Questions — Edexcel (9685 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 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 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
Sort by: Default | Easiest first | Hardest first
Edexcel S4 2012 June Q1
9 marks Standard +0.3
  1. A medical student is investigating whether there is a difference in a person's blood pressure when sitting down and after standing up. She takes a random sample of 12 people and measures their blood pressure, in mmHg , when sitting down and after standing up.
The results are shown below.
PersonA\(B\)CD\(E\)F\(G\)\(H\)I\(J\)\(K\)\(L\)
Sitting down135146138146141158136135146161119151
Standing up131147132140138160127136142154130144
The student decides to carry out a paired \(t\)-test to investigate whether, on average, the blood pressure of a person when sitting down is more than their blood pressure after standing up.
  1. State clearly the hypotheses that should be used and any necessary assumption that needs to be made.
  2. Carry out the test at the \(1 \%\) level of significance.
    \section*{L}
Edexcel S4 2012 June Q2
16 marks Challenging +1.2
  1. A biologist investigating the shell size of turtles takes random samples of adult female and adult male turtles and records the length, \(x \mathrm {~cm}\), of the shell. The results are summarised below.
Number in sampleSample mean \(\bar { x }\)\(\sum x ^ { 2 }\)
Female619.62308.01
Male1213.72262.57
You may assume that the samples come from independent normal distributions with the same variance. The biologist claims that the mean shell length of adult female turtles is 5 cm longer than the mean shell length of adult male turtles.
  1. Test the biologist's claim at the \(5 \%\) level of significance.
  2. Given that the true values for the variance of the population of adult male turtles and adult female turtles are both \(0.9 \mathrm {~cm} ^ { 2 }\),
    1. show that when samples of size 6 and 12 are used with a \(5 \%\) level of significance, the biologist's claim will be accepted if \(4.07 < \bar { X } _ { F } - \bar { X } _ { M } < 5.93\) where \(\bar { X } _ { F }\) and \(\bar { X } _ { M }\) are the mean shell lengths of females and males respectively.
    2. Hence find the probability of a type II error for this test if in fact the true mean shell length of adult female turtles is 6 cm more than the mean shell length of adult male turtles.
Edexcel S4 2012 June Q3
5 marks Standard +0.8
  1. The sample variance of the lengths of a random sample of 9 paving slabs sold by a builders' merchant is \(36 \mathrm {~mm} ^ { 2 }\). The sample variance of the lengths of a random sample of 11 paving slabs sold by a second builders' merchant is \(225 \mathrm {~mm} ^ { 2 }\). Test at the \(10 \%\) significance level whether or not there is evidence that the lengths of paving slabs sold by these builders' merchants differ in variability. State your hypotheses clearly.
    (You may assume the lengths of paving slabs are normally distributed.)
  2. A newspaper runs a daily Sudoku. A random sample of 10 people took the following times, in minutes, to complete the Sudoku.
\section*{\(\begin{array} { l l l l l l l l l l } 5.0 & 4.5 & 4.7 & 5.3 & 5.2 & 4.1 & 5.3 & 4.8 & 5.5 & 4.6 \end{array}\)} Given that the times to complete the Sudoku follow a normal distribution,
  1. calculate a 95\% confidence interval for
    1. the mean,
    2. the variance,
      of the times taken by people to complete the Sudoku. The newspaper requires the average time needed to complete the Sudoku to be 5 minutes with a standard deviation of 0.7 minutes.
  2. Comment on whether or not the Sudoku meets this requirement. Give a reason for your answer.
Edexcel S4 2012 June Q5
13 marks Standard +0.3
5. Boxes of chocolates manufactured by Philippe have a mean weight of \(\mu\) grams and a standard deviation of \(\sigma\) grams. A random sample of 25 of these boxes are weighed. Using this sample, the unbiased estimate of \(\mu\) is 455 and the unbiased estimate of \(\sigma ^ { 2 }\) is 55 .
