Questions — Edexcel (10514 questions)

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Edexcel S3 2004 June Q5
12 marks Standard +0.3
5. A random sample of 500 adults completed a questionnaire on how often they took part in some form of exercise. They gave a response of 'never', 'sometimes' or 'regularly'. Of those asked, \(52 \%\) were females of whom \(10 \%\) never exercised and \(35 \%\) exercised regularly. Of the males, \(12.5 \%\) never exercised and \(55 \%\) sometimes exercised. Test, at the \(5 \%\) level of significance, whether or not there is any association between gender and the amount of exercise. State your hypotheses clearly.
Edexcel S3 2004 June Q6
15 marks Standard +0.3
6 Three six-sided dice, which were assumed to be fair, were rolled 250 times. On each occasion the number \(X\) of sixes was recorded. The results were as follows.
Number of sixes0123
Frequency125109133
  1. Write down a suitable model for \(X\).
  2. Test, at the \(1 \%\) level of significance, the suitability of your model for these data.
  3. Explain how the test would have been modified if it had not been assumed that the dice were fair.
Edexcel S3 2004 June Q7
16 marks Standard +0.3
7. The random variable \(D\) is defined as $$D = A - 3 B + 4 C$$ where \(A \sim \mathrm {~N} \left( 5,2 ^ { 2 } \right) , B \sim \mathrm {~N} \left( 7,3 ^ { 2 } \right)\) and \(C \sim \mathrm {~N} \left( 9,4 ^ { 2 } \right)\), and \(A , B\) and \(C\) are independent.
  1. Find \(\mathrm { P } ( \mathrm { D } < 44 )\). The random variables \(B _ { 1 } , B _ { 2 }\) and \(B _ { 3 }\) are independent and each has the same distribution as \(B\). The random variable \(X\) is defined as $$X = A - \sum _ { i = 1 } ^ { 3 } B _ { i } + 4 C .$$
  2. Find \(\mathrm { P } ( X > 0 )\). \section*{END}
Edexcel S3 2007 June Q1
10 marks Standard +0.3
  1. During a village show, two judges, \(P\) and \(Q\), had to award a mark out of 30 to some flower displays. The marks they awarded to a random sample of 8 displays were as follows:
Display\(A\)\(B\)\(C\)\(D\)\(E\)\(F\)\(G\)\(H\)
Judge \(P\)2519212328171620
Judge \(Q\)209211317141115
  1. Calculate Spearman's rank correlation coefficient for the marks awarded by the two judges. After the show, one competitor complained about the judges. She claimed that there was no positive correlation between their marks.
  2. Stating your hypotheses clearly, test whether or not this sample provides support for the competitor's claim. Use a \(5 \%\) level of significance.
Edexcel S3 2007 June Q2
10 marks Standard +0.3
  1. The Director of Studies at a large college believed that students' grades in Mathematics were independent of their grades in English. She examined the results of a random group of candidates who had studied both subjects and she recorded the number of candidates in each of the 6 categories shown.
Maths grade A or BMaths grade C or DMaths grade E or U
English grade A or B252510
English grade C to U153015
  1. Stating your hypotheses clearly, test the Director's belief using a \(10 \%\) level of significance. You must show each step of your working. The Head of English suggested that the Director was losing accuracy by combining the English grades C to U in one row. He suggested that the Director should split the English grades into two rows, grades C or D and grades E or U as for Mathematics.
  2. State why this might lead to problems in performing the test.
Edexcel S3 2007 June Q3
7 marks Moderate -0.3
  1. The time, in minutes, it takes Robert to complete the puzzle in his morning newspaper each day is normally distributed with mean 18 and standard deviation 3. After taking a holiday, Robert records the times taken to complete a random sample of 15 puzzles and he finds that the mean time is 16.5 minutes. You may assume that the holiday has not changed the standard deviation of times taken to complete the puzzle.
Stating your hypotheses clearly test, at the \(5 \%\) level of significance, whether or not there has been a reduction in the mean time Robert takes to complete the puzzle.
