Questions S3 (597 questions)

Browse by module
All questions AEA AS Paper 1 AS Paper 2 AS Pure 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 Further AS Paper 1 Further AS Paper 2 Discrete Further AS Paper 2 Mechanics Further AS Paper 2 Statistics Further Additional Pure Further Additional Pure AS Further Discrete Further Discrete AS Further Extra Pure Further Mechanics Further Mechanics A AS Further Mechanics AS Further Mechanics B AS Further Mechanics Major Further Mechanics Minor Further Numerical Methods Further Paper 1 Further Paper 2 Further Paper 3 Further Paper 3 Discrete Further Paper 3 Mechanics Further Paper 3 Statistics Further Paper 4 Further Pure Core Further Pure Core 1 Further Pure Core 2 Further Pure Core AS Further Pure with Technology Further Statistics Further Statistics A AS Further Statistics AS Further Statistics B AS Further Statistics Major Further Statistics Minor Further Unit 1 Further Unit 2 Further Unit 3 Further Unit 4 Further Unit 5 Further Unit 6 H240/01 H240/02 H240/03 M1 M2 M3 M4 M5 Mechanics 1 P1 P2 P3 P4 PMT Mocks PURE Paper 1 Paper 2 Paper 3 Pure 1 S1 S2 S3 S4 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 Stats 1 Unit 1 Unit 2 Unit 3 Unit 4
Browse by board
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 2015 June Q4
  1. A farm produces potatoes. The potatoes are packed into sacks.
The weight of a sack of potatoes is modelled by a normal distribution with mean 25.6 kg and standard deviation 0.24 kg
  1. Find the probability that two randomly chosen sacks of potatoes differ in weight by more than 0.5 kg
    (6) Sacks of potatoes are randomly selected and packed onto pallets.
    The weight of an empty pallet is modelled by a normal distribution with mean 20.0 kg and standard deviation 0.32 kg Each full pallet of potatoes holds 30 sacks of potatoes.
  2. Find the probability that the total weight of a randomly chosen full pallet of potatoes is greater than 785 kg
Edexcel S3 2015 June Q5
  1. A Head of Department at a large university believes that gender is independent of the grade obtained by students on a Business Foundation course. A random sample was taken of 200 male students and 160 female students who had studied the course.
The results are summarised below.
\cline { 3 - 4 } \multicolumn{2}{c|}{}MaleFemale
\multirow{3}{*}{Grade}Distinction\(18.5 \%\)\(27.5 \%\)
\cline { 2 - 4 }Merit\(63.5 \%\)\(60.0 \%\)
\cline { 2 - 4 }Unsatisfactory\(18.0 \%\)\(12.5 \%\)
Stating your hypotheses clearly, test the Head of Department's belief using a 5\% level of significance. Show your working clearly.
Edexcel S3 2015 June Q6
  1. As part of an investigation, a random sample was taken of 50 footballers who had completed an obstacle course in the early morning. The time taken by each of these footballers to complete the obstacle course, \(x\) minutes, was recorded and the results are summarised by
$$\sum x = 1570 \quad \text { and } \quad \sum x ^ { 2 } = 49467.58$$
  1. Find unbiased estimates for the mean and variance of the time taken by footballers to complete the obstacle course in the early morning. An independent random sample was taken of 50 footballers who had completed the same obstacle course in the late afternoon. The time taken by each of these footballers to complete the obstacle course, \(y\) minutes, was recorded and the results are summarised as $$\bar { y } = 30.9 \quad \text { and } \quad s _ { y } ^ { 2 } = 3.03$$
  2. Test, at the \(5 \%\) level of significance, whether or not the mean time taken by footballers to complete the obstacle course in the early morning, is greater than the mean time taken by footballers to complete the obstacle course in the late afternoon. State your hypotheses clearly.
  3. Explain the relevance 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 2015 June Q7
  1. A fair six-sided die is labelled with the numbers \(1,2,3,4,5\) and 6 The die is rolled 40 times and the score, \(S\), for each roll is recorded.