  1. Test, at the \(5 \%\) level of significance, whether or not \(\sigma\) is greater than 6 . State your hypotheses clearly.
  2. Test, at the \(5 \%\) level of significance, whether or not \(\mu\) is more than 450 .
  3. State an assumption you have made in order to carry out the above tests.
Edexcel S4 2012 June Q6
16 marks Standard +0.8
6. When a tree seed is planted the probability of it germinating is \(p\). A random sample of size \(n\) is taken and the number of tree seeds, \(X\), which germinate is recorded.
    1. Show that \(\hat { p } _ { 1 } = \frac { X } { n }\) is an unbiased estimator of \(p\).
    2. Find the variance of \(\hat { p } _ { 1 }\). A second sample of size \(m\) is taken and the number of tree seeds, \(Y\), which germinate is recorded. Given that \(\hat { p } _ { 2 } = \frac { Y } { m }\) and that \(\hat { p } _ { 3 } = a \left( 3 \hat { p } _ { 1 } + 2 \hat { p } _ { 2 } \right)\) is an unbiased estimator of \(p\),
  2. show that
    1. \(\quad a = \frac { 1 } { 5 }\),
    2. \(\operatorname { Var } \left( \hat { p } _ { 3 } \right) = \frac { p ( 1 - p ) } { 25 } \left( \frac { 9 } { n } + \frac { 4 } { m } \right)\).
  3. Find the range of values of \(\frac { n } { m }\) for which $$\operatorname { Var } \left( \hat { p } _ { 3 } \right) < \operatorname { Var } \left( \hat { p } _ { 1 } \right) \text { and } \operatorname { Var } \left( \hat { p } _ { 3 } \right) < \operatorname { Var } \left( \hat { p } _ { 2 } \right)$$
  4. Given that \(n = 20\) and \(m = 60\), explain which of \(\hat { p } _ { 1 } , \hat { p } _ { 2 }\) or \(\hat { p } _ { 3 }\) is the best estimator.
Edexcel S4 2013 June Q1
4 marks Standard +0.3
  1. (a) Find the value of the constant \(a\) such that
$$\mathrm { P } \left( 1.690 < \chi _ { 7 } ^ { 2 } < a \right) = 0.95$$ The random variable \(Y\) follows an \(F\)-distribution with 6 and 4 degrees of freedom.
(b) (i) Find the upper \(1 \%\) critical value for \(Y\).
(ii) Find the lower \(1 \%\) critical value for \(Y\).
Edexcel S4 2013 June Q2
7 marks Standard +0.3
2. The time, \(t\) hours, that a typist can sit before incurring back pain is modelled by \(\mathrm { N } \left( \mu , \sigma ^ { 2 } \right)\). A random sample of 30 typists gave unbiased estimates for \(\mu\) and \(\sigma ^ { 2 }\) as shown below. $$\hat { \mu } = 2.5 \quad s ^ { 2 } = 0.36$$
  1. Find a 95\% confidence interval for \(\sigma ^ { 2 }\)
  2. State with a reason whether or not the confidence interval supports the assertion that \(\sigma ^ { 2 } = 0.495\)
Edexcel S4 2013 June Q3
10 marks Challenging +1.2
3. The number of houses sold per week by a firm of estate agents follows a Poisson distribution with mean 2 . The firm believes that the appointment of a new salesman will increase the number of houses sold. The firm tests its belief by recording the number of houses sold, \(x\), in the week following the appointment. The firm sets up the hypotheses \(\mathrm { H } _ { 0 } : \lambda = 2\) and \(\mathrm { H } _ { 1 } : \lambda > 2\), where \(\lambda\) is the mean number of houses sold per week, and rejects the null hypothesis if \(x \geqslant 3\)
