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Edexcel S4 2009 June Q2
12 marks Standard +0.3
2. An emission-control device is tested to see if it reduces \(\mathrm { CO } _ { 2 }\) emissions from cars. The emissions from 6 randomly selected cars are measured with and without the device. The results are as follows.
CarA\(B\)CDE\(F\)
Emissions without device151.4164.3168.5148.2139.4151.2
Emissions with device148.9162.7166.9150.1140.0146.7
  1. State an assumption that needs to be made in order to carry out a \(t\)-test in this case.
  2. State why a paired \(t\)-test is suitable for use with these data.
  3. Using a \(5 \%\) level of significance, test whether or not there is evidence that the device reduces \(\mathrm { CO } _ { 2 }\) emissions from cars.
  4. Explain, in context, what a type II error would be in this case.
Edexcel S4 2009 June Q3
12 marks Standard +0.3
Define, in terms of \(\mathrm { H } _ { 0 }\) and/or \(\mathrm { H } _ { 1 }\),
  1. the size of a hypothesis test,
  2. the power of a hypothesis test. The probability of getting a head when a coin is tossed is denoted by \(p\).
    This coin is tossed 12 times in order to test the hypotheses \(\mathrm { H } _ { 0 } : p = 0.5\) against \(\mathrm { H } _ { 1 } : p \neq 0.5\), using a 5\% level of significance.
  3. Find the largest critical region for this test, such that the probability in each tail is less than 2.5\%.
  4. Given that \(p = 0.4\)
    1. find the probability of a type II error when using this test,
    2. find the power of this test.
  5. Suggest two ways in which the power of the test can be increased.
Edexcel S4 2009 June Q4
14 marks Standard +0.3
  1. A farmer set up a trial to assess whether adding water to dry feed increases the milk yield of his cows. He randomly selected 22 cows. Thirteen of the cows were given dry feed and the other 9 cows were given the feed with water added. The milk yields, in litres per day, were recorded with the following results.
\cline { 2 - 4 } \multicolumn{1}{c|}{}Sample sizeMean\(s ^ { 2 }\)
Dry feed1325.542.45
Feed with water added927.941.02
You may assume that the milk yield from cows given the dry feed and the milk yield from cows given the feed with water added are from independent normal distributions.
  1. Test, at the \(10 \%\) level of significance, whether or not the variances of the populations from which the samples are drawn are the same. State your hypotheses clearly.
  2. Calculate a \(95 \%\) confidence interval for the difference between the two mean milk yields.
  3. Explain the importance of the test in part (a) to the calculation in part (b).
Edexcel S4 2009 June Q5
14 marks Challenging +1.2
  1. A machine fills jars with jam. The weight of jam in each jar is normally distributed. To check the machine is working properly the contents of a random sample of 15 jars are weighed in grams. Unbiased estimates of the mean and variance are obtained as
$$\hat { \mu } = 560 \quad s ^ { 2 } = 25.2$$ Calculate a 95\% confidence interval for,
  1. the mean weight of jam,
  2. the variance of the weight of jam. A weight of more than 565 g is regarded as too high and suggests the machine is not working properly.
  3. Use appropriate confidence limits from parts (a) and (b) to find the highest estimate of the proportion of jars that weigh too much.
Edexcel S4 2009 June Q6
15 marks Standard +0.8
6. A continuous uniform distribution on the interval \([ 0 , k ]\) has mean \(\frac { k } { 2 }\) and variance \(\frac { k ^ { 2 } } { 12 }\). A random sample of three independent variables \(X _ { 1 } , X _ { 2 }\) and \(X _ { 3 }\) is taken from this distribution.
  1. Show that \(\frac { 2 } { 3 } X _ { 1 } + \frac { 1 } { 2 } X _ { 2 } + \frac { 5 } { 6 } X _ { 3 }\) is an unbiased estimator for \(k\). An unbiased estimator for \(k\) is given by \(\hat { k } = a X _ { 1 } + b X _ { 2 }\) where \(a\) and \(b\) are constants.
  2. Show that \(\operatorname { Var } ( \hat { k } ) = \left( a ^ { 2 } - 2 a + 2 \right) \frac { k ^ { 2 } } { 6 }\)
  3. Hence determine the value of \(a\) and the value of \(b\) for which \(\hat { k }\) has minimum variance, and calculate this minimum variance.
Edexcel S4 2010 June Q1
13 marks Standard +0.3
  1. A teacher wishes to test whether playing background music enables students to complete a task more quickly. The same task was completed by 15 students, divided at random into two groups. The first group had background music playing during the task and the second group had no background music playing.
    The times taken, in minutes, to complete the task are summarised below.