Edexcel S3 2007 June Q4
13 marks Standard +0.3
4. A quality control manager regularly samples 20 items from a production line and records the number of defective items \(x\). The results of 100 such samples are given in table 1 below. \begin{table}[h]
\(x\)01234567 or more
Frequency173119149730
\captionsetup{labelformat=empty} \caption{Table 1}
\end{table}
  1. Estimate the proportion of defective items from the production line. The manager claimed that the number of defective items in a sample of 20 can be modelled by a binomial distribution. He used the answer in part (a) to calculate the expected frequencies given in Table 2. \begin{table}[h]
    \(x\)01234567 or more
    Expected
    frequency
    		12.227.0\(r\)19.0\(s\)3.20.90.2
    \captionsetup{labelformat=empty} \caption{Table 2}
    \end{table}
  2. Find the value of \(r\) and the value of \(s\) giving your answers to 1 decimal place.
  3. Stating your hypotheses clearly, use a \(5 \%\) level of significance to test the manager's claim.
  4. Explain what the analysis in part (c) tells the manager about the occurrence of defective items from this production line.
Edexcel S3 2007 June Q5
14 marks Standard +0.3
  1. In a trial of \(\operatorname { diet } A\) a random sample of 80 participants were asked to record their weight loss, \(x \mathrm {~kg}\), after their first week of using the diet. The results are summarised by
$$\sum x = 361.6 \text { and } \sum x ^ { 2 } = 1753.95$$
  1. Find unbiased estimates for the mean and variance of weight lost after the first week of using diet \(A\). The designers of diet \(A\) believe it can achieve a greater mean weight loss after the first week than a standard diet \(B\). A random sample of 60 people used diet \(B\). After the first week they had achieved a mean weight loss of 4.06 kg , with an unbiased estimate of variance of weight loss of \(2.50 \mathrm {~kg} ^ { 2 }\).
  2. Test, at the \(5 \%\) level of significance, whether or not the mean weight loss after the first week using \(\operatorname { diet } A\) is greater than that using diet \(B\). State your hypotheses clearly.
  3. Explain the significance of the central limit theorem to the test in part (b).
  4. State an assumption you have made in carrying out the test in part (b).
Edexcel S3 2007 June Q6
6 marks Standard +0.3
  1. A random sample of the daily sales (in £s) of a small company is taken and, using tables of the normal distribution, a 99\% confidence interval for the mean daily sales is found to be
    (123.5, 154.7)
Find a \(95 \%\) confidence interval for the mean daily sales of the company.
(6)
Edexcel S3 2007 June Q7
15 marks Standard +0.8
7. A set of scaffolding poles come in two sizes, long and short. The length \(L\) of a long pole has the normal distribution \(\mathrm { N } \left( 19.7,0.5 ^ { 2 } \right)\). The length \(S\) of a short pole has the normal distribution \(\mathrm { N } \left( 4.9,0.2 ^ { 2 } \right)\). The random variables \(L\) and \(S\) are independent. A long pole and a short pole are selected at random.
  1. Find the probability that the length of the long pole is more than 4 times the length of the short pole. Four short poles are selected at random and placed end to end in a row. The random variable \(T\) represents the length of the row.
  2. Find the distribution of \(T\).
  3. Find \(\mathrm { P } ( | L - T | < 0.1 )\).
Edexcel S3 2008 June Q1
8 marks Moderate -0.8
  1. Some biologists were studying a large group of wading birds. A random sample of 36 were measured and the wing length, \(x \mathrm {~mm}\) of each wading bird was recorded. The results are summarised as follows
$$\sum x = 6046 \quad \sum x ^ { 2 } = 1016338$$
  1. Calculate unbiased estimates of the mean and the variance of the wing lengths of these birds. Given that the standard deviation of the wing lengths of this particular type of bird is actually 5.1 mm ,
  2. find a \(99 \%\) confidence interval for the mean wing length of the birds from this group.
Edexcel S3 2008 June Q2
11 marks Standard +0.3
2. Students in a mixed sixth form college are classified as taking courses in either Arts, Science or Humanities. A random sample of students from the college gave the following results
\cline { 3 - 4 } \multicolumn{2}{c|}{}Course
\cline { 3 - 5 } \multicolumn{2}{c|}{}ArtsScienceHumanities
EsuderBoy305035
\cline { 2 - 5 }Girl402042
Showing your working clearly, test, at the \(1 \%\) level of significance, whether or not there is an association between gender and the type of course taken. State your hypotheses clearly.
Edexcel S3 2008 June Q3
14 marks Standard +0.3
The product moment correlation coefficient is denoted by \(r\) and Spearman's rank correlation coefficient is denoted by \(r _ { s }\).
  1. Sketch separate scatter diagrams, with five points on each diagram, to show
    1. \(r = 1\),
    2. \(r _ { s } = - 1\) but \(r > - 1\). Two judges rank seven collie dogs in a competition. The collie dogs are labelled \(A\) to \(G\) and the rankings are as follows
      Rank1234567
      Judge 1\(A\)\(C\)\(D\)\(B\)\(E\)\(F\)\(G\)
      Judge 2\(A\)\(B\)\(D\)\(C\)\(E\)\(G\)\(F\)
    (b)
    1. Calculate Spearman's rank correlation coefficient for these data.
    2. Stating your hypotheses clearly, test, at the \(5 \%\) level of significance, whether or not the judges are generally in agreement.