    1. Find the mean and the variance of \(S\).
    2. Find an approximation for the probability that the mean of the 40 scores is less than 3 (3)
    3. A factory produces steel sheets whose weights \(X \mathrm {~kg}\), are such that \(X \sim \mathrm {~N} \left( \mu , \sigma ^ { 2 } \right)\)
    A random sample of these sheets is taken and a \(95 \%\) confidence interval for \(\mu\) is found to be (29.74, 31.86)
  2. Find, to 2 decimal places, the standard error of the mean.
  3. Hence, or otherwise, find a \(90 \%\) confidence interval for \(\mu\) based on the same sample of sheets. Using four different random samples, four \(90 \%\) confidence intervals for \(\mu\) are to be found.
  4. Calculate the probability that at least 3 of these intervals will contain \(\mu\).
Edexcel S3 2016 June Q1
  1. The table below shows the distance travelled by car and the amount of commission earned by each of 8 salespersons in 2015
SalespersonDistance travelled (in 1000's of km)Commission earned (in \\(1000's)
A20.417.7
B22.224.1
C29.920.3
D37.828.3
E25.534.9
\)F$30.229.3
G35.323.6
H16.526.8
  1. Find Spearman's rank correlation coefficient for these data.
  2. Stating your hypotheses clearly, test, at the \(5 \%\) level of significance, whether or not there is evidence of a positive correlation between the distance travelled by car and the amount of commission earned.
Edexcel S3 2016 June Q2
2. A researcher investigates the results of candidates who took their driving test at one of three driving test centres. A random sample of 620 candidates gave the following results.
\multirow{2}{*}{}Driving test centre\multirow{2}{*}{Total}
\(\boldsymbol { A }\)BC
\multirow{2}{*}{Result}Pass9911068277
Fail108116119343
Total207226187620
  1. Test, at the \(5 \%\) level of significance, whether there is an association between the results of candidates' driving tests and the driving test centre. State your hypotheses and show your working clearly. You should state your expected frequencies correct to 2 decimal places. The researcher decides to conduct a further investigation into the results of candidates' driving tests.
  2. State which driving test centre you would recommend for further investigation. Give a reason for your answer.
Edexcel S3 2016 June Q3
3. A company wants to survey its employees' attitudes to work. The company's workforce is located at three offices. The number of employees at each location is summarised in the table below.
Office locationNumber of employees
Bristol856
Dudley429
Glasgow1215
Each employee is located at only one office. A personnel assistant plans to survey the first 50 employees who arrive for work at the Bristol office on a Monday morning.
  1. Give two reasons why this survey is likely to lead to a biased response. A personnel manager has access to the company's information system that holds details of each employee including their place of work. The manager decides to take a stratified sample of 150 employees.
  2. Describe how to choose employees for this stratified sample.
  3. Explain an advantage of using a stratified sample rather than a quota sample.
Edexcel S3 2016 June Q4
4. A random sample of 60 children and a random sample of 50 adults were taken and each person was given the same task to complete. The table below summarises the times taken, \(t\) seconds, to complete the task.
Mean, \(\overline { \boldsymbol { t } }\)Standard deviation, \(\boldsymbol { s }\)\(\boldsymbol { n }\)
Children61.25.960
Adults59.15.250
  1. Stating your hypotheses clearly, test, at the \(5 \%\) level of significance, whether or not there is evidence that the mean time taken to complete the task by children is greater than the mean time taken by adults.
    (6)
  2. Explain the relevance of the Central Limit Theorem to your calculation in part (a).
  3. State an assumption you have made to carry out the test in part (a).
Edexcel S3 2016 June Q5
5. Kylie used video technology to monitor the direction of flight, as a bearing, \(x\) degrees, for 450 honeybees that left her beehive during a particular morning. Kylie's results are summarised in the table below.