Sample size \(n\)Standard deviation \(s\)Mean \(\bar { x }\)
With background music84.115.9
Without background music75.217.9
You may assume that the times taken to complete the task by the students are two independent random samples from normal distributions.
  1. Stating your hypotheses clearly, test, at the \(10 \%\) level of significance, whether or not the variances of the times taken to complete the task with and without background music are equal.
  2. Find a 99\% confidence interval for the difference in the mean times taken to complete the task with and without background music. Experiments like this are often performed using the same people in each group.
  3. Explain why this would not be appropriate in this case.
Edexcel S4 2010 June Q2
9 marks Standard +0.8
  1. As part of an investigation, a random sample of 10 people had their heart rate, in beats per minute, measured whilst standing up and whilst lying down. The results are summarised below.
Person12345678910
Heart rate lying down66705965726662695668
Heart rate standing up75766367807565746375
  1. State one assumption that needs to be made in order to carry out a paired \(t\)-test.
  2. Test, at the \(5 \%\) level of significance, whether or not there is any evidence that standing up increases people's mean heart rate by more than 5 beats per minute. State your hypotheses clearly.
Edexcel S4 2010 June Q3
12 marks Standard +0.3
A manager in a sweet factory believes that the machines are working incorrectly and the proportion \(p\) of underweight bags of sweets is more than \(5 \%\). He decides to test this by randomly selecting a sample of 5 bags and recording the number \(X\) that are underweight. The manager sets up the hypotheses \(\mathrm { H } _ { 0 } : p = 0.05\) and \(\mathrm { H } _ { 1 } : p > 0.05\) and rejects the null hypothesis if \(x > 1\).
  1. Find the size of the test.
  2. Show that the power function of the test is $$1 - ( 1 - p ) ^ { 4 } ( 1 + 4 p )$$ The manager goes on holiday and his deputy checks the production by randomly selecting a sample of 10 bags of sweets. He rejects the hypothesis that \(p = 0.05\) if more than 2 underweight bags are found in the sample.
  3. Find the probability of a Type I error using the deputy's test. \section*{Question 3 continues on page 12} The table below gives some values, to 2 decimal places, of the power function for the deputy's test.
    \(p\)0.100.150.200.25
    Power0.07\(s\)0.320.47
  4. Find the value of \(s\). The graph of the power function for the manager's test is shown in Figure 1. \begin{figure}[h]
    \includegraphics[alt={},max width=\textwidth]{0bc6c296-9cbe-498b-89d9-c034b1b246e4-08_1157_1436_847_260} \captionsetup{labelformat=empty} \caption{Figure 1}
    \end{figure}
  5. On the same axes, draw the graph of the power function for the deputy's test.
    1. State the value of \(p\) where these graphs intersect.
    2. Compare the effectiveness of the two tests if \(p\) is greater than this value. The deputy suggests that they should use his sampling method rather than the manager's.
  7. Give a reason why the manager might not agree to this change.
Edexcel S4 2010 June Q4
16 marks Challenging +1.2
4. A random sample of 15 strawberries is taken from a large field and the weight \(x\) grams of each strawberry is recorded. The results are summarised below. $$\sum x = 291 \quad \sum x ^ { 2 } = 5968$$ Assume that the weights of strawberries are normally distributed. Calculate a 95\% confidence interval for
    1. the mean of the weights of the strawberries in the field,
    2. the variance of the weights of the strawberries in the field. Strawberries weighing more than 23 g are considered to be less tasty.
  2. Use appropriate confidence limits from part (a) to find the highest estimate of the proportion of strawberries that are considered to be less tasty.
Edexcel S4 2010 June Q5
11 marks Standard +0.3
  1. A car manufacturer claims that, on a motorway, the mean number of miles per gallon for the Panther car is more than 70 . To test this claim a car magazine measures the number of miles per gallon, \(x\), of each of a random sample of 20 Panther cars and obtained the following statistics.
$$\bar { x } = 71.2 \quad s = 3.4$$ The number of miles per gallon may be assumed to be normally distributed.
  1. Stating your hypotheses clearly and using a \(5 \%\) level of significance, test the manufacturer's claim. The standard deviation of the number of miles per gallon for the Tiger car is 4 .
  2. Stating your hypotheses clearly, test, at the \(5 \%\) level of significance, whether or not there is evidence that the variance of the number of miles per gallon for the Panther car is different from that of the Tiger car.