  1. Find the size of the test.
  2. Show that the power function for this test is $$1 - \frac { 1 } { 2 } e ^ { - \lambda } \left( 2 + 2 \lambda + \lambda ^ { 2 } \right)$$ The table below gives the values of the power function to 2 decimal places. \begin{table}[h]
    \(\lambda\)2.53.03.54.05.07.0
    Power0.46\(r\)0.68\(s\)0.880.97
    \captionsetup{labelformat=empty} \caption{Table 1}
    \end{table}
  3. Calculate the values of \(r\) and \(s\).
  4. Draw a graph of the power function on the graph paper provided on page 6
  5. Find the range of values of \(\lambda\) for which the power of this test is greater than 0.6 For your convenience Table 1 is repeated here.
    \(\lambda\)2.53.03.54.05.07.0
    Power0.46\(r\)0.68\(s\)0.880.97
    \begin{figure}[h]
    \captionsetup{labelformat=empty} \caption{Table 1} \includegraphics[alt={},max width=\textwidth]{4f096806-33da-453f-a4c1-12be20d1a96d-06_2125_1603_614_166}
    \end{figure} \includegraphics[max width=\textwidth, alt={}, center]{4f096806-33da-453f-a4c1-12be20d1a96d-07_72_47_2615_1886}
Edexcel S4 2013 June Q4
15 marks Challenging +1.2
4. A company carries out an investigation into the strengths of rods from two different suppliers, Ardo and Bards. Independent random samples of rods were taken from each supplier and the force, \(x \mathrm { kN }\), needed to break each rod was recorded. The company wrote the results on a piece of paper but unfortunately spilt ink on it so some of the results can not be seen.
The paper with the results on is shown below. \includegraphics[max width=\textwidth, alt={}, center]{4f096806-33da-453f-a4c1-12be20d1a96d-08_435_1522_541_244}
    1. Use the data from Ardo to calculate an unbiased estimate, \(s _ { A } ^ { 2 }\), of the variance.
    2. Hence find an unbiased estimate, \(s _ { B } ^ { 2 }\), of the variance for the sample of 9 values from Bards.
  2. Stating your hypotheses clearly, test at the \(10 \%\) level of significance whether or not there is a difference in variability of strength between the rods from Ardo and the rods from Bards.
    (You may assume the two samples come from independent normal distributions.)
  3. Use a \(5 \%\) level of significance to test whether the mean strength of rods from Bards is more than 0.9 kN greater than the mean strength of rods from Ardo.
    (6)
Edexcel S4 2013 June Q5
8 marks Standard +0.3
  1. Students studying for their Mathematics GCSE are assessed by two examination papers. A teacher believes that on average the score on paper I is more than 1 mark higher than the score on paper II. To test this belief the scores of 8 randomly selected students are recorded. The results are given in the table below.
Student\(A\)\(B\)\(C\)\(D\)\(E\)\(F\)\(G\)\(H\)
Score on paper I5763688143655231
Score on paper II5362617844644329
Assuming that the scores are normally distributed and stating your hypotheses clearly, test at the \(5 \%\) level of significance whether or not there is evidence to support the teacher's belief.
Edexcel S4 2013 June Q6
10 marks Standard +0.3
  1. A machine fills bottles with water. The amount of water in each bottle is normally distributed. To check the machine is working properly, a random sample of 12 bottles is selected and the amount of water, in ml, in each bottle is recorded. Unbiased estimates for the mean and variance are
$$\hat { \mu } = 502 \quad s ^ { 2 } = 5.6$$ Stating your hypotheses clearly, test at the 1\% level of significance
  1. whether or not the mean amount of water in a bottle is more than 500 ml ,
  2. whether or not the standard deviation of the amount of water in a bottle is less than 3 ml .
Edexcel S4 2013 June Q7
9 marks Challenging +1.2
7. A machine produces bricks. The lengths, \(x \mathrm {~mm}\), of the bricks are distributed \(\mathrm { N } \left( \mu , 2 ^ { 2 } \right)\). At the start of each week a random sample of \(n\) bricks is taken to check the machine is working correctly.
