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Edexcel S3 Q6
14 marks Standard +0.3
6. A market researcher recorded the number of adverts for vehicles in each of three categories on ITV, Channel 4 and Channel 5 over a period of time. The results are shown in the table below.
ITVChannel 4Channel 5
Family Saloon693528
Sports Car202818
Off-road Vehicle12228
  1. Stating your hypotheses clearly, test at the \(5 \%\) level of significance whether or not there is evidence of the proportion of adverts for each type of vehicle being dependent on the channel.
  2. Suggest a reason for your result in part (a).
Edexcel S3 Q7
14 marks Standard +0.3
7. (a) Briefly state the central limit theorem. A student throws ten dice and records the number of sixes showing. The dice are fair, numbered 1 to 6 on the faces.
(b) Write down the distribution of the number of sixes obtained when the ten dice are thrown.
(c) Find the mean and variance of this distribution. The student throws the ten dice 100 times, recording the number of sixes showing each time.
(d) Find the probability that the mean number of sixes obtained is more than 1.8
Edexcel S3 Q1
5 marks Easy -1.8
  1. A personnel manager has details on all company employees and wishes to consult a sample of them on a possible change to the company's hours of business. She decides to take a stratified sample based on different age groups.
    1. Give one advantage of using stratified sampling in this situation.
    The manager needs to select a sample of size 10 , without replacement, from a list of 65 employees aged 16 to 25 . She numbers these employees from 01 to 65 in alphabetical order and uses the table of random numbers given in the formula book. She starts with the top of the sixth two-digit column and works down. The first two numbers she writes down are 30 and 47.
  2. Find the other eight numbers in the sample.
  3. Suggest another factor that might be useful to consider in deciding on the strata.
    (1 mark)
Edexcel S3 Q2
6 marks Standard +0.3
2. A Geography teacher is interested in the link between mathematical ability and the ability to visualise three-dimensional situations. He gives a group of 15 students a test and records each student's score, \(m\), on the mathematics questions and each student's score, \(v\), on the visiospatial questions. He calculates the following summary statistics: $$S _ { m m } = 3747.73 , \quad S _ { v v } = 2791.33 , \quad S _ { m v } = 2564.33$$
  1. Calculate the product moment correlation coefficient for these data.
  2. Stating your hypotheses clearly and using a \(5 \%\) level of significance test the theory that students who are good at Mathematics tend to have better visio-spatial awareness.
    (4 marks)
Edexcel S3 Q3
9 marks Standard +0.3
3. A random variable \(X\) is distributed normally with a standard deviation of 6.8 Sixty observations of \(X\) are made and found to have a mean of 31.4
  1. Find a 90\% confidence interval for the mean of \(X\).
  2. How many observations of \(X\) would be needed in order to obtain a \(90 \%\) confidence interval for the mean of \(X\) with a width of less than 1.5
    (5 marks)
Edexcel S3 Q4
12 marks Standard +0.3
4. A paranormal investigator invites couples who believe they have a telepathic connection to participate in a trial. With each couple one person looks at a card with one of five shapes on it and the other person says which of the shapes they think it is. This is repeated six times and the number of correct answers recorded. The results from 120 couples are given below.
Number Correct0123456
Number of Couples2656288200
The investigator wishes to see if this data fits a binomial distribution with parameters \(n = 6\) and \(p = \frac { 1 } { 5 }\) and calculates to 2 decimal places the expected frequencies given below.
Number Correct0123456
Expected Frequency9.831.840.180.01
  1. Find the other expected frequencies.
  2. Stating your hypotheses clearly, test at the \(5 \%\) level of significance whether or not the distribution is an appropriate model.
  3. Comment on your findings.
Edexcel S3 Q5
13 marks Standard +0.3
5. A Policy Unit wished to find out whether attitudes to the European Union varied with age. It conducted a survey asking 200 individuals to which of three age groups they belonged and whether they regarded themselves as generally pro-Europe or Eurosceptic. The results are shown in the table below.
\cline { 2 - 3 } \multicolumn{1}{c|}{}Pro-EuropeEurosceptic
\(18 - 34\) years4321
\(35 - 54\) years3036
55 years or over2743
  1. Stating your hypotheses clearly, test at the \(5 \%\) level of significance whether attitudes to Europe are associated with age.
