y _ { 4 } , Y _ { 5 }\) are each taken from a normal population with mean \(\mu\) and standard deviation \(\sigma\).
      1. Find the distribution of the random variable \(D = Y _ { 1 } - \bar { X }\)
    3. Hence show that \(\mathrm { P } \left( Y _ { 1 } > \bar { X } + \sigma \right) = 0.181\) correct to 3 decimal places.
    Ankit believes that \(\mathrm { P } \left( U _ { 1 } > \bar { U } + \sigma \right) = 0.181\) correct to 3 decimal places, for any random sample \(U _ { 1 } , U _ { 2 } , U _ { 3 } , U _ { 4 } , U _ { 5 }\) taken from a normal population with mean \(\mu\) and standard deviation \(\sigma\).
  2. Explain briefly why the result from part (b) should not be used to confirm Ankit's belief.
  3. Find, correct to 3 decimal places, the actual value of \(\mathrm { P } \left( U _ { 1 } > \bar { U } + \sigma \right)\).
Edexcel S3 2015 June Q6
19 marks Standard +0.3
6. \begin{figure}[h]
\includegraphics[alt={},max width=\textwidth]{740f7555-3a9a-4526-9048-39908aa8f8dd-10_684_694_239_625} \captionsetup{labelformat=empty} \caption{Figure 1}
\end{figure} The sketch in Figure 1 represents a target which consists of 4 regions formed from 4 concentric circles of radii \(4 \mathrm {~cm} , 7 \mathrm {~cm} , 9 \mathrm {~cm}\) and 10 cm . The regions are coloured as labelled in Figure 1.
A random sample of 100 children each choose a point on the target and their results are summarised in the table below. (b) Find the value of \(r\) and the value of \(s\). Henry obtained a test statistic of 6.188 and no groups were pooled.
(c) State what conclusion Henry should make about his claim. Phoebe believes that the children chose the region of the target according to colour. She believes that boys and girls would favour different colours and splits the original data by gender to obtain the following table. \section*{Observed frequencies}
Colour of regionGreenRedBlueYellowTotal
Boys101210335
Girls1227151165
(d) State suitable hypotheses to test Phoebe's belief. Phoebe calculated the following expected frequencies to carry out a suitable test. \section*{Expected frequencies}
Colour of regionGreenRedBlueYellow
Boys7.713.658.754.9
Girls14.325.3516.259.1
(e) Show how the value of 25.35 was obtained. Phoebe carried out the test using 2 degrees of freedom and a \(10 \%\) level of significance. She obtained a test statistic of 1.411
(f) Explain clearly why Phoebe used 2 degrees of freedom.
(g) Stating your critical value clearly, determine whether or not these data support Phoebe's belief.
Edexcel S3 2017 June Q1
6 marks Easy -1.3
  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
10 marks Standard +0.3
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
10 marks Standard +0.3
  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
14 marks Standard +0.3
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
10 marks Moderate -0.3
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
9 marks Standard +0.3
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
16 marks Challenging +1.2
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
13 marks Standard +0.3
  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
13 marks Standard +0.3
  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
10 marks Standard +0.3
  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
9 marks Standard +0.3
  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
12 marks Challenging +1.2
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
18 marks Standard +0.3
  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
6 marks Moderate -0.5
  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
7 marks Easy -1.3
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
11 marks Standard +0.3
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
11 marks Standard +0.3
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
12 marks Standard +0.3
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
12 marks Standard +0.3
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
16 marks Standard +0.3
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
4 marks Easy -1.2
  1. A Veterinary Surgeon wishes to survey a stratified sample of size 100 from those people who have pets registered at her surgery. The list below shows the strata to be used and the number in each group.
  • people who own just dogs - 165 ,
  • people who own just cats - 140 ,
  • people who own just small mammals - 105,
  • others, including those who own more than one type of pet - 90 .
    1. Find how many members of each group should be included in the sample.
    2. Give two advantages of using stratified sampling.
Edexcel S3 Q2
9 marks Standard +0.3
  1. A psychologist is investigating the numbers people choose when asked to pick a number at random in a given interval. He finds that when asked to pick a number between 0 and 100 people are less likely to pick certain numbers, such as multiples of ten. He believes, however that if people are asked to pick an odd number between 0 and 100 they are equally likely to pick a number ending in any of the digits \(1,3,5,7\) or 9 .
To test this theory he asks 80 people to pick an odd number between 0 and 100 and records the last digit of the numbers chosen. His results are shown in the table below.
Last Digit13579
Frequency1620141713
Stating your hypotheses clearly and using a 10\% level of significance test the psychologist's theory.
(9 marks)
Edexcel S3 Q3
10 marks Standard +0.3
3. A clothes manufacturer wishes to find out if adult females have become taller on average since twenty years ago when their mean height was 5 ft 6 inches. Studies over time have shown that the standard deviation of the height of adult females has been fairly constant at 2.3 inches. The manager wishes to test if the mean height is now more than 5 ft 6 inches and takes a sample of 150 adult females.
  1. Stating your hypotheses clearly, find the critical region for the mean height of the sample for a test at the \(5 \%\) level of significance. The total height of the females in the sample is 832 ft .
  2. Carry out the test making your conclusion clear.