A test is then carried out at the \(1 \%\) level of significance with $$\mathrm { H } _ { 0 } : \mu = 202 \text { and } \mathrm { H } _ { 1 } : \mu < 202$$
  1. Find, in terms of \(n\), the critical region of the test. The probability of a type II error, when \(\mu = 200\), is less than 0.05
  2. Find the minimum value of \(n\).
Edexcel S4 2013 June Q8
12 marks Challenging +1.2
8. A random sample \(W _ { 1 } , W _ { 2 } \ldots , W _ { n }\) is taken from a distribution with mean \(\mu\) and variance \(\sigma ^ { 2 }\)
  1. Write down \(\mathrm { E } \left( \sum _ { i = 1 } ^ { n } W _ { i } \right)\) and show that \(\mathrm { E } \left( \sum _ { i = 1 } ^ { n } W _ { i } ^ { 2 } \right) = n \left( \sigma ^ { 2 } + \mu ^ { 2 } \right)\) An estimator for \(\mu\) is $$\bar { X } = \frac { 1 } { n } \sum _ { i = 1 } ^ { n } W _ { i }$$
  2. Show that \(\bar { X }\) is a consistent estimator for \(\mu\). An estimator of \(\sigma ^ { 2 }\) is $$U = \frac { 1 } { n } \sum _ { i = 1 } ^ { n } W _ { i } ^ { 2 } - \left( \frac { 1 } { n } \sum _ { i = 1 } ^ { n } W _ { i } \right) ^ { 2 }$$
  3. Find the bias of \(U\).
  4. Write down an unbiased estimator of \(\sigma ^ { 2 }\) in the form \(k U\), where \(k\) is in terms of \(n\).
Edexcel S4 2013 June Q1
7 marks Standard +0.3
  1. George owns a garage and he records the mileage of cars, \(x\) thousands of miles, between services. The results from a random sample of 10 cars are summarised below.
$$\sum x = 113.4 \quad \sum x ^ { 2 } = 1414.08$$ The mileage of cars between services is normally distributed and George believes that the standard deviation is 2.4 thousand miles. Stating your hypotheses clearly, test, at the \(5 \%\) level of significance, whether or not these data support George’s belief.
Edexcel S4 2013 June Q2
10 marks Standard +0.3
2. Every 6 months some engineers are tested to see if their times, in minutes, to assemble a particular component have changed. The times taken to assemble the component are normally distributed. A random sample of 8 engineers was chosen and their times to assemble the component were recorded in January and in July. The data are given in the table below.
Engineer\(A\)\(B\)\(C\)\(D\)\(E\)\(F\)\(G\)\(H\)
January1719222615281821
July1918252417251619
  1. Calculate a \(95 \%\) confidence interval for the mean difference in times.
  2. Use your confidence interval to state, giving a reason, whether or not there is evidence of a change in the mean time to assemble a component. State your hypotheses clearly.
Edexcel S4 2013 June Q3
12 marks Standard +0.8
3. An archaeologist is studying the compression strength of bricks at some ancient European sites. He took random samples from two sites \(A\) and \(B\) and recorded the compression strength of these bricks in appropriate units. The results are summarised below.
SiteSample size \(( n )\)Sample mean \(( \bar { x } )\)Standard deviation \(( s )\)
\(A\)78.434.24
\(B\)1314.314.37
It can be assumed that the compression strength of bricks is normally distributed.
  1. Test, at the \(2 \%\) level of significance, whether or not there is evidence of a difference in the variances of compression strength of the bricks between these two sites. State your hypotheses clearly.
    (5) Site \(A\) is older than site \(B\) and the archaeologist claims that the mean compression strength of the bricks was greater at the younger site.