Direction of flightFrequency
\(0 \leqslant x < 72\)78
\(72 \leqslant x < 140\)69
\(140 \leqslant x < 190\)51
\(190 \leqslant x < 260\)108
\(260 \leqslant x < 360\)144
Kylie believes that a continuous uniform distribution over the interval [0,360] is a suitable model for the direction of flight. Stating your hypotheses clearly, use a 1\% level of significance to test Kylie's belief. Show your working clearly.
Edexcel S3 2016 June Q6
6. The random variable \(W\) is defined as $$W = 3 X - 4 Y$$ where \(X \sim \mathrm {~N} \left( 21,2 ^ { 2 } \right)\) and \(Y \sim \mathrm {~N} \left( 8.5 , \sigma ^ { 2 } \right)\) and \(X\) and \(Y\) are independent.
Given that \(\mathrm { P } ( W < 44 ) = 0.9\)
  1. find the value of \(\sigma\), giving your answer to 2 decimal places. The random variables \(A _ { 1 } , A _ { 2 }\) and \(A _ { 3 }\) each have the same distribution as \(A\), where \(A \sim \mathrm {~N} \left( 28,5 ^ { 2 } \right)\) The random variable \(B\) is defined as $$B = 2 X + \sum _ { i = 1 } ^ { 3 } A _ { i }$$ where \(X , A _ { 1 } , A _ { 2 }\) and \(A _ { 3 }\) are independent.
  2. Find \(\mathrm { P } ( B \leqslant 145 \mid B > 120 )\)
Edexcel S3 2016 June Q7
7. A random sample of 8 apples is taken from an orchard and the weight, in grams, of each apple is measured. The results are given below. $$\begin{array} { l l l l l l l l } 143 & 131 & 165 & 122 & 137 & 155 & 148 & 151 \end{array}$$
  1. Calculate unbiased estimates for the mean and the variance of the weights of apples. A population has an unknown mean \(\mu\) and an unknown variance \(\sigma ^ { 2 }\)
    A random sample represented by \(X _ { 1 } , X _ { 2 } , X _ { 3 } , \ldots , X _ { 8 }\) is taken from this population.
  2. Explain why \(\sum _ { i = 1 } ^ { 8 } \left( X _ { i } - \mu \right) ^ { 2 }\) is not a statistic. Given that \(\mathrm { E } \left( S ^ { 2 } \right) = \sigma ^ { 2 }\), where \(S ^ { 2 }\) is an unbiased estimator of \(\sigma ^ { 2 }\) and the statistic $$Y = \frac { 1 } { 8 } \left( \sum _ { i = 1 } ^ { 8 } X _ { i } ^ { 2 } - 8 \bar { X } ^ { 2 } \right)$$
  3. find \(\mathrm { E } ( Y )\) in terms of \(\sigma ^ { 2 }\)
  4. Hence find the bias, in terms of \(\sigma ^ { 2 }\), when \(Y\) is used as an estimator of \(\sigma ^ { 2 }\)
Edexcel S3 2016 June Q8
8. A six-sided die is labelled with the numbers \(1,2,3,4,5\) and 6 A group of 50 students want to test whether or not the die is fair for the number six.
The 50 students each roll the die 30 times and record the number of sixes they each obtain.
Given that \(\bar { X }\) denotes the mean number of sixes obtained by the 50 students, and using $$\mathrm { H } _ { 0 } : p = \frac { 1 } { 6 } \text { and } \mathrm { H } _ { 1 } : p \neq \frac { 1 } { 6 }$$ where \(p\) is the probability of rolling a 6 ,
  1. use the Central Limit Theorem to find an approximate distribution for \(\bar { X }\), if \(\mathrm { H } _ { 0 }\) is true.
