Questions — Edexcel S3 (313 questions)

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Edexcel S3 2014 June Q4
  1. The random variable \(A\) is defined as
$$A = B + 4 C - 3 D$$ where \(B\), \(C\) and \(D\) are independent random variables with $$B \sim \mathrm {~N} \left( 6,2 ^ { 2 } \right) \quad C \sim \mathrm {~N} \left( 7,3 ^ { 2 } \right) \quad D \sim \mathrm {~N} \left( 4,1.5 ^ { 2 } \right)$$ Find \(\mathrm { P } ( A < 45 )\)
Edexcel S3 2014 June Q5
5. A research station is doing some work on the germination of a new variety of genetically modified wheat. They planted 120 rows containing 7 seeds in each row.
The number of seeds germinating in each row was recorded. The results are as follows
Number of seeds germinating in each row01234567
Observed number of rows2611192532169
  1. Write down two reasons why a binomial distribution may be a suitable model.
  2. Show that the probability of a randomly selected seed from this sample germinating is 0.6 The research station used a binomial distribution with probability 0.6 of a seed germinating. The expected frequencies were calculated to 2 decimal places. The results are as follows
    Number of seeds germinating in each row01234567
    Expected number of rows0.202.06\(s\)23.22\(t\)31.3515.683.36
  3. Find the value of \(s\) and the value of \(t\).
  4. Stating your hypotheses clearly, test, at the \(1 \%\) level of significance, whether or not the data can be modelled by a binomial distribution.
Edexcel S3 2014 June Q6
6. A random sample \(X _ { 1 } , X _ { 2 } , \ldots , X _ { n }\) is taken from a population with mean \(\mu\).
  1. Show that \(\bar { X } = \frac { 1 } { n } \left( X _ { 1 } + X _ { 2 } + \ldots + X _ { n } \right)\) is an unbiased estimator of the population mean \(\mu\). A company produces small jars of coffee. Five jars of coffee were taken at random and weighed. The weights, in grams, were as follows $$\begin{array} { l l l l l } 197 & 203 & 205 & 201 & 195 \end{array}$$
  2. Calculate unbiased estimates of the population mean and variance of the weights of the jars produced by the company. It is known from previous results that the weights are normally distributed with standard deviation 4.8 g . The manager is going to take a second random sample. He wishes to ensure that there is at least a \(95 \%\) probability that the estimate of the population mean is within 1.25 g of its true value.
  3. Find the minimum sample size required.
Edexcel S3 2014 June Q7
7. A machine fills packets with \(X\) grams of powder where \(X\) is normally distributed with mean \(\mu\). Each packet is supposed to contain 1 kg of powder. To comply with regulations, the weight of powder in a randomly selected packet should be such that \(\mathrm { P } ( X < \mu - 30 ) = 0.0005\)
  1. Show that this requires the standard deviation to be 9.117 g to 3 decimal places. A random sample of 10 packets is selected from the machine. The weight, in grams, of powder in each packet is as follows 999.8991 .61000 .31006 .11008 .2997 .0993 .21000 .0997 .11002 .1
  2. Assuming that the standard deviation of the population is 9.117 g , test, at the \(1 \%\) significance level, whether or not the machine is delivering packets with mean weight of less than 1 kg . State your hypotheses clearly.
Edexcel S3 2014 June Q8
8. The heights, in metres, and weights, in kilograms, of a random sample of 9 men are shown in the table below
Man\(A\)\(B\)\(C\)\(D\)\(E\)\(F\)\(G\)\(H\)\(I\)
Height \(( x )\)1.681.741.751.761.781.821.841.881.98
Weight \(( y )\)757610077909511096120
  1. Given that \(\mathrm { S } _ { x x } = 0.0632 , \mathrm {~S} _ { y y } = 1957.5556\) and \(\mathrm { S } _ { x y } = 9.3433\) calculate, to 3 decimal places, the product moment correlation coefficient between height and weight for these men.
  2. Use your value of the product moment correlation coefficient to test whether or not there is evidence of a positive correlation between the height and weight of men. Use a \(5 \%\) significance level. State your hypotheses clearly. Peter does not know the heights or weights of the 9 men. He is given photographs of them and asked to put them in order of increasing weight. He puts them in the order $$A C E B G D I F H$$
  3. Find, to 3 decimal places, Spearman's rank correlation coefficient between Peter's order and the actual order.
  4. Use your value of Spearman’s rank correlation coefficient to test for evidence of Peter's ability to correctly order men, by their weight, from their photographs. Use a 5\% significance level and state your hypotheses clearly.
Edexcel S3 2015 June Q1
  1. A mobile library has 160 books for children on its records. The librarian believes that books with fewer pages are borrowed more often. He takes a random sample of 10 books for children.
    1. Explain how the librarian should select this random sample.
      (2)
    The librarian ranked the 10 books according to how often they had been borrowed, with 1 for the book borrowed the most and 10 for the book borrowed the least. He also recorded the number of pages in each book. The results are in the table below.
    Book\(A\)\(B\)\(C\)\(D\)\(E\)\(F\)\(G\)\(H\)\(I\)\(J\)
    Borrowing rank12345678910
    Number of pages502121158030190356283152317
  2. Calculate Spearman's rank correlation coefficient for these data.
  3. Test the librarian's belief using a \(5 \%\) level of significance. State your hypotheses clearly.
Edexcel S3 2015 June Q2
2. A researcher believes that the mean weight loss of those people using a slimming plan as part of a group is more than 1.5 kg a year greater than the mean weight loss of those using the plan on their own. The mean weight loss of a random sample of 80 people using the plan as part of a group is 8.7 kg with a standard deviation of 2.1 kg . The mean weight loss of a random sample of 65 people using the plan on their own is 6.6 kg with a standard deviation of 1.4 kg .