    (11 marks)
    The survey also asked people if they voted at the last election. When the above test was repeated using only the results from those who had voted a value of 4.872 was calculated for \(\sum \frac { ( O - E ) ^ { 2 } } { E }\). No classes were combined.
  2. Find if this value leads to a different result.
Edexcel S3 Q6
14 marks Challenging +1.2
6. Four swimmers, \(A , B , C\) and \(D\), are to be used in a \(4 \times 100\) metres freestyle relay. The time for each swimmer to complete a leg follows a normal distribution. The mean and standard deviation, in seconds, of the time for each swimmer to complete a leg and the order in which they are to swim are shown in the table below.
meanstandard deviation
\(1 ^ { \text {st } }\) leg \(- A\)63.11.2
\(2 ^ { \text {nd } }\) leg \(- B\)65.71.5
\(3 ^ { \text {rd } } \operatorname { leg } - C\)65.41.8
\(4 ^ { \text {th } }\) leg - \(D\)62.50.9
  1. Find the probability that the total time for first two legs is less than the total time for the last two.
    (6 marks)
    The total time for another team to complete this relay is normally distributed with a mean of 259.0 seconds and a standard deviation of 3.4 seconds. The two teams are to compete over four races.
  2. Find the probability that the first team wins all four races, assuming that the team's performances are not affected by previous results.
    (8 marks)
Edexcel S3 Q7
16 marks Standard +0.3
7. A telephone company believes that, for young people, the average length of a telephone call on a land line is longer than on a mobile, due to the difference in price. The company collected data on the time, \(t\) minutes, of 500 calls made by young people on mobiles and the data is summarised by $$\Sigma t = 7335 , \quad \Sigma t ^ { 2 } = 172040 .$$
  1. Calculate unbiased estimates of the mean and variance of \(t\). For 200 calls made on land lines by the same young people, unbiased estimates of the mean and variance of the call length were 15.9 minutes and 108.5 minutes \({ } ^ { 2 }\) respectively.
  2. Stating your hypotheses clearly, test at the \(5 \%\) level whether or not there is evidence that longer calls are made on land lines than on mobiles.
    (9 marks)
  3. Explain the importance of the central limit theorem in carrying out the test in part (b).
Edexcel S4 2006 January Q1
8 marks Standard +0.3
  1. A diabetic patient records her blood glucose readings in \(\mathrm { mmol } / \mathrm { l }\) at random times of day over several days. Her readings are given below.
$$\begin{array} { l l l l l l l } 5.3 & 5.7 & 8.4 & 8.7 & 6.3 & 8.0 & 7.2 \end{array}$$ Assuming that the blood glucose readings are normally distributed calculate
  1. an unbiased estimate for the variance \(\sigma ^ { 2 }\) of the blood glucose readings,
  2. a \(90 \%\) confidence interval for the variance \(\sigma ^ { 2 }\) of blood glucose readings.
  3. State whether or not the confidence interval supports the assertion that \(\sigma = 0.9\). Give a reason for your answer.
Edexcel S4 2006 January Q2
13 marks Standard +0.3
2. (a) Define
  1. a Type I error,
  2. a Type II error. A manufacturer sells socks in boxes of 50 .
    The mean number of faulty socks per box is 7.5 . In order to reduce the number of faulty socks a new machine is tried. A box of socks made on the new machine was tested and the number of faulty socks was 2.
    (b) (i) Assuming that the number of faulty socks per box follows a binomial distribution derive a critical region needed to test whether or not there is evidence that the new machine has reduced the mean number of faulty socks per box. Use a \(5 \%\) significance level.
  3. Stating your hypotheses clearly, carry out the test in part (i).
    (c) Find the probability of the Type I error for this test.
    (d) Given that the true mean number of faulty socks per box on the new machine is 5 , calculate the probability of a Type II error for this test.
    (e) Explain what would have been the effect of changing the significance level for the test in part (b) to \(2 \frac { 1 } { 2 } \%\).
Edexcel S4 2006 January Q3
7 marks Standard +0.3
3. A population has mean \(\mu\) and variance \(\sigma ^ { 2 }\). A random sample of size 3 is to be taken from this population and \(\bar { X }\) denotes its sample mean. A second random sample of size 4 is to be taken from this population and \(\bar { Y }\) denotes its sample mean.
