Questions S3 (597 questions)

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Edexcel S3 2007 June Q6
  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
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
  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
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
  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.
Edexcel S3 2008 June Q4
  1. 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
  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
  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
  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 2009 June Q1
  1. A telephone directory contains 50000 names. A researcher wishes to select a systematic sample of 100 names from the directory.
    1. Explain in detail how the researcher should obtain such a sample.
    2. Give one advantage and one disadvantage of
      1. quota sampling,
      2. systematic sampling.
    3. The heights of a random sample of 10 imported orchids are measured. The mean height of the sample is found to be 20.1 cm . The heights of the orchids are normally distributed.
    Given that the population standard deviation is 0.5 cm ,
  2. estimate limits between which \(95 \%\) of the heights of the orchids lie,
  3. find a 98\% confidence interval for the mean height of the orchids. A grower claims that the mean height of this type of orchid is 19.5 cm .
  4. Comment on the grower's claim. Give a reason for your answer.
Edexcel S3 2009 June Q3
3. A doctor is interested in the relationship between a person's Body Mass Index (BMI) and their level of fitness. She believes that a lower BMI leads to a greater level of fitness. She randomly selects 10 female 18 year-olds and calculates each individual's BMI. The females then run a race and the doctor records their finishing positions. The results are shown in the table.
Individual\(A\)\(B\)\(C\)\(D\)\(E\)\(F\)\(G\)\(H\)\(I\)\(J\)
BMI17.421.418.924.419.420.122.618.425.828.1
Finishing position35196410278
  1. Calculate Spearman's rank correlation coefficient for these data.
  2. Stating your hypotheses clearly and using a one tailed test with a \(5 \%\) level of significance, interpret your rank correlation coefficient.
  3. Give a reason to support the use of the rank correlation coefficient rather than the product moment correlation coefficient with these data.
Edexcel S3 2009 June Q4
4. A sample of size 8 is to be taken from a population that is normally distributed with mean 55 and standard deviation 3. Find the probability that the sample mean will be greater than 57.
Edexcel S3 2009 June Q5
5. The number of goals scored by a football team is recorded for 100 games. The results are summarised in Table 1 below. \begin{table}[h]
Number of goalsFrequency
040
133
214
38
45
\captionsetup{labelformat=empty} \caption{Table 1}
\end{table}
  1. Calculate the mean number of goals scored per game. The manager claimed that the number of goals scored per match follows a Poisson distribution. He used the answer in part (a) to calculate the expected frequencies given in Table 2. \begin{table}[h]
    Number of goalsExpected Frequency
    034.994
    1\(r\)
    2\(s\)
    36.752
    \(\geqslant 4\)2.221
    \captionsetup{labelformat=empty} \caption{Table 2}
    \end{table}
  2. Find the value of \(r\) and the value of \(s\) giving your answers to 3 decimal places.
  3. Stating your hypotheses clearly, use a \(5 \%\) level of significance to test the manager's claim.
Edexcel S3 2009 June Q6
  1. The lengths of a random sample of 120 limpets taken from the upper shore of a beach had a mean of 4.97 cm and a standard deviation of 0.42 cm . The lengths of a second random sample of 150 limpets taken from the lower shore of the same beach had a mean of 5.05 cm and a standard deviation of 0.67 cm .
    1. Test, using a \(5 \%\) level of significance, whether or not the mean length of limpets from the upper shore is less than the mean length of limpets from the lower shore. State your hypotheses clearly.
    2. State two assumptions you made in carrying out the test in part (a).
    3. A company produces climbing ropes. The lengths of the climbing ropes are normally distributed. A random sample of 5 ropes is taken and the length, in metres, of each rope is measured. The results are given below.
      119.9
      120.3
      120.1
      120.4
      120.2
    4. Calculate unbiased estimates for the mean and the variance of the lengths of the climbing ropes produced by the company.
    The lengths of climbing rope are known to have a standard deviation of 0.2 m . The company wants to make sure that there is a probability of at least 0.90 that the estimate of the population mean, based on a random sample size of \(n\), lies within 0.05 m of its true value.
  2. Find the minimum sample size required.
Edexcel S3 2009 June Q8
  1. The random variable \(A\) is defined as
$$A = 4 X - 3 Y$$ where \(X \sim \mathrm {~N} \left( 30,3 ^ { 2 } \right) , Y \sim \mathrm {~N} \left( 20,2 ^ { 2 } \right)\) and \(X\) and \(Y\) are independent. Find
  1. \(\mathrm { E } ( A )\),
  2. \(\operatorname { Var } ( A )\). The random variables \(Y _ { 1 } , Y _ { 2 } , Y _ { 3 }\) and \(Y _ { 4 }\) are independent and each has the same distribution as \(Y\). The random variable \(B\) is defined as $$B = \sum _ { i = 1 } ^ { 4 } Y _ { i }$$
  3. Find \(\mathrm { P } ( B > A )\).
Edexcel S3 2010 June Q3
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
  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
  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
  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
  1. 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
  2. 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.