  2. Stating your hypotheses clearly and using a \(1 \%\) level of significance, test the archaeologist's claim.
  3. Explain briefly the importance of the test in part (a) to the test in part (b).
Edexcel S4 2013 June Q4
16 marks Standard +0.3
  1. A random sample of size \(2 , X _ { 1 }\) and \(X _ { 2 }\), is taken from the random variable \(X\) which has a continuous uniform distribution over the interval \([ - a , 2 a ] , a > 0\)
    1. Show that \(\bar { X } = \frac { X _ { 1 } + X _ { 2 } } { 2 }\) is a biased estimator of \(a\) and find the bias.
    The random variable \(Y = k \bar { X }\) is an unbiased estimator of \(a\).
  2. Write down the value of the constant \(k\).
  3. Find \(\operatorname { Var } ( Y )\). The random variable \(M\) is the maximum of \(X _ { 1 }\) and \(X _ { 2 }\) The probability density function, \(m ( x )\), of \(M\) is given by $$m ( x ) = \left\{ \begin{array} { c l } \frac { 2 ( x + a ) } { 9 a ^ { 2 } } & - a \leqslant x \leqslant 2 a \\ 0 & \text { otherwise } \end{array} \right.$$
  4. Show that \(M\) is an unbiased estimator of \(a\). Given that \(\mathrm { E } \left( M ^ { 2 } \right) = \frac { 3 } { 2 } a ^ { 2 }\)
  5. find \(\operatorname { Var } ( M )\).
  6. State, giving a reason, whether you would use \(Y\) or \(M\) as an estimator of \(a\). A random sample of two values of \(X\) are 5 and - 1
  7. Use your answer to part (f) to estimate \(a\).
Edexcel S4 2013 June Q5
17 marks Challenging +1.8
5. Water is tested at various stages during a purification process by an environmental scientist. A certain organism occurs randomly in the water at a rate of \(\lambda\) every 10 ml . The scientist selects a random sample of 20 ml of water to check whether there is evidence that \(\lambda\) is greater than 1 . The criterion the scientist uses for rejecting the hypothesis that \(\lambda = 1\) is that there are 4 or more organisms in the sample of 20 ml .
  1. Find the size of the test.
  2. When \(\lambda = 2.5\) find P (Type II error). A statistician suggests using an alternative test. The statistician's test involves taking a random sample of 10 ml and rejecting the hypothesis that \(\lambda = 1\) if 2 or more organisms are present but accepting the hypothesis if no organisms are in the sample. If only 1 organism is found then a second random sample of 10 ml is taken and the hypothesis is rejected if 2 or more organisms are present, otherwise the hypothesis is accepted.
  3. Show that the power of the statistician's test is given by $$1 - \mathrm { e } ^ { - \lambda } - \lambda ( 1 + \lambda ) \mathrm { e } ^ { - 2 \lambda }$$ Table 1 below gives some values, to 2 decimal places, of the power function of the statistician’s test. \begin{table}[h] \end{table} Table 1 Figure 1 shows a graph of the power function for the scientist's test.
  4. On the same axes draw the graph of the power function for the statistician’s test. Given that it takes 20 minutes to collect and test a 20 ml sample and 15 minutes to collect and test a 10 ml sample
  5. show that the expected time of the statistician's test is slower than the scientist's test for \(\lambda \mathrm { e } ^ { - \lambda } > \frac { 1 } { 3 }\)
  6. By considering the times when \(\lambda = 1\) and \(\lambda = 2\) together with the power curves in part (e) suggest, giving a reason, which test you would use.
