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Edexcel S3 2015 June Q3
11 marks Moderate -0.8
3. A nursery has 16 staff and 40 children on its records. In preparation for an outing the manager needs an estimate of the mean weight of the people on its records and decides to take a stratified sample of size 14 .
  1. Describe how this stratified sample should be taken. The weights, \(x \mathrm {~kg}\), of each of the 14 people selected are summarised as $$\sum x = 437 \text { and } \sum x ^ { 2 } = 26983$$
  2. Find unbiased estimates of the mean and the variance of the weights of all the people on the nursery's records.
  3. Estimate the standard error of the mean. The estimates of the standard error of the mean for the staff and for the children are 5.11 and 1.10 respectively.
  4. Comment on these values with reference to your answer to part (c) and give a reason for any differences.
Edexcel S3 2015 June Q4
9 marks Standard +0.3
The weights of bags of rice, \(X \mathrm {~kg}\), have a normal distribution with unknown mean \(\mu \mathrm { kg }\) and known standard deviation \(\sigma \mathrm { kg }\). A random sample of 100 bags of rice gave a \(90 \%\) confidence interval for \(\mu\) of \(( 0.4633,0.5127 )\).
  1. Without carrying out any further calculations, use this confidence interval to test whether or not \(\mu = 0.5\) State your hypotheses clearly and write down the significance level you have used. A second random sample, of 150 of these bags of rice, had a mean weight of 0.479 kg .
  2. Calculate a \(95 \%\) confidence interval for \(\mu\) based on this second sample.
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 \(D = Y _ { 1 } - \bar { X }\)
  2. 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 \(\sigma\).
  3. Explain briefly why the result from part (b) should not be used to confirm Ankit's belief.
  4. 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
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
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 D1 2015 January Q1
6 marks Moderate -0.8
1.
ABCDEFGH
A-981317111210
B9-11211524137
C811-2023171715
D132120-15281118
E17152315-312330
F1124172831-1315
G121317112313-23
H1071518301523-
The table represents a network that shows the time taken, in minutes, to travel by car between eight villages, \(\mathrm { A } , \mathrm { B } , \mathrm { C } , \mathrm { D } , \mathrm { E } , \mathrm { F } , \mathrm { G }\) and H .
  1. Use Prim's algorithm, starting at A , to find a minimum spanning tree for this network. You must list the arcs that form your tree in the order in which you select them.
  2. Draw your minimum spanning tree using the vertices given in Diagram 1 in the answer book and state the weight of the tree.
  3. State whether your minimum spanning tree is unique. Justify your answer.
Edexcel D1 2015 January Q2
8 marks Moderate -0.8
2. \begin{figure}[h]
\includegraphics[alt={},max width=\textwidth]{00abfcc0-63b3-4784-a4b5-06aba234068c-3_616_561_278_356} \captionsetup{labelformat=empty} \caption{Figure 1}
\end{figure} \begin{figure}[h]
\includegraphics[alt={},max width=\textwidth]{00abfcc0-63b3-4784-a4b5-06aba234068c-3_606_552_285_1146} \captionsetup{labelformat=empty} \caption{Figure 2}
\end{figure} Figure 1 shows the possible allocations of six workers, Amrit (A), Bernard (B), Cameron (C), David (D), Emily (E) and Francis (F), to six tasks, 1, 2, 3, 4, 5 and 6
  1. Explain why it is not possible to find a complete matching. Figure 2 shows an initial matching. Starting from this initial matching,
  2. find the two alternating paths that start at C .
  3. List the two improved matchings generated by using the two alternating paths found in (b). After training, task 5 is added to Bernard's possible allocation.
    Starting from either of the two improved matchings found in (c),
  4. use the maximum matching algorithm to obtain a complete matching. You must list the additional alternating path that you use, and state the complete matching.
    (3)
Edexcel D1 2015 January Q3
14 marks Easy -1.2
3. $$\begin{array} { l l l l l l l l l l } 1.1 & 0.7 & 1.9 & 0.9 & 2.1 & 0.2 & 2.3 & 0.4 & 0.5 & 1.7 \end{array}$$
  1. Use the first-fit bin packing algorithm to determine how the numbers listed above can be packed into bins of size 3 The list is to be sorted into descending order.
    1. Starting at the left-hand end of the list, perform one pass through the list using a bubble sort. Write down the list that results at the end of your first pass.
    2. Write down the number of comparisons and the number of swaps performed during your first pass. After a second pass using this bubble sort, the updated list is $$\begin{array} { l l l l l l l l l l } 1.9 & 1.1 & 2.1 & 0.9 & 2.3 & 0.7 & 0.5 & 1.7 & 0.4 & 0.2 \end{array}$$
  3. Use a quick sort on this updated list to obtain the fully sorted list. You must make your pivots clear.
  4. Apply the first-fit decreasing bin packing algorithm to your fully sorted list to pack the numbers into bins of size 3
Edexcel D1 2015 January Q4
16 marks Challenging +1.2
4. \begin{figure}[h]
\includegraphics[alt={},max width=\textwidth]{00abfcc0-63b3-4784-a4b5-06aba234068c-5_889_1591_258_239} \captionsetup{labelformat=empty} \caption{Figure 3
[0pt] [The total weight of the network is 100]}
\end{figure} Figure 3 represents a network of pipes in a building. The number on each arc represents the length, in metres, of the corresponding pipe.