  1. Stating your hypotheses clearly, test the researcher's claim. Use a \(1 \%\) level of significance.
  2. For the test in part (a), state whether or not it is necessary to assume that the weight loss of a person using this plan has a normal distribution. Give a reason for your answer.
Edexcel S3 2015 June Q3
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
  1. 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
    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
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 2016 June Q1
  1. (a) State two reasons why stratified sampling might be a more suitable sampling method than simple random sampling.
    (b) State two reasons why stratified sampling might be a more suitable sampling method than quota sampling.
  2. A new drug to vaccinate against influenza was given to 110 randomly chosen volunteers. The volunteers were given the drug in one of 3 different concentrations, \(A , B\) and \(C\), and then were monitored to see if they caught influenza. The results are shown in the table below.
\cline { 2 - 4 } \multicolumn{1}{c|}{}ABC
Influenza12299
No influenza152322
Test, at the \(10 \%\) level of significance, whether or not there is an association between catching influenza and the concentration of the new drug. State your hypotheses and show your working clearly. You should state your expected frequencies to 2 decimal places.
(10)
Edexcel S3 2016 June Q3
3. (a) Describe when you would use Spearman's rank correlation coefficient rather than the product moment correlation coefficient to measure the strength of the relationship between two variables.
(1) A shop sells sunglasses and ice cream. For one week in the summer the shopkeeper ranked the daily sales of ice cream and sunglasses. The ranks are shown in the table below.
SunMonTuesWedsThursFriSat
Ice cream6475321
Sunglasses6572341
(b) Calculate Spearman's rank correlation coefficient for these data.
(c) Test, at the \(5 \%\) level of significance, whether or not there is a positive correlation between sales of ice cream and sales of sunglasses. State your hypotheses clearly. The shopkeeper calculates the product moment correlation coefficient from his raw data and finds \(r = 0.65\)
(d) Using this new coefficient, test, at the \(5 \%\) level of significance, whether or not there is a positive correlation between sales of ice cream and sales of sunglasses.
(e) Using your answers to part (c) and part (d), comment on the nature of the relationship between sales of sunglasses and sales of ice cream.
Edexcel S3 2016 June Q4
4. The weights of eggs are normally distributed with mean 60 g and standard deviation 5 g Sairah chooses 2 eggs at random.
  1. Find the probability that the difference in weight of these 2 eggs is more than 2 g
    (5) Sairah is packing eggs into cartons. The weight of an empty egg carton is normally distributed with mean 40 g and standard deviation 1.5 g
  2. Find the distribution of the total weight of a carton filled with 12 randomly chosen eggs.
  3. Find the probability that a randomly chosen carton, filled with 12 randomly chosen eggs, weighs more than 800 g
Edexcel S3 2016 June Q5
5. A doctor claims there is a higher mean lung capacity in people who exercise regularly compared to people who do not exercise regularly. He measures the lung capacity, \(x\), of 35 people who exercise regularly and 42 people who do not exercise regularly. His results are summarised in the table below.
\cline { 2 - 4 } \multicolumn{1}{c|}{}\(n\)\(\bar { x }\)\(s ^ { 2 }\)
Exercise regularly3526.312.2
Do not exercise regularly4224.810.1
  1. Test, at the \(5 \%\) level of significance, the doctor's claim. State your hypotheses clearly.
  2. State any assumptions you have made in testing the doctor's claim. The doctor decides to add another person who exercises regularly to his data. He measures the person's lung capacity and finds \(x = 31.7\)
  3. Find the unbiased estimate of the variance for the sample of 36 people who exercise regularly. Give your answer to 3 significant figures.
Edexcel S3 2016 June Q6
6. An airport manager carries out a survey of families and their luggage. Each family is allowed to check in a maximum of 4 suitcases. She observes 50 families at the check-in desk and counts the total number of suitcases each family checks in. The data are summarised in the table below.
Number of suitcases01234
Frequency6251261
The manager claims that the data can be modelled by a binomial distribution with \(p = 0.3\)
  1. Test the manager's claim at the \(5 \%\) level of significance. State your hypotheses clearly.
    Show your working clearly and give your expected frequencies to 2 decimal places.
    (8) The manager also carries out a survey of the time taken by passengers to check in. She records the number of passengers that check in during each of 100 five-minute intervals. The manager makes a new claim that these data can be modelled by a Poisson distribution. She calculates the expected frequencies given in the table below.
    Number of passengers012345 or more
    Observed frequency540311860
    Expected frequency16.5329.75\(r\)\(s\)7.233.64
  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 \(1 \%\) level of significance to test the manager's new claim.
Edexcel S3 2016 June Q7
7. A restaurant states that its hamburgers contain \(20 \%\) fat. Paul claims that the mean fat content of their hamburgers is less than \(20 \%\). Paul takes a random sample of 50 hamburgers from the restaurant and finds that they contain a mean fat content of 19.5\% with a standard deviation of 1.5\% You may assume that the fat content of hamburgers is normally distributed.
  1. Find the \(90 \%\) confidence interval for the mean fat content of hamburgers from the restaurant.
  2. State, with a reason, what action Paul should recommend the restaurant takes over the stated fat content of their hamburgers. The restaurant changes the mean fat content of their hamburgers to \(\mu \%\) and adjusts the standard deviation to \(2 \%\). Paul takes a sample of size \(n\) from this new batch of hamburgers. He uses the sample mean \(\bar { X }\) as an estimator of \(\mu\).
  3. Find the minimum value of \(n\) such that \(\mathrm { P } ( | \bar { X } - \mu | < 0.5 ) \geqslant 0.9\)
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.