    (2) \begin{figure}[h]
    \includegraphics[alt={},max width=\textwidth]{399f7507-4878-45ad-b77e-02ebd807ed75-10_1185_1157_1452_392} \captionsetup{labelformat=empty} \caption{Figure 1}
    \end{figure} \includegraphics[max width=\textwidth, alt={}, center]{399f7507-4878-45ad-b77e-02ebd807ed75-11_81_47_2622_1886}
Edexcel S4 2013 June Q6
13 marks Challenging +1.2
6. The carbon content, measured in suitable units, of steel is normally distributed. Two independent random samples of steel were taken from a refining plant at different times and their carbon content recorded. The results are given below. Sample A: \(\quad 1.5 \quad 0.9 \quad 1.3 \quad 1.2\) \(\begin{array} { l l l l l l l } \text { Sample } B : & 0.4 & 0.6 & 0.8 & 0.3 & 0.5 & 0.4 \end{array}\)
  1. Stating your hypotheses clearly, carry out a suitable test, at the \(10 \%\) level of significance, to show that both samples can be assumed to have come from populations with a common variance \(\sigma ^ { 2 }\).
  2. Showing your working clearly, find the \(99 \%\) confidence interval for \(\sigma ^ { 2 }\) based on both samples.
Edexcel S4 2014 June Q1
9 marks Standard +0.3
  1. In a trial for a new cough medicine, a random sample of 8 healthy patients were given steadily increasing doses of a pepper extract until they started coughing. The level of pepper that triggered the coughing was recorded. Each patient completed the trial after taking a standard cough medicine and, at a later time, after taking the new medicine. The results are given in the table below.
Level of pepper extract that triggers coughing
Patient\(A\)\(B\)\(C\)\(D\)\(E\)\(F\)\(G\)\(H\)
Standard medicine461218312316279
New medicine5316134911343822
  1. Using a suitable test, at the \(5 \%\) level of significance, state whether or not, on the basis of this trial, you would recommend using the new medicine. State your hypotheses clearly.
  2. State an assumption needed to carry out this test.
Edexcel S4 2014 June Q2
7 marks Challenging +1.8
2. The cloth produced by a certain manufacturer has defects that occur randomly at a constant rate of \(\lambda\) per square metre. If \(\lambda\) is thought to be greater than 1.5 then action has to be taken. Using \(\mathrm { H } _ { 0 } : \lambda = 1.5\) and \(\mathrm { H } _ { 1 } : \lambda > 1.5\) a quality control officer takes a \(4 \mathrm {~m} ^ { 2 }\) sample of cloth and rejects \(\mathrm { H } _ { 0 }\) if there are 11 or more defects. If there are 8 or fewer defects she accepts \(\mathrm { H } _ { 0 }\). If there are 9 or 10 defects a second sample of \(4 \mathrm {~m} ^ { 2 }\) is taken and \(\mathrm { H } _ { 0 }\) is rejected if there are 11 or more defects in this second sample, otherwise it is accepted.
  1. Find the size of this test.
  2. Find the power of this test when \(\lambda = 2\)
Edexcel S4 2014 June Q3
12 marks Standard +0.8
3. A farmer is investigating the milk yields of two breeds of cow. He takes a random sample of 9 cows of breed \(A\) and an independent random sample of 12 cows of breed \(B\). For a 5 day period he measures the amount of milk, \(x\) gallons, produced by each cow. The results are summarised in the table below.
BreedSample sizeMean \(( \overline { \boldsymbol { x } } )\)Standard deviation \(\left( \boldsymbol { s } _ { \boldsymbol { x } } \right)\)
\(A\)96.232.98
\(B\)127.132.33
The amount of milk produced by each cow can be assumed to follow a normal distribution.
  1. Use a two-tail test to show, at the \(10 \%\) level of significance, that the variances of the yields of the two breeds can be assumed to be equal. State your hypotheses clearly.
  2. Stating your hypotheses clearly, test, at the \(5 \%\) level of significance, whether or not there is a difference in the mean yields of the two breeds of cow.
