Questions S3 (597 questions)

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AQA AS Paper 1 AS Paper 2 C1 C2 C3 C4 D1 D2 FP1 FP2 FP3 Further AS Paper 1 Further AS Paper 2 Discrete Further AS Paper 2 Mechanics Further AS Paper 2 Statistics Further Paper 1 Further Paper 2 Further Paper 3 Discrete Further Paper 3 Mechanics Further Paper 3 Statistics M1 M2 M3 Paper 1 Paper 2 Paper 3 S1 S2 S3 CAIE FP1 FP2 Further Paper 1 Further Paper 2 Further Paper 3 Further Paper 4 M1 M2 P1 P2 P3 S1 S2 Edexcel AEA AS Paper 1 AS Paper 2 C1 C12 C2 C3 C34 C4 CP AS CP1 CP2 D1 D2 F1 F2 F3 FD1 FD1 AS FD2 FD2 AS FM1 FM1 AS FM2 FM2 AS FP1 FP1 AS FP2 FP2 AS FP3 FS1 FS1 AS FS2 FS2 AS M1 M2 M3 M4 M5 P1 P2 P3 P4 PMT Mocks Paper 1 Paper 2 Paper 3 S1 S2 S3 S4 OCR AS Pure C1 C2 C3 C4 D1 D2 FD1 AS FM1 AS FP1 FP1 AS FP2 FP3 FS1 AS Further Additional Pure Further Additional Pure AS Further Discrete Further Discrete AS Further Mechanics Further Mechanics AS Further Pure Core 1 Further Pure Core 2 Further Pure Core AS Further Statistics Further Statistics AS H240/01 H240/02 H240/03 M1 M2 M3 M4 Mechanics 1 PURE Pure 1 S1 S2 S3 S4 Stats 1 OCR MEI AS Paper 1 AS Paper 2 C1 C2 C3 C4 D1 D2 FP1 FP2 FP3 Further Extra Pure Further Mechanics A AS Further Mechanics B AS Further Mechanics Major Further Mechanics Minor Further Numerical Methods Further Pure Core Further Pure Core AS Further Pure with Technology Further Statistics A AS Further Statistics B AS Further Statistics Major Further Statistics Minor M1 M2 M3 M4 Paper 1 Paper 2 Paper 3 S1 S2 S3 S4 SPS SPS ASFM SPS ASFM Mechanics SPS ASFM Pure SPS ASFM Statistics SPS FM SPS FM Mechanics SPS FM Pure SPS FM Statistics SPS SM SPS SM Mechanics SPS SM Pure SPS SM Statistics WJEC Further Unit 1 Further Unit 2 Further Unit 3 Further Unit 4 Further Unit 5 Further Unit 6 Unit 1 Unit 2 Unit 3 Unit 4
Edexcel S3 2003 June Q5
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
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
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
  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
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
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
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
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
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
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 2005 June Q1
  1. (a) State two reasons why stratified sampling might be chosen as a method of sampling when carrying out a statistical survey.
    (b) State one advantage and one disadvantage of quota sampling.
  2. A sample of size 5 is taken from a population that is normally distributed with mean 10 and standard deviation 3 . Find the probability that the sample mean lies between 7 and 10 .
    (Total 6 marks)
  3. A researcher carried out a survey of three treatments for a fruit tree disease. The contingency table below shows the results of a survey of a random sample of 60 diseased trees.
No actionRemove diseased branchesSpray with chemicals
Tree died within 1 year1056
Tree survived for 1-4 years597
Tree survived beyond 4 years567
Test, at the \(5 \%\) level of significance, whether or not there is any association between the treatment of the trees and their survival. State your hypotheses and conclusion clearly.
(Total 11 marks)
Edexcel S3 2005 June Q4
4. Over a period of time, researchers took 10 blood samples from one patient with a blood disease. For each sample, they measured the levels of serum magnesium, \(s \mathrm { mg } / \mathrm { dl }\), in the blood and the corresponding level of the disease protein, \(d \mathrm { mg } / \mathrm { dl }\). The results are shown in the table.
\(s\)1.21.93.23.92.54.55.74.01.15.9
\(d\)3.87.011.012.09.012.013.512.22.013.9
$$\text { [Use } \sum s ^ { 2 } = 141.51 , \sum d ^ { 2 } = 1081.74 \text { and } \sum s d = 386.32 \text { ] }$$
  1. Draw a scatter diagram to represent these data.
  2. State what is measured by the product moment correlation coefficient.
  3. Calculate \(S _ { x x } , S _ { d d }\) and \(S _ { s d }\).
  4. Calculate the value of the product moment correlation coefficient \(r\) between \(s\) and \(d\).
  5. Stating your hypotheses clearly, test, at the \(1 \%\) significance level, whether or not the correlation coefficient is greater than zero.