  2. Hence find, in terms of \(\bar { X }\), the critical region for this test. Use a \(5 \%\) level of significance.
Edexcel S3 2017 June Q1
  1. The ages, in years, of a random sample of 8 parrots are shown in the table below.
Parrot\(A\)\(B\)\(C\)\(D\)\(E\)\(F\)\(G\)\(H\)
Age10413152186
A parrot breeder does not know the ages of these 8 parrots. She examines each of these 8 parrots and is asked to put them in order of decreasing age. She puts them in the order $$\begin{array} { l l l l l l l l } D & G & H & C & A & B & F & E \end{array}$$
  1. Find, to 3 decimal places, Spearman's rank correlation coefficient between the breeder's order and the actual order.
    (5)
  2. Use your value of Spearman's rank correlation coefficient to test for evidence of the breeder's ability to order parrots correctly, by their age, after examining them. Use a \(1 \%\) significance level and state your hypotheses clearly.
Edexcel S3 2017 June Q2
2. A school uses online report cards to promote both hard work and good behaviour of its pupils. Each card details a pupil's recent achievement and contains exactly one of three inspirational messages \(A , B\) or \(C\), chosen by the pupil's teacher. The headteacher believes that there is an association between the pupil's gender and the inspirational message chosen. He takes a random sample of 225 pupils and examines the card for each pupil. His results are shown in Table 1. \begin{table}[h]
\cline { 2 - 5 } \multicolumn{2}{c|}{}Inspirational message\multirow{2}{*}{Total}
\cline { 3 - 5 } \multicolumn{2}{c|}{}\(\boldsymbol { A }\)\(\boldsymbol { B }\)\(\boldsymbol { C }\)
\multirow{2}{*}{
Pupil's
gender
}		Male253745107
\cline { 2 - 6 }Female325036118
Total578781225
\captionsetup{labelformat=empty} \caption{Table 1}
\end{table} Stating your hypotheses clearly, test, at the \(10 \%\) level of significance, whether or not there is evidence to support the headteacher's belief. Show your working clearly. You should state your expected frequencies correct to 2 decimal places.
Edexcel S3 2017 June Q3
3. The manager of a gym claimed that the mean age of its customers is 30 years. A random sample of 75 customers is taken and their ages have a mean of 28.2 years and a standard deviation, \(s\), of 8.5 years.
  1. Stating your hypotheses clearly and using a 10\% level of significance, test whether or not the manager's claim is supported by the data.
  2. Explain the relevance of the Central Limit Theorem to your calculation in part (a).
  3. State an additional assumption needed to carry out the test in part (a).
Edexcel S3 2017 June Q4
4. The number of emergency plumbing calls received per day by a local council was recorded over a period of 80 days. The results are summarised in the table below.
Number of calls, \(\boldsymbol { x }\)012345678
Frequency3131415108863
  1. Show that the mean number of emergency plumbing calls received per day is 3.5 A council officer suggests that a Poisson distribution can be used to model the number of emergency plumbing calls received per day. He uses the mean from the sample above and calculates the expected frequencies shown in the table below.
    \(\boldsymbol { x }\)01234567
    8 or
    more
    Expected
    frequency
    		2.428.4614.80\(r\)15.1010.576.173.08\(s\)
  2. Calculate the value of \(r\) and the value of \(s\), giving your answers correct to 2 decimal places.
  3. Test, at the \(5 \%\) level of significance, whether or not the Poisson distribution is a suitable model for the number of emergency plumbing calls received per day. State your hypotheses clearly.
Edexcel S3 2017 June Q5
5. A dance studio has 800 dancers of which \begin{displayquote} 452 are beginners
251 are intermediates
97 are professionals
  1. Explain in detail how a stratified sample of size 50 could be taken.
  2. State an advantage of stratified sampling rather than simple random sampling in this situation. \end{displayquote} Independent random samples of 80 beginners and 60 intermediates are chosen. Each of these dancers is given an assessment score, \(x\), based on the quality of their dancing. The results are summarised in the table below.
    \(\bar { x }\)\(s ^ { 2 }\)\(n\)
    Beginners31.757.380
    Intermediates36.938.160
    The studio manager believes that the mean score of intermediates is more than 3 points greater than the mean score of beginners.
