Questions — Edexcel S3 (313 questions)

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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.
Edexcel S3 2011 June Q7
  1. Roastie's Coffee is sold in packets with a stated weight of 250 g . A supermarket manager claims that the mean weight of the packets is less than the stated weight. She weighs a random sample of 90 packets from their stock and finds that their weights have a mean of 248 g and a standard deviation of 5.4 g .
    1. Using a \(5 \%\) level of significance, test whether or not the manager's claim is justified. State your hypotheses clearly.
    2. Find the \(98 \%\) confidence interval for the mean weight of a packet of coffee in the supermarket's stock.
    3. State, with a reason, the action you would recommend the manager to take over the weight of a packet of Roastie's Coffee.
    Roastie's Coffee company increase the mean weight of their packets to \(\mu \mathrm { g }\) and reduce the standard deviation to 3 g . The manager takes a sample of size \(n\) from these new packets. She uses the sample mean \(\bar { X }\) as an estimator of \(\mu\).
  2. Find the minimum value of \(n\) such that \(\mathrm { P } ( | \bar { X } - \mu ! | < 1 ) \geqslant 0.98\)
Edexcel S3 2012 June Q1
  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
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
3. (a) 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 \(\sum x = 1740000\)
(b) Find a 95\% confidence interval for the mean lifetime of a brake pad.
(c) 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.
(d) Comment on the hire company's complaint. Give a reason for your answer.
Edexcel S3 2012 June Q4
  1. Two breeds of chicken are surveyed to measure their egg yield. The results are shown in the table below.
\backslashbox{Breed}{Egg yield}LowMediumHigh
Leghorn225226
Cornish14324
Showing each stage of your working clearly, test, at the \(5 \%\) significance level, whether or not there is an association between egg yield and breed of chicken. State your hypotheses clearly.
Edexcel S3 2012 June Q5
5. Mr Alan and Ms Burns are two Mathematics teachers teaching mixed ability groups of students in a large college. At the end of the college year all students took the same examination. A random sample of 29 of Mr Alan's students and a random sample of 26 of Ms Burns' students are chosen. The results are summarised in the table below.
Sample Size, \(n\)Mean, \(\bar { x }\)Standard Deviation, \(s\)
Mr Alan298010
Ms Burns267415
  1. Stating your hypotheses clearly, test, at the \(10 \%\) level of significance whether there is evidence that there is a difference in the mean scores of their students. Ms Burns thinks the comparison was unfair as the examination was set by Mr Alan. She looks up a different set of examination results for these students and, although Mr Alan's sample has a higher mean, she calculates the test statistic for this new set of results to be 1.6 However, Mr Alan now claims that the mean marks of his students are higher than the mean marks of Ms Burns' students.
  2. Test Mr Alan's claim, stating the hypotheses and critical values you would use. Use a \(10 \%\) level of significance.
Edexcel S3 2012 June Q6
6. A total of 100 random samples of 6 items are selected from a production line in a factory and the number of defective items in each sample is recorded. The results are summarised in the table below.
Number of
defective
items
0123456
Number of
samples
616202317108
  1. Show that the mean number of defective items per sample is 2.91 A factory manager suggests that the data can be modelled by a binomial distribution with \(n = 6\). He uses the mean from the sample above and calculates expected frequencies as shown in the table below.
    Number of
    defective
    items
    		0123456
    Expected
    frequency
    		1.8710.5424.82\(a\)22.018.29\(b\)
  2. Calculate the value of \(a\) and the value of \(b\) giving your answers to 2 decimal places.
  3. Test, at the \(5 \%\) level, whether or not the binomial distribution is a suitable model for the number of defective items in samples of 6 items. State your hypotheses clearly.
Edexcel S3 2012 June Q7
7. The heights, in cm, of the male employees in a large company follow a normal distribution with mean 177 and standard deviation 5 The heights, in cm, of the female employees follow a normal distribution with mean 163 and standard deviation 4 A male employee and a female employee are chosen at random.
  1. Find the probability that the male employee is taller than the female employee. Six male employees and four female employees are chosen at random.
  2. Find the probability that their total height is less than 17 m .