Questions — Edexcel S4 (144 questions)

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Edexcel S4 2013 June Q5
  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
  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
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
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
  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
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
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
  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
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
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
  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
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
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
  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
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
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.
Edexcel S4 2014 June Q1
  1. A production line is designed to fill bottles with oil. The amount of oil placed in a bottle is normally distributed and the mean is set to 100 ml .
The amount of oil, \(x \mathrm { ml }\), in each of 8 randomly selected bottles is recorded, and the following statistics are obtained. $$\bar { x } = 92.875 \quad s = 8.3055$$ Malcolm believes that the mean amount of oil placed in a bottle is less than 100 ml .
Stating your hypotheses clearly, test, at the \(5 \%\) significance level, whether or not Malcolm's belief is supported.
Edexcel S4 2014 June Q2
2. (a) Define
  1. a Type I error,
  2. a Type II error. Rolls of material, manufactured by a machine, contain defects at a mean rate of 6 per roll. The machine is modified. A single roll is selected at random and a test is carried out to see whether or not the mean number of defects per roll has decreased. The significance level is chosen to be as close as possible to \(5 \%\).
    (b) Calculate the probability of a Type I error for this test.
    (c) Given that the true mean number of defects per roll of material made by the machine is now 4, calculate the probability of a Type II error.
Edexcel S4 2014 June Q3
3. A large number of chicks were fed a special diet for 10 days. A random sample of 9 of these chicks is taken and the weight gained, \(x\) grams, by each chick is recorded. The results are summarised below. $$\sum x = 181 \quad \sum x ^ { 2 } = 3913$$ You may assume that the weights gained by the chicks are normally distributed.
Calculate a 95\% confidence interval for
    1. the mean of the weights gained by the chicks,
    2. the variance of the weights gained by the chicks. A chick which gains less than \(16 g\) has to be given extra feed.
  2. Using appropriate confidence limits from part (a), find the lowest estimate of the proportion of chicks that need extra feed.
Edexcel S4 2014 June Q4
  1. A random sample of 8 people were given a new drug designed to help people sleep.
In a two-week period the drug was given for one week and a placebo (a tablet that contained no drug) was given for one week. In the first week 4 people, selected at random, were given the drug and the other 4 people were given the placebo. Those who were given the drug in the first week were given the placebo in the second week. Those who were given the placebo in the first week were given the drug in the second week. The mean numbers of hours of sleep per night for each of the people are shown in the table.
Person\(A\)\(B\)\(C\)\(D\)\(E\)\(F\)\(G\)\(H\)
Hours of sleep with drug10.87.28.76.89.410.911.17.6
Hours of sleep with placebo10.06.59.05.68.78.09.86.8
  1. State one assumption that needs to be made in order to carry out a paired \(t\)-test.
  2. Stating your hypotheses clearly, test, at the \(1 \%\) level of significance, whether or not the drug increases the mean number of hours of sleep per night by more than 10 minutes. State the critical value for this test.
Edexcel S4 2014 June Q5
5. A statistician believes a coin is biased and the probability, \(p\), of getting a head when the coin is tossed is less than 0.5 The statistician decides to test this by tossing the coin 10 times and recording the number, \(X\), of heads. He sets up the hypotheses \(\mathrm { H } _ { 0 } : p = 0.5\) and \(\mathrm { H } _ { 1 } : p < 0.5\) and rejects the null hypothesis if \(x < 3\)
  1. Find the size of the test.
  2. Show that the power function of this test is $$( 1 - p ) ^ { 8 } \left( 36 p ^ { 2 } + 8 p + 1 \right)$$ Table 1 gives values, to 2 decimal places, of the power function for the statistician's test. \begin{table}[h] \end{table} Table 1
  3. On the axes below draw the graph of the power function for the statistician's test.
  4. Find the range of values of \(p\) for which the probability of accepting the coin as unbiased, when in fact it is biased, is less than or equal to 0.4
    \includegraphics[max width=\textwidth, alt={}, center]{1d84c9fc-be67-45be-b439-3111c48ff1cb-09_1143_1209_945_402}
Edexcel S4 2014 June Q6
  1. (a) Explain what is meant by the sampling distribution of an estimator \(T\) of the population parameter \(\theta\).