  3. Stating your hypotheses clearly and using a \(5 \%\) level of significance, test whether or not these data support the studio manager's belief.
Edexcel S3 2017 June Q6
6. A company produces a certain type of mug. The masses of these mugs are normally distributed with mean \(\mu\) and standard deviation 1.2 grams. A random sample of 5 mugs is taken and the mass, in grams, of each mug is measured. The results are given below. \section*{\(\begin{array} { l l l l l } 229.1 & 229.6 & 230.9 & 231.2 & 231.7 \end{array}\)}
  1. Find a \(95 \%\) confidence interval for \(\mu\), giving your limits correct to 1 decimal place. Sonia plans to take 20 random samples, each of 5 mugs. A 95\% confidence interval for \(\mu\) is to be determined for each sample.
  2. Find the probability that more than 3 of these intervals will not contain \(\mu\).
Edexcel S3 2017 June Q7
7. The independent random variables \(X\) and \(Y\) are such that $$X \sim \mathrm {~N} \left( 30,4.5 ^ { 2 } \right) \text { and } Y \sim \mathrm {~N} \left( 20,3.5 ^ { 2 } \right)$$ The random variables \(X _ { 1 } , X _ { 2 }\) and \(X _ { 3 }\) are independent and each has the same distribution as \(X\). The random variables \(Y _ { 1 }\) and \(Y _ { 2 }\) are independent and each has the same distribution as \(Y\). Given that the random variable \(A\) is defined as $$A = \frac { X _ { 1 } + X _ { 2 } + X _ { 3 } + Y _ { 1 } + Y _ { 2 } } { 5 }$$
  1. find \(\mathrm { P } ( A < 24 )\) The random variable \(W\) is such that \(W \sim \mathrm {~N} \left( \mu , 2.8 ^ { 2 } \right)\) Given that \(\mathrm { P } ( W - X < 4 ) = 0.1\) and that \(W\) and \(X\) are independent,
  2. find the value of \(\mu\), giving your answer to 3 significant figures.
Edexcel S3 2017 June Q8
8. The random variable \(X\) has a continuous uniform distribution over the interval \([ \alpha + 3,2 \alpha + 9 ]\) where \(\alpha\) is a constant. The mean of a random sample of size \(n\), taken from this distribution, is denoted by \(\bar { X }\)
  1. Show that \(\bar { X }\) is a biased estimator of \(\alpha\)
  2. Hence find the bias, in terms of \(\alpha\), when \(\bar { X }\) is used as an estimator of \(\alpha\) Given that \(Y = \frac { 2 \bar { X } } { 3 } + k\) is an unbiased estimator of \(\alpha\)
  3. find the value of the constant \(k\) A random sample of 8 values of \(X\) is taken and the results are as follows
    4.8
    5.8
    6.5
    7.1
    8.2
    9.5
    9.9
    10.6
  4. Use the sample to estimate the maximum value that \(X\) can take.
    \includegraphics[max width=\textwidth, alt={}]{3b6aaa8a-aeac-4a44-820b-e82f317d0c85-28_2633_1828_119_121}
Edexcel S3 2018 June Q1
  1. A random sample of 9 footballers is chosen to participate in an obstacle course. The time taken, \(y\) seconds, for each footballer to complete the obstacle course is recorded, together with the footballer's Body Mass Index, \(x\). The results are shown in the table below.
FootballerBody Mass Index, \(\boldsymbol { x }\)Time taken to complete the obstacle course, \(y\) seconds
A18.7690
B19.5801
C20.2723
D20.4633
E20.8660
F21.9655
G23.2711
H24.3642
I24.8607
Russell claims, that for footballers, as Body Mass Index increases the time taken to complete the obstacle course tends to decrease.
  1. Find, to 3 decimal places, Spearman's rank correlation coefficient between \(x\) and \(y\).
  2. Use your value of Spearman's rank correlation coefficient to test Russell's claim. Use a 5\% significance level and state your hypotheses clearly. The product moment correlation coefficient for these data is - 0.5594
  3. Use the value of the product moment correlation coefficient to test for evidence of a negative correlation between Body Mass Index and the time taken to complete the obstacle course. Use a 5\% significance level.