  3. Explain briefly the importance of the test in part (a) for the test in part (b).
Edexcel S4 2014 June Q4
12 marks Challenging +1.8
  1. At the start of each academic year, a large college carries out a diagnostic test on a random sample of new students. Past experience has shown that the standard deviation of the scores on this test is 19.71
The admissions tutor claimed that the new students in 2013 would have more varied scores than usual. The scores for the students taking the test can be assumed to come from a normal distribution. A random sample of 10 new students was taken and the score \(x\), for each student was recorded. The data are summarised as \(\sum x = 619 \sum x ^ { 2 } = 42397\)
  1. Stating your hypotheses clearly, and using a \(5 \%\) level of significance, test the admission tutor's claim. The admissions tutor decides that in future he will use the same hypotheses but take a larger sample of size 30 and use a significance level of 1\%.
  2. Use the tables to show that, to 3 decimal places, the critical region for \(S ^ { 2 }\) is \(S ^ { 2 } > 664.281\)
  3. Find the probability of a type II error using this test when the true value of the standard deviation is in fact 22.20
Edexcel S4 2014 June Q5
16 marks Standard +0.3
5. A large company has designed an aptitude test for new recruits. The score, \(S\), for an individual taking the test, has a normal distribution with mean \(\mu\) and standard deviation \(\sigma\). In order to estimate \(\mu\) and \(\sigma\), a random sample of 15 new recruits were given the test and their scores, \(x\), are summarised as $$\sum x = 880 \quad \sum x ^ { 2 } = 54892$$
  1. Calculate a 95\% confidence interval for
    1. \(\mu\),
    2. \(\sigma\). The company wants to ensure that no more than \(80 \%\) of new recruits pass the test.
  2. Using values from your confidence intervals in part (a), estimate the lowest pass mark they should set.
Edexcel S4 2014 June Q6
19 marks Challenging +1.2
6. Emily is monitoring the level of pollution in a river. Over a period of time she has found that the amount of pollution, \(X\), in a 100 ml sample of river water has a continuous distribution with probability density function \(\mathrm { f } ( x )\) given by $$f ( x ) = \left\{ \begin{array} { c c } \frac { 2 x } { a ^ { 2 } } & 0 \leqslant x \leqslant a \\ 0 & \text { otherwise } \end{array} \right.$$ where \(a\) is a constant. Emily takes a random sample \(X _ { 1 } , X _ { 2 } , X _ { 3 } , \ldots , X _ { n }\) to try to estimate the value of \(a\).
  1. Show that \(\mathrm { E } ( \bar { X } ) = \frac { 2 a } { 3 }\) and \(\operatorname { Var } ( \bar { X } ) = \frac { a ^ { 2 } } { 18 n }\) The random variable \(S = p \bar { X }\), where \(p\) is a constant, is an unbiased estimator of \(a\).
  2. Write down the value of \(p\) and find \(\operatorname { Var } ( S )\). Felix suggests using the statistic \(M = \max \left\{ X _ { 1 } , X _ { 2 } , X _ { 3 } , \ldots , X _ { n } \right\}\) as an estimator of \(a\).
    He calculates \(\mathrm { E } ( M ) = \frac { 2 n } { 2 n + 1 } a\) and \(\operatorname { Var } ( M ) = \frac { n } { ( n + 1 ) ( 2 n + 1 ) ^ { 2 } } a ^ { 2 }\)
  3. State, giving your reasons, whether or not \(M\) is a consistent estimator of \(a\). The random variable \(T = q M\), where \(q\) is a constant, is an unbiased estimator of \(a\).
  4. Write down, in terms of \(n\), the value of \(q\) and find \(\operatorname { Var } ( T )\).
  5. State, giving your reasons, which of \(S\) or \(T\) you would recommend Emily use as an estimator of \(a\). Emily took a sample of 5 values of \(X\) and obtained the following:
    5.3
    4.3 \(\begin{array} { l l } 5.7 & 7.8 \end{array}\) 6.9
  6. Calculate the estimate of \(a\) using your recommended estimator from part (e).
  7. Find the standard error of your estimate, giving your answer to 2 decimal places.