Edexcel S3 2008 June Q4
11 marks Standard +0.8
The weights of adult men are normally distributed with a mean of 84 kg and a standard deviation of 11 kg .
  1. Find the probability that the total weight of 4 randomly chosen adult men is less than 350 kg . The weights of adult women are normally distributed with a mean of 62 kg and a standard deviation of 10 kg .
  2. Find the probability that the weight of a randomly chosen adult man is less than one and a half times the weight of a randomly chosen adult woman.
Edexcel S3 2008 June Q5
10 marks Easy -1.2
  1. A researcher is hired by a cleaning company to survey the opinions of employees on a proposed pension scheme. The company employs 55 managers and 495 cleaners.
To collect data the researcher decides to give a questionnaire to the first 50 cleaners to leave at the end of the day.
  1. Give 2 reasons why this method is likely to produce biased results.
  2. Explain briefly how the researcher could select a sample of 50 employees using
    1. a systematic sample,
    2. a stratified sample. Using the random number tables in the formulae book, and starting with the top left hand corner (8) and working across, 50 random numbers between 1 and 550 inclusive were selected. The first two suitable numbers are 384 and 100 .
  3. Find the next two suitable numbers.
Edexcel S3 2008 June Q6
13 marks Standard +0.3
  1. Ten cuttings were taken from each of 100 randomly selected garden plants. The numbers of cuttings that did not grow were recorded.
The results are as follows
No. of cuttings
which did
not grow
012345678,9 or 10
Frequency11213020123210
  1. Show that the probability of a randomly selected cutting, from this sample, not growing is 0.223 A gardener believes that a binomial distribution might provide a good model for the number of cuttings, out of 10 , that do not grow. He uses a binomial distribution, with the probability 0.2 of a cutting not growing. The calculated expected frequencies are as follows
    No. of cuttings which did
    not grow
    		012345 or more
    Expected frequency\(r\)26.84\(s\)20.138.81\(t\)
  2. Find the values of \(r , s\) and \(t\).
  3. State clearly the hypotheses required to test whether or not this binomial distribution is a suitable model for these data. The test statistic for the test is 4.17 and the number of degrees of freedom used is 4 .
  4. Explain fully why there are 4 degrees of freedom.
  5. Stating clearly the critical value used, carry out the test using a \(5 \%\) level of significance.
Edexcel S3 2008 June Q7
8 marks Standard +0.3
  1. A sociologist is studying how much junk food teenagers eat. A random sample of 100 female teenagers and an independent random sample of 200 male teenagers were asked to estimate what their weekly expenditure on junk food was. The results are summarised below.
\(n\)means.d.
Female teenagers100\(\pounds 5.48\)\(\pounds 3.62\)
Male teenagers200\(\pounds 6.86\)\(\pounds 4.51\)
  1. Using a 5\% significance level, test whether or not there is a difference in the mean amounts spent on junk food by male teenagers and female teenagers. State your hypotheses clearly.
  2. Explain briefly the importance of the central limit theorem in this problem.
Edexcel S3 2010 June Q3
10 marks Moderate -0.3
3. A woodwork teacher measures the width, \(w \mathrm {~mm}\), of a board. The measured width, \(X \mathrm {~mm}\), is normally distributed with mean \(w \mathrm {~mm}\) and standard deviation 0.5 mm .
  1. Find the probability that \(X\) is within 0.6 mm of \(w\). The same board is measured 16 times and the results are recorded.
  2. Find the probability that the mean of these results is within 0.3 mm of \(w\). Given that the mean of these 16 measurements is 35.6 mm ,
  3. find a \(98 \%\) confidence interval for \(w\).
Edexcel S3 2010 June Q4
10 marks Standard +0.3
  1. A researcher claims that, at a river bend, the water gradually gets deeper as the distance from the inner bank increases. He measures the distance from the inner bank, \(b \mathrm {~cm}\), and the depth of a river, \(s \mathrm {~cm}\), at seven positions. The results are shown in the table below.