    (b) Explain what you understand by the statement that \(T\) is a biased estimator of \(\theta\).
A population has mean \(\mu\) and variance \(\sigma ^ { 2 }\)
A random sample \(X _ { 1 } , X _ { 2 } , \ldots , X _ { 10 }\) is taken from this population.
(c) Calculate the bias of each of the following estimators of \(\mu\). $$\begin{aligned} & \hat { \mu } _ { 1 } = \frac { X _ { 3 } + X _ { 5 } + X _ { 7 } } { 3 }
& \hat { \mu } _ { 2 } = \frac { 5 X _ { 1 } + 2 X _ { 2 } + X _ { 9 } } { 6 }
& \hat { \mu } _ { 3 } = \frac { 3 X _ { 10 } - X _ { 1 } } { 3 } \end{aligned}$$ (d) Find the variance of each of these three estimators.
(e) State, giving a reason, which of these three estimators for \(\mu\) is
  1. the best estimator,
  2. the worst estimator.
Edexcel S4 2014 June Q7
7. Two groups of students take the same examination. A random sample of students is taken from each of the groups. The marks of the 9 students from Group 1 are as follows $$\begin{array} { l l l l l l l l l } 30 & 29 & 35 & 27 & 23 & 33 & 33 & 35 & 28 \end{array}$$ The marks, \(x\), of the 7 students from Group 2 gave the following statistics $$\bar { x } = 31.29 \quad s ^ { 2 } = 12.9$$ A test is to be carried out to see whether or not there is a difference between the mean marks of the two groups of students. You may assume that the samples are taken from normally distributed populations and that they are independent.
  1. State one other assumption that must be made in order to apply this test and show that this assumption is reasonable by testing it at a \(10 \%\) level of significance. State your hypotheses clearly.
  2. Stating your hypotheses clearly, test, using a significance level of \(5 \%\), whether or not there is a difference between the mean marks of the two groups of students.
Edexcel S4 2015 June Q1
  1. The Sales Manager of a large chain of convenience stores is studying the sale of lottery tickets in her stores. She randomly selects 8 of her stores. From these stores she collects data for the total sales of lottery tickets in the previous January and July. The data are shown below
StoreABCDEFGH
January ticket sales \(( \boldsymbol { \pounds } )\)10801639710110891510661322819
July ticket sales \(( \boldsymbol { \pounds } )\)11131702831104886110901303852
  1. Use a paired \(t\)-test to determine whether or not there is evidence, at the \(5 \%\) level of significance, that the mean sales of lottery tickets in this chain's stores are higher in July than in January. You should state your hypotheses and show your working clearly.
  2. State what assumption the Sales Manager needs to make about the sales of lottery tickets in her stores for the test in part (a) to be valid.
Edexcel S4 2015 June Q2
  1. Fred is a new employee in a delicatessen. He is asked to cut cheese into 100 g blocks. A random sample of 8 of these blocks of cheese is selected. The weight, in grams, of each block of cheese is given below
$$94 , \quad 106 , \quad 115 , \quad 98 , \quad 111 , \quad 104 , \quad 113 , \quad 102$$
  1. Calculate a \(90 \%\) confidence interval for the standard deviation of the weights of the blocks of cheese cut by Fred. Given that the weights of the blocks of cheese are independent,
  2. state what further assumption is necessary for this confidence interval to be valid. The delicatessen manager expects the standard deviation of the weights of the blocks of cheese cut by an employee to be less than 5 g. Any employee who does not achieve this target is given training.
  3. Use your answer from part (a) to comment on Fred's results. A second employee, Olga, has just been given training. Olga is asked to cut cheese into 100 g blocks. A random sample of 20 of these blocks of cheese is selected. The weight of each block of cheese, \(x\) grams, is recorded and the results are summarised below. $$\bar { x } = 102.6 \quad s ^ { 2 } = 19.4$$ Given that the assumption in part (b) is also valid in this case,
  4. test, at a \(10 \%\) level of significance, whether or not the mean weight of the blocks of cheese cut by Olga after her training is 100 g . State your hypotheses clearly.
    (6)