  6. With reference to your scatter diagram, comment on your result in part (e).
Edexcel S3 2005 June Q5
5. The number of times per day a computer fails and has to be restarted is recorded for 200 days. The results are summarised in the table.
Number of restartsFrequency
099
165
222
312
42
Test whether or not a Poisson model is suitable to represent the number of restarts per day. Use a \(5 \%\) level of significance and state your hypothesis clearly.
(Total 12 marks)
Edexcel S3 2005 June Q6
6. A computer company repairs large numbers of PCs and wants to estimate the mean time to repair a particular fault. Five repairs are chosen at random from the company's records and the times taken, in seconds, are $$\begin{array} { l l l l l } 205 & 310 & 405 & 195 & 320 \end{array} .$$
  1. Calculate unbiased estimates of the mean and the variance of the population of repair times from which this sample has been taken. It is known from previous results that the standard deviation of the repair time for this fault is 100 seconds. The company manager wants to ensure that there is a probability of at least 0.95 that the estimate of the population mean lies within 20 seconds of its true value.
  2. Find the minimum sample size required.
    (Total 10 marks)
Edexcel S3 2005 June Q7
7. A manufacturer produces two flavours of soft drink, cola and lemonade. The weights, \(C\) and \(L\), in grams, of randomly selected cola and lemonade cans are such that \(C \sim \mathrm {~N} ( 350,8 )\) and \(L \sim \mathrm {~N} ( 345,17 )\).
  1. Find the probability that the weights of two randomly selected cans of cola will differ by more than 6 g . One can of each flavour is selected at random.
  2. Find the probability that the can of cola weighs more than the can of lemonade. Cans are delivered to shops in boxes of 24 cans. The weights of empty boxes are normally distributed with mean 100 g and standard deviation 2 g .
  3. Find the probability that a full box of cola cans weighs between 8.51 kg and 8.52 kg .
  4. State an assumption you made in your calculation in part (c).
Edexcel S3 2006 June Q1
\begin{enumerate} \item Describe one advantage and one disadvantage of
  1. quota sampling,
  2. simple random sampling. \item A report on the health and nutrition of a population stated that the mean height of three-year old children is 90 cm and the standard deviation is 5 cm . A sample of 100 three-year old children was chosen from the population.
Edexcel S3 2006 June Q5
5. The workers in a large office block use a lift that can carry a maximum load of 1090 kg . The weights of the male workers are normally distributed with mean 78.5 kg and standard deviation 12.6 kg . The weights of the female workers are normally distributed with mean 62.0 kg and standard deviation 9.8 kg . Random samples of 7 males and 8 females can enter the lift.
  1. Find the mean and variance of the total weight of the 15 people that enter the lift.
  2. Comment on any relationship you have assumed in part (a) between the two samples.
  3. Find the probability that the maximum load of the lift will be exceeded by the total weight of the 15 people.
    (4)
Edexcel S3 2006 June Q6
6. A research worker studying colour preference and the age of a random sample of 50 children obtained the results shown below.
Age in yearsRedBlueTotals
412618
810717
126915
Totals282250
Using a \(5 \%\) significance level, carry out a test to decide whether or not there is an association between age and colour preference. State your hypotheses clearly.
Edexcel S3 2006 June Q7
7. A machine produces metal containers. The weights of the containers are normally distributed. A random sample of 10 containers from the production line was weighed, to the nearest 0.1 kg , and gave the following results $$\begin{array} { l l l l l } 49.7 , & 50.3 , & 51.0 , & 49.5 , & 49.9
50.1 , & 50.2 , & 50.0 , & 49.6 , & 49.7 . \end{array}$$
  1. Find unbiased estimates of the mean and variance of the weights of the population of metal containers. The machine is set to produce metal containers whose weights have a population standard deviation of 0.5 kg .
  2. Estimate the limits between which \(95 \%\) of the weights of metal containers lie.
  3. Determine the \(99 \%\) confidence interval for the mean weight of metal containers.
Edexcel S3 2006 June Q8
8. Five coins were tossed 100 times and the number of heads recorded. The results are shown in the table below.
Number
of heads
012345
Frequency6182934103
  1. Suggest a suitable distribution to model the number of heads when five unbiased coins are tossed.
  2. Test, at the \(10 \%\) level of significance, whether or not the five coins are unbiased. State your hypotheses clearly.
Edexcel S3 2007 June Q1
  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
  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
  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
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
  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).