  4. Using your conclusions to part (b) and part (c), describe the relationship between Body Mass Index and the time taken to complete the obstacle course.
Edexcel S3 2018 June Q2
  1. A random sample of 75 packets of seeds is selected from a production line. Each packet contains 12 seeds. The seeds are planted and the number of seeds that germinate from each packet is recorded. The results are as follows.
Number of seeds that
germinate from each packet
6 or
fewer
789101112
Number of packets0351828174
  1. Show that the probability of a randomly selected seed from this sample germinating is 0.82 A gardener suggests that a binomial distribution can be used to model the number of seeds that germinate from a packet of 12 seeds. She uses a binomial distribution with the estimated probability 0.82 of a seed germinating. Some of the calculated expected frequencies are shown in the table below.
    Number of seeds that
    germinate from each packet
    6 or
    fewer
    		789101112
    Expected frequency\(s\)2.807.97\(r\)22.0418.266.93
  2. Calculate the value of \(r\) and the value of \(s\), giving your answers correct to 2 decimal places.
  3. Test, at the \(10 \%\) level of significance, whether or not these data suggest that the binomial distribution is a suitable model for the number of seeds that germinate from a packet of 12 seeds. State your hypotheses clearly and show your working.
Edexcel S3 2018 June Q3
3. Star Farm produces duck eggs. Xander takes a random sample of 20 duck eggs from Star Farm and their widths, \(x \mathrm {~cm}\), are recorded. Xander's results are summarised as follows. $$\sum x = 92.0 \quad \sum x ^ { 2 } = 433.4974$$
  1. Calculate unbiased estimates of the mean and the variance of the width of duck eggs produced by Star Farm. Yinka takes an independent random sample of 30 duck eggs from Star Farm and their widths, \(y \mathrm {~cm}\), are recorded. Yinka's results are summarised as follows. $$\sum y = 142.5 \quad \sum y ^ { 2 } = 689.5078$$
  2. Treating the combined sample of 50 duck eggs as a single sample, estimate the standard error of the mean.
    (5) Research shows that the population of duck egg widths is normally distributed with standard deviation 0.71 cm . The farmer claims that the mean width of duck eggs produced by Star Farm is greater than 4.5 cm .
  3. Using your combined mean, test, at the \(5 \%\) level of significance, the farmer's claim. State your hypotheses clearly.
Edexcel S3 2018 June Q4
4. A company selects a random sample of five of its warehouses. The table below summarises the number of employees, in thousands, at each warehouse and the number of reported first aid incidents at each warehouse during 2017
WarehouseA\(B\)CDE
Number of employees, (in thousands)213.832.2
Number of reported first aid incidents1510402623
The personnel manager claims that the mean number of reported first aid incidents per 1000 employees is the same at each of the company's warehouses.
  1. Stating your hypotheses clearly, use a \(5 \%\) level of significance to test the manager's claim. Jean, the safety officer at warehouse \(C\), kept a record of each reported first aid incident at warehouse \(C\) in 2017. Jean wishes to select a systematic sample of 10 records from warehouse \(C\).
  2. Explain, in detail, how Jean should obtain such a sample.
Edexcel S3 2018 June Q5
5. A factory produces steel sheets whose weights, \(X \mathrm {~kg}\), have a normal distribution with an unknown mean \(\mu \mathrm { kg }\) and known standard deviation \(\sigma \mathrm { kg }\). A random sample of 25 sheets gave both a
  • \(95 \%\) confidence interval for \(\mu\) of \(( 30.612,31.788 )\)
  • \(c \%\) confidence interval for \(\mu\) of \(( 30.66,31.74 )\)
    1. Find the value of \(\sigma\)
    2. Find the value of \(c\), giving your answer correct to 3 significant figures.