Questions — Edexcel S3 (332 questions)

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Edexcel S3 2006 January Q7
12 marks Standard +0.3
7. The numbers of deaths from pneumoconiosis and lung cancer in a developing country are given in the table.
Age group (years)20-2930-3940-4950-5960-6970 and over
Deaths from pneumoconiosis (1000s)12.55.918.519.431.231.0
Deaths from lung cancer (1000s)3.79.010.219.013.018.0
The correlation between the number of deaths in the different age groups for each disease is to be investigated.
  1. Give one reason why Spearman's rank correlation coefficient should be used.
  2. Calculate Spearman's rank correlation coefficient for these data.
  3. Use a suitable test, at the \(5 \%\) significance level, to interpret your result. State your hypotheses clearly.
    (5)
Edexcel S3 2003 June Q1
8 marks Easy -1.8
  1. Explain how to obtain a sample from a population using
    1. stratified sampling,
    2. quota sampling.
    Give one advantage and one disadvantage of each sampling method.
Edexcel S3 2003 June Q2
8 marks Moderate -0.5
2. A random sample of 30 apples was taken from a batch. The mean weight of the sample was 124 g with standard deviation 20 g .
  1. Find a \(99 \%\) confidence interval for the mean weight \(\mu\) grams of the population of apples. Write down any assumptions you made in your calculations. Given that the actual value of \(\mu\) is 140 ,
  2. state, with a reason, what you can conclude about the sample of 30 apples.
Edexcel S3 2003 June Q3
9 marks Moderate -0.5
3. Given the random variables \(X \sim \mathrm {~N} ( 20,5 )\) and \(Y \sim \mathrm {~N} ( 10,4 )\) where \(X\) and \(Y\) are independent, find
  1. \(\mathrm { E } ( X - Y )\),
  2. \(\operatorname { Var } ( X - Y )\),
  3. \(\mathrm { P } ( 13 < X - Y < 16 )\).
Edexcel S3 2003 June Q4
11 marks Moderate -0.3
4. A new drug to treat the common cold was used with a randomly selected group of 100 volunteers. Each was given the drug and their health was monitored to see if they caught a cold. A randomly selected control group of 100 volunteers was treated with a dummy pill. The results are shown in the table below.
\cline { 2 - 3 } \multicolumn{1}{c|}{}ColdNo cold
Drug3466
Dummy pill4555
Using a \(5 \%\) significance level, test whether or not the chance of catching a cold is affected by taking the new drug. State your hypotheses clearly.
Edexcel S3 2003 June Q5
11 marks Standard +0.3
5. A scientist monitored the levels of river pollution near a factory. Before the factory was closed down she took 100 random samples of water from different parts of the river and found an average weight of pollutants of \(10 \mathrm { mg } \mathrm { l } ^ { - 1 }\) with a standard deviation of \(2.64 \mathrm { mg } \mathrm { l } ^ { - 1 }\). After the factory was closed down the scientist collected a further 120 random samples and found that they contained \(8 \mathrm { mg } \mathrm { l } ^ { - 1 }\) of pollutants on average with a standard deviation of \(1.94 \mathrm { mg } \mathrm { l } ^ { - 1 }\). Test, at the \(5 \%\) level of significance, whether or not the mean river pollution fell after the factory closed down.
Edexcel S3 2003 June Q6
11 marks Standard +0.3
6. Two judges ranked 8 ice skaters in a competition according to the table below.
\backslashbox{Judge}{Skater}(i)(ii)(iii)(iv)(v)(vi)(vii)(viii)
A25378146
B32657418
  1. Evaluate Spearman's rank correlation coefficient between the ranks of the two judges.
  2. Use a suitable test, at the \(5 \%\) level of significance, to interpret this result.
Edexcel S3 2003 June Q7
17 marks Standard +0.3
7. A bag contains a large number of coins of which \(30 \%\) are 50 p coins, \(20 \%\) are 10 p coins and the rest are 2 p coins.
  1. Find the mean \(\mu\) and the variance \(\sigma ^ { 2 }\) of this population of coins. A random sample of 2 coins is drawn from the bag one after the other.
  2. List all possible samples that could be drawn.
  3. Find the sampling distribution of \(\bar { X }\), the mean of the coins drawn.
  4. Find \(\mathrm { P } ( 2 \leq \bar { X } < 7 )\).
  5. Use the sampling distribution of \(\bar { X }\) to verify \(\mathrm { E } ( \bar { X } ) = \mu\) and \(\operatorname { Var } ( \bar { X } ) = \frac { 1 } { 2 } \sigma ^ { 2 }\). END
Edexcel S3 2004 June Q1
6 marks Easy -2.0
  1. There are 64 girls and 56 boys in a school.
Explain briefly how you could take a random sample of 15 pupils using
  1. a simple random sample,
  2. a stratified sample.
Edexcel S3 2004 June Q2
8 marks Standard +0.3
2. A random sample of 8 students sat examinations in Geography and Statistics. The product moment correlation coefficient between their results was 0.572 and the Spearman rank correlation coefficient was 0.655 .
  1. Test both of these values for positive correlation. Use a \(5 \%\) level of significance.
  2. Comment on your results.
Edexcel S3 2004 June Q3
8 marks Standard +0.3
3. It is known from past evidence that the weight of coffee dispensed into jars by machine \(A\) is normally distributed with mean \(\mu _ { \mathrm { A } }\) and standard deviation 2.5 g . Machine \(B\) is known to dispense the same nominal weight of coffee into jars with mean \(\mu _ { B }\) and standard deviation 2.3 g . A random sample of 10 jars filled by machine \(A\) contained a mean weight of 249 g of coffee. A random sample of 15 jars filled by machine \(B\) contained a mean weight of 251 g .
  1. Test, at the \(5 \%\) level of significance, whether or not there is evidence that the population mean weight dispensed by machine B is greater than that of machine A .
  2. Write down an assumption needed to carry out this test.
Edexcel S3 2004 June Q4
10 marks Moderate -0.3
4. Kylie regularly travels from home to visit a friend. On 10 randomly selected occasions the journey time \(x\) minutes was recorded. The results are summarised as follows. $$\Sigma x = 753 , \quad \Sigma x ^ { 2 } = 57455 .$$
  1. Calculate unbiased estimates of the mean and the variance of the population of journey times. After many journeys, a random sample of 100 journeys gave a mean of 74.8 minutes and a variance of 84.6 minutes \({ } ^ { 2 }\).
  2. Calculate a 95\% confidence interval for the mean of the population of journey times.
  3. Write down two assumptions you made in part (b).
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
  1. 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\(A\)\(B\)\(D\)\(C\)\(E\)\(G\)\(F\)
    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.