Position\(A\)\(B\)\(C\)\(D\)\(E\)\(F\)\(G\)
Distance from
inner bank \(b \mathrm {~cm}\)
100200300400500600700
Depth
\(s \mathrm {~cm}\)
60758576110120104
  1. Calculate Spearman's rank correlation coefficient between \(b\) and \(s\).
  2. Stating your hypotheses clearly, test whether or not the data provides support for the researcher's claim. Use a \(1 \%\) level of significance.
Edexcel S3 2010 June Q5
10 marks Standard +0.3
  1. A random sample of 100 people were asked if their finances were worse, the same or better than this time last year. The sample was split according to their annual income and the results are shown in the table below.
Annual income FinancesWorseSameBetter
Under \(\pounds 15000\)14119
\(\pounds 15000\) and above172029
Test, at the \(5 \%\) level of significance, whether or not the relative state of their finances is independent of their income range. State your hypotheses and show your working clearly.
Edexcel S3 2010 June Q6
12 marks Standard +0.8
  1. A total of 228 items are collected from an archaeological site. The distance from the centre of the site is recorded for each item. The results are summarised in the table below.
Distance from the
centre of the site \(( \mathrm { m } )\)
\(0 - 1\)\(1 - 2\)\(2 - 4\)\(4 - 6\)\(6 - 9\)\(9 - 12\)
Number of items221544375258
Test, at the \(5 \%\) level of significance, whether or not the data can be modelled by a continuous uniform distribution. State your hypotheses clearly.
Edexcel S3 2010 June Q7
17 marks Moderate -0.3
A large company surveyed its staff to investigate the awareness of company policy. The company employs 6000 full time staff and 4000 part time staff.
  1. Describe how a stratified sample of 200 staff could be taken.
  2. Explain an advantage of using a stratified sample rather than a simple random sample. A random sample of 80 full time staff and an independent random sample of 80 part time staff were given a test of policy awareness. The results are summarised in the table below.
    Mean score \(( \bar { x } )\)
    Variance of
    scores \(\left( s ^ { 2 } \right)\)
    Full time staff5221
    Part time staff5019
  3. Stating your hypotheses clearly, test, at the \(1 \%\) level of significance, whether or not the mean policy awareness scores for full time and part time staff are different.
  4. Explain the significance of the Central Limit Theorem to the test in part (c).
  5. State an assumption you have made in carrying out the test in part (c). After all the staff had completed a training course the 80 full time staff and the 80 part time staff were given another test of policy awareness. The value of the test statistic \(z\) was 2.53
  6. Comment on the awareness of company policy for the full time and part time staff in light of this result. Use a \(1 \%\) level of significance.
  7. Interpret your answers to part (c) and part (f).
Edexcel S3 2012 June Q1
12 marks Standard +0.3
  1. Interviews for a job are carried out by two managers. Candidates are given a score by each manager and the results for a random sample of 8 candidates are shown in the table below.
Candidate\(A\)\(B\)\(C\)\(D\)\(E\)\(F\)\(G\)\(H\)
Manager \(X\)6256875465151210
Manager \(Y\)5447715049253044
  1. Calculate Spearman's rank correlation coefficient for these data.
  2. Test, at the \(5 \%\) level of significance, whether there is agreement between the rankings awarded by each manager. State your hypotheses clearly. Manager \(Y\) later discovered he had miscopied his score for candidate \(D\) and it should be 54 .
  3. Without carrying out any further calculations, explain how you would calculate Spearman's rank correlation in this case.
Edexcel S3 2012 June Q2
8 marks Moderate -0.8
2. A lake contains 3 species of fish. There are estimated to be 1400 trout, 600 bass and 450 pike in the lake. A survey of the health of the fish in the lake is carried out and a sample of 30 fish is chosen.
  1. Give a reason why stratified random sampling cannot be used.
  2. State an appropriate sampling method for the survey.
  3. Give one advantage and one disadvantage of this sampling method.
  4. Explain how this sampling method could be used to select the sample of 30 fish. You must show your working.
Edexcel S3 2012 June Q3
11 marks Standard +0.3
3.
  1. Explain what you understand by the Central Limit Theorem. A garage services hire cars on behalf of a hire company. The garage knows that the lifetime of the brake pads has a standard deviation of 5000 miles. The garage records the lifetimes, \(x\) miles, of the brake pads it has replaced. The garage takes a random sample of 100 brake pads and finds that \(\sum x = 1740000\)
  2. Find a 95\% confidence interval for the mean lifetime of a brake pad.
  3. Explain the relevance of the Central Limit Theorem in part (b). Brake pads are made to be changed every 20000 miles on average.
    The hire car company complain that the garage is changing the brake pads too soon.
  4. Comment on the hire company's complaint. Give a reason for your answer.