  1. Use Dijkstra's algorithm to find the shortest path from A to J . State your path and its length. On a particular day Kim needs to check each pipe. A route of minimum length, which traverses each pipe at least once and starts and finishes at A, needs to be found.
  2. Use an appropriate algorithm to find the arcs that will need to be traversed twice. You must make your method and working clear.
  3. Write down a possible route, giving its length. All the pipes directly attached to B are removed. Kim needs to check all the remaining pipes and may now start at any vertex and finish at any vertex. A route is required that excludes all those pipes directly attached to B .
  4. State all possible combinations of starting and finishing points so that the length of Kim's route is minimised. State the length of Kim's route.
Edexcel D1 2015 January Q5
7 marks Moderate -0.8
5.
ActivityImmediately preceding activities
A-
B-
CA
DA
EA, B
FC
GC, D
HE
IE
JH, I
KF, G
  1. Draw the activity network described in the precedence table, using activity on arc. Your activity network must contain only the minimum number of dummies.
    (5)
  2. Explain why, in general, dummies may be required in an activity network.
Edexcel D1 2015 January Q6
12 marks Moderate -0.8
6. Jonathan is going to make hats to sell at a fete. He can make red hats and green hats. Jonathan can use linear programming to determine the number of each colour of hat that he should make. Let \(x\) be the number of red hats he makes and \(y\) be the number of green hats he makes.
One of the constraints is that there must be at least 30 hats.
  1. Write down an inequality, in terms of \(x\) and \(y\), to model this constraint. Two further constraints are $$\begin{aligned} & 2 y + x \geqslant 40 \\ & 2 y - x \geqslant - 30 \end{aligned}$$
  2. Write down two more constraints which apply.
  3. Represent all these constraints on Diagram 1 in the answer book. Hence determine, and label, the feasible region R . The cost of making a green hat is three times the cost of making a red hat. Jonathan wishes to minimise the total cost.
  4. Use the objective line (ruler) method to determine the number of red hats and number of green hats that Jonathan should make. You must clearly draw and label your objective line. Given that the minimum total cost of making the hats is \(\pounds 107.50\)
  5. determine the cost of making one green hat and the cost of making one red hat. You must make your method clear.
Edexcel D1 2015 January Q7
12 marks Moderate -0.5
7. \begin{figure}[h]
\includegraphics[alt={},max width=\textwidth]{00abfcc0-63b3-4784-a4b5-06aba234068c-8_980_1577_229_268} \captionsetup{labelformat=empty} \caption{Figure 4
[0pt] [The sum of all the activity durations is 99 days]}
\end{figure} The network in Figure 4 shows the activities that need to be undertaken to complete a project. Each activity is represented by an arc and the duration of the activity, in days, is shown in brackets. The early event times and late event times are to be shown at each vertex and some have been completed for you. Given that activity F is a critical activity and that the total float on activity G is 2 days,
  1. write down the value of \(x\) and the value of \(y\),
  2. calculate the missing early event times and late event times and hence complete Diagram 1 in your answer book. Each activity requires one worker and the project must be completed in the shortest possible time.
  3. Calculate a lower bound for the number of workers needed to complete the project in the shortest possible time.
  4. Draw a cascade (Gantt) chart for this project on Grid 1 in the answer book.
  5. Use your cascade chart to determine the minimum number of workers needed to complete the project in the shortest possible time. You must make specific reference to times and activities. (You do not need to provide a schedule of the activities.)
Edexcel D1 2016 January Q1
8 marks Moderate -0.3
1. \begin{figure}[h]
\includegraphics[alt={},max width=\textwidth]{a3ca2743-2311-4225-8b78-dcd5eb592704-2_741_558_370_372} \captionsetup{labelformat=empty} \caption{Figure 1}
\end{figure} \begin{figure}[h]
\includegraphics[alt={},max width=\textwidth]{a3ca2743-2311-4225-8b78-dcd5eb592704-2_734_561_374_1114} \captionsetup{labelformat=empty} \caption{Figure 2}
\end{figure} Figure 1 shows the possible allocations of six workers, \(\mathrm { A } , \mathrm { B } , \mathrm { C } , \mathrm { D } , \mathrm { E }\) and F , to six tasks, 1, 2, 3, 4, 5 and 6. Figure 2 shows an initial matching.
  1. Starting from this initial matching, use the maximum matching algorithm to find an improved matching. You must list the alternating path used and state your improved matching.
  2. Explain why it is not possible to find a complete matching. Now, exactly one worker may be trained so that a complete matching becomes possible.
    Either worker A can be trained to do task 1 or worker E can be trained to do task 4.
  3. Decide which worker, A or E , should be trained. Give a reason for your answer. You may now assume that the worker you identified in (c) has been trained.
  4. Starting from the improved matching found in (a), use the maximum matching algorithm to find a complete matching. You must list the alternating path used and state your complete matching.