Edexcel S4 2010 June Q6
14 marks Standard +0.8
6. Faults occur in a roll of material at a rate of \(\lambda\) per \(\mathrm { m } ^ { 2 }\). To estimate \(\lambda\), three pieces of material of sizes \(3 \mathrm {~m} ^ { 2 } , 7 \mathrm {~m} ^ { 2 }\) and \(10 \mathrm {~m} ^ { 2 }\) are selected and the number of faults \(X _ { 1 } , X _ { 2 }\) and \(X _ { 3 }\) respectively are recorded. The estimator \(\hat { \lambda }\), where $$\hat { \lambda } = k \left( X _ { 1 } + X _ { 2 } + X _ { 3 } \right)$$ is an unbiased estimator of \(\lambda\).
  1. Write down the distributions of \(X _ { 1 } , X _ { 2 }\) and \(X _ { 3 }\) and find the value of \(k\).
  2. Find \(\operatorname { Var } ( \hat { \lambda } )\). A random sample of \(n\) pieces of this material, each of size \(4 \mathrm {~m} ^ { 2 }\), was taken. The number of faults on each piece, \(Y\), was recorded.
  3. Show that \(\frac { 1 } { 4 } \bar { Y }\) is an unbiased estimator of \(\lambda\).
  4. Find \(\operatorname { Var } \left( \frac { 1 } { 4 } \bar { Y } \right)\).
  5. Find the minimum value of \(n\) for which \(\frac { 1 } { 4 } \bar { Y }\) becomes a better estimator of \(\lambda\) than \(\hat { \lambda }\).
Edexcel S4 2011 June Q1
2 marks Challenging +1.2
  1. Find the value of the constant \(a\) such that
  2. Find the value of the constant \(a\) such that
$$\mathrm { P } \left( a < F _ { 8,10 } < 3.07 \right) = 0.94$$
Edexcel S4 2011 June Q2
5 marks Standard +0.8
  1. Two independent random samples \(X _ { 1 } , X _ { 2 } , \ldots , X _ { 7 }\) and \(Y _ { 1 } , Y _ { 2 } , Y _ { 3 } , Y _ { 4 }\) were taken from different normal populations with a common standard deviation \(\sigma\). The following sample statistics were calculated.
$$s _ { x } = 14.67 \quad s _ { y } = 12.07$$ Find the \(99 \%\) confidence interval for \(\sigma ^ { 2 }\) based on these two samples.
Edexcel S4 2011 June Q3
8 marks Standard +0.3
3. Manuel is planning to buy a new machine to squeeze oranges in his cafe and he has two models, at the same price, on trial. The manufacturers of machine \(B\) claim that their machine produces more juice from an orange than machine \(A\). To test this claim Manuel takes a random sample of 8 oranges, cuts them in half and puts one half in machine \(A\) and the other half in machine \(B\). The amount of juice, in ml , produced by each machine is given in the table below.
Orange12345678
Machine \(A\)6058555352515456
Machine \(B\)6160585255505258
Stating your hypotheses clearly, test, at the \(10 \%\) level of significance, whether or not the mean amount of juice produced by machine \(B\) is more than the mean amount produced by machine \(A\).
Edexcel S4 2011 June Q4
12 marks Challenging +1.2
4. A proportion \(p\) of letters sent by a company are incorrectly addressed and if \(p\) is thought to be greater than 0.05 then action is taken.
Using \(\mathrm { H } _ { 0 } : p = 0.05\) and \(\mathrm { H } _ { 1 } : p > 0.05\), a manager from the company takes a random sample of 40 letters and rejects \(\mathrm { H } _ { 0 }\) if the number of incorrectly addressed letters is more than 3 .
  1. Find the size of this test.
  2. Find the probability of a Type II error in the case where \(p\) is in fact 0.10 Table 1 below gives some values, to 2 decimal places, of the power function of this test. \begin{table}[h]
    \(p\)0.0750.1000.1250.1500.1750.2000.225
    Power0.35\(s\)0.750.870.940.970.99
    \captionsetup{labelformat=empty} \caption{Table 1}
    \end{table}
  3. Write down the value of \(s\). A visiting consultant uses an alternative system to test the same hypotheses. A sample of 15 letters is taken. If these are all correctly addressed then \(\mathrm { H } _ { 0 }\) is accepted. If 2 or more are found to have been incorrectly addressed then \(\mathrm { H } _ { 0 }\) is rejected. If only one is found to be incorrectly addressed then a further random sample of 15 is taken and \(\mathrm { H } _ { 0 }\) is rejected if 2 or more are found to have been incorrectly addressed in this second sample, otherwise \(\mathrm { H } _ { 0 }\) is accepted.