  3. Explain the significance of the Central Limit Theorem to the test in part (c).
  4. 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
  5. 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.
  6. Interpret your answers to part (c) and part (f).
Edexcel S3 2011 June Q1
  1. Explain what you understand by the Central Limit Theorem.
  2. A county councillor is investigating the level of hardship, \(h\), of a town and the number of calls per 100 people to the emergency services, \(c\). He collects data for 7 randomly selected towns in the county. The results are shown in the table below.
TownABCDE\(F\)G
\(h\)14201618371924
\(c\)52454342618255
  1. Calculate the Spearman's rank correlation coefficient between \(h\) and \(c\). After collecting the data, the councillor thinks there is no correlation between hardship and the number of calls to the emergency services.
  2. Test, at the \(5 \%\) level of significance, the councillor's claim. State your hypotheses clearly.
Edexcel S3 2011 June Q3
3. A factory manufactures batches of an electronic component. Each component is manufactured in one of three shifts. A component may have one of two types of defect, \(D _ { 1 }\) or \(D _ { 2 }\), at the end of the manufacturing process. A production manager believes that the type of defect is dependent upon the shift that manufactured the component. He examines 200 randomly selected defective components and classifies them by defect type and shift. The results are shown in the table below.
\backslashbox{Shift}{Defect type}\(D _ { 1 }\)\(D _ { 2 }\)
First shift4518
Second shift5520
Third shift5012
Stating your hypotheses, test, at the \(10 \%\) level of significance, whether or not there is evidence to support the manager's belief. Show your working clearly.
Edexcel S3 2011 June Q4
  1. A shop manager wants to find out if customers spend more money when music is playing in the shop. The amount of money spent by a customer in the shop is \(\pounds x\). A random sample of 80 customers, who were shopping without music playing, and an independent random sample of 60 customers, who were shopping with music playing, were surveyed. The results of both samples are summarised in the table below.
\(\sum x\)\(\sum x ^ { 2 }\)Unbiased estimate of meanUnbiased estimate of variance
Customers shopping without music5320392000\(\bar { x }\)\(s ^ { 2 }\)
Customers shopping with music414031200069.0446.44
  1. Find the values of \(\bar { x }\) and \(s ^ { 2 }\).
  2. Test, at the \(5 \%\) level of significance, whether or not the mean money spent is greater when music is playing in the shop. State your hypotheses clearly.
Edexcel S3 2011 June Q5
  1. The number of hurricanes per year in a particular region was recorded over 80 years. The results are summarised in Table 1 below.
\begin{table}[h]
No of
hurricanes,
\(h\)
01234567
Frequency0251720121212
\captionsetup{labelformat=empty} \caption{Table 1}
\end{table}
  1. Write down two assumptions that will support modelling the number of hurricanes per year by a Poisson distribution.
  2. Show that the mean number of hurricanes per year from Table 1 is 4.4875
  3. Use the answer in part (b) to calculate the expected frequencies \(r\) and \(s\) given in Table 2 below to 2 decimal places. \begin{table}[h]
    \(h\)01234567 or more
    Expected
    frequency
    		0.904.04\(r\)13.55\(s\)13.6510.2113.39
    \captionsetup{labelformat=empty} \caption{Table 2}
    \end{table}
  4. Test, at the \(5 \%\) level of significance, whether or not the data can be modelled by a Poisson distribution. State your hypotheses clearly.
Edexcel S3 2011 June Q6
  1. The lifetimes of batteries from manufacturer \(A\) are normally distributed with mean 20 hours and standard deviation 5 hours when used in a camera.
    1. Find the mean and standard deviation of the total lifetime of a pack of 6 batteries from manufacturer \(A\).
    Judy uses a camera that takes one battery at a time. She takes a pack of 6 batteries from manufacturer \(A\) to use in her camera on holiday.
  2. Find the probability that the batteries will last for more than 110 hours on her holiday. The lifetimes of batteries from manufacturer \(B\) are normally distributed with mean 35 hours and standard deviation 8 hours when used in a camera.
  3. Find the probability that the total lifetime of a pack of 6 batteries from manufacturer \(A\) is more than 4 times the lifetime of a single battery from manufacturer \(B\) when used in a camera.