  4. Find the size of the test used by the consultant. \section*{Question 4 continues on page 8} For your convenience Table 1 is repeated here \begin{table}[h]
    \(p\)0.0750.1000.1250.1500.1750.2000.225
    Power0.35\(s\)0.750.870.940.970.99
    \captionsetup{labelformat=empty} \caption{Table 1}
    \end{table} Figure 1 shows the graph of the power function of the test used by the consultant. \begin{figure}[h]
    \includegraphics[alt={},max width=\textwidth]{8dfc721d-4782-4482-9976-11189370f3b7-07_1712_1673_660_130} \captionsetup{labelformat=empty} \caption{Figure 1}
    \end{figure}
  5. On Figure 1 draw the graph of the power function of the manager's test.
    (2)
  6. State, giving your reasons, which test you would recommend.
    (2)
Edexcel S4 2011 June Q5
14 marks Standard +0.3
  1. The weights of the contents of breakfast cereal boxes are normally distributed.
A manufacturer changes the style of the boxes but claims that the weight of the contents remains the same.
A random sample of 6 old style boxes had contents with the following weights (in grams). $$\begin{array} { l l l l l l } 512 & 503 & 514 & 506 & 509 & 515 \end{array}$$ The weights, \(y\) grams, of the contents of an independent random sample of 5 new style boxes gave $$\bar { y } = 504.8 \text { and } s _ { y } = 3.420$$
  1. Use a two-tail test to show, at the \(10 \%\) level of significance, that the variances of the weights of the contents of the old and new style boxes can be assumed to be equal. State your hypotheses clearly.
  2. Showing your working clearly, find a \(90 \%\) confidence interval for \(\mu _ { x } - \mu _ { y }\), where \(\mu _ { x }\) and \(\mu _ { y }\) are the mean weights of the contents of old and new style boxes respectively.
  3. With reference to your confidence interval comment on the manufacturer's claim.
Edexcel S4 2011 June Q6
16 marks Challenging +1.2
  1. A random sample \(X _ { 1 } , X _ { 2 } , \ldots , X _ { n }\) is taken from a population where each of the \(X _ { i }\) have a continuous uniform distribution over the interval \([ 0 , \beta ]\).
    The random variable \(Y = \max \left\{ X _ { 1 } , X _ { 2 } , \ldots , X _ { n } \right\}\).
    The probability density function of \(Y\) is given by
$$f ( y ) = \left\{ \begin{array} { c c } \frac { n } { \beta ^ { n } } y ^ { n - 1 } & 0 \leqslant y \leqslant \beta \\ 0 & \text { otherwise } \end{array} \right.$$
  1. Show that \(\mathrm { E } \left( Y ^ { m } \right) = \frac { n } { n + m } \beta ^ { m }\).
  2. Write down \(\mathrm { E } ( Y )\).
  3. Using your answers to parts (a) and (b), or otherwise, show that $$\operatorname { Var } ( Y ) = \frac { n } { ( n + 1 ) ^ { 2 } ( n + 2 ) } \beta ^ { 2 }$$
  4. State, giving your reasons, whether or not \(Y\) is a consistent estimator of \(\beta\). The random variables \(M = 2 \bar { X }\), where \(\bar { X } = \frac { 1 } { n } \left( X _ { 1 } + X _ { 2 } + \ldots + X _ { n } \right)\), and \(S = k Y\), where \(k\) is a constant, are both unbiased estimators of \(\beta\).
  5. Find the value of \(k\) in terms of \(n\).
  6. State, giving your reasons, which of \(M\) and \(S\) is the better estimator of \(\beta\) in this case. Five observations of \(X\) are: \(\quad \begin{array} { l l l l l } 8.5 & 6.3 & 5.4 & 9.1 & 7.6 \end{array}\)
  7. Calculate the better estimate of \(\beta\).
Edexcel S4 2011 June Q7
18 marks Standard +0.8
  1. A machine produces components whose lengths are normally distributed with mean 102.3 mm and standard deviation 2.8 mm . After the machine had been serviced, a random sample of 20 components were tested to see if the mean and standard deviation had changed. The lengths, \(x \mathrm {~mm}\), of each of these 20 components are summarised as
$$\sum x = 2072 \quad \sum x ^ { 2 } = 214856$$
  1. Stating your hypotheses clearly, test, at the \(5 \%\) level of significance, whether or not there is evidence of a change in standard deviation.
  2. Stating your hypotheses clearly, test, at the \(5 \%\) level of significance, whether or not the mean length of the components has changed from the original value of 102.3 mm using
    1. a normal distribution,
    2. a \(t\) distribution.
  3. Comment on the mean length of components produced after the service in the light of the tests from part (a) and part (b). Give a reason for your answer.
Edexcel S4 2013 June Q1
4 marks Standard +0.3
  1. 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.
    1. Find the upper \(1 \%\) critical value for \(Y\).
    2. 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 \(n\).