  1. Show that unbiased estimators for \(\mu\) are given by
    1. \(\hat { \mu } _ { 1 } = \frac { 1 } { 3 } \bar { X } + \frac { 2 } { 3 } \bar { Y }\),
    2. \(\hat { \mu } _ { 2 } = \frac { 5 \bar { X } + 4 \bar { Y } } { 9 }\).
  2. Calculate Var \(\left( \hat { \mu } _ { 1 } \right)\)
  3. Given that \(\operatorname { Var } \left( \hat { \mu } _ { 2 } \right) = \frac { 37 } { 243 } \sigma ^ { 2 }\), state, giving a reason, which of these two estimators should be
    used. used.
Edexcel S4 2006 January Q4
6 marks Standard +0.3
4. The number of accidents that occur at a crossroads has a mean of 3 per month. In order to improve the flow of traffic the priority given to traffic is changed. Colin believes that since the change in priority the number of accidents has increased. He tests his belief by recording the number of accidents \(x\) in the month following the change. Colin sets up the hypotheses \(\mathrm { H } _ { 0 } : \lambda = 3\) and \(\mathrm { H } _ { 1 } : \lambda > 3\), where \(\lambda\) is the mean number of accidents per month, and rejects the null hypothesis if \(x > 4\).
  1. Find the size of the test. The table gives the values of the power function of the test to two decimal places.
    \(\lambda\)4567
    Power\(r\)0.56\(s\)0.83
  2. Calculate the value of \(r\) and the value of \(s\).
  3. Comment on the suitability of the test when \(\lambda = 4\).
Edexcel S4 2006 January Q5
13 marks Standard +0.3
5. Seven pipes of equal length are selected at random. Each pipe is cut in half. One piece of each pipe is coated with protective paint and the other is left uncoated. All of the pieces of pipe are buried to the same depth in various soils for 6 months. The table gives the percentage area of the pieces of pipe in the various soils that are subject to corrosion.
SoilABCDEFG
\% Corrosion
coated pipe
39404332423336
\% Corrosion
uncoated pipe
41366148424845
  1. Stating your hypotheses clearly and using a \(5 \%\) significance level, carry out a paired \(t\)-test to assess whether or not there is a difference between the mean percentage of corrosion on the coated pipes and the mean percentage of corrosion on the uncoated pipes.
    1. State an assumption that has been made in order to carry out this test.
    2. Comment on the validity of this assumption.
  3. State what difference would be made to the conclusion in part (a) if the test had been to determine whether or not the percentage of corrosion on the uncoated pipes was higher than the mean percentage of corrosion on the coated pipes. Justify your answer.
Edexcel S4 2006 January Q6
12 marks Standard +0.3
6. A tree is cut down and sawn into pieces. Half of the pieces are stored outside and half of the pieces are stored inside. After a year, a random sample of pieces is taken from each location and the hardness is measured. The hardness \(x\) units are summarised in the following table.
Number of
pieces sampled
\(\Sigma x\)\(\Sigma x ^ { 2 }\)
Stored outside202340274050
Stored inside374884645282
  1. Show that unbiased estimates for the variance of the values of hardness for wood stored outside and for the wood stored inside are 14.2 and 16.5 , to 1 decimal place, respectively.
    (2) The hardness of wood stored outside and the hardness of wood stored inside can be assumed to be normally distributed with equal variances.
  2. Calculate \(95 \%\) confidence limits for the difference in mean hardness between the wood that was stored outside and the wood that was stored inside.
    (8)
  3. Using your answer to part (b), comment on the means of the hardness of wood stored outside and inside. Give a reason for your answer.
    (2)
    (Total 12 marks)
Edexcel S4 2006 January Q7
16 marks Standard +0.3
7. A psychologist gives a test to students from two different schools, \(A\) and \(B\). A group of 9 students is randomly selected from school \(A\) and given instructions on how to do the test.
A group of 7 students is randomly selected from school \(B\) and given the test without the instructions. The table shows the time taken, to the nearest second, to complete the test by the two groups.
\(A\)111212131415161717
\(B\)8101113131414
Stating your hypotheses clearly,
  1. test at the \(10 \%\) significance level, whether or not the variance of the times taken to complete the test by students from school \(A\) is the same as the variance of the times taken to complete the test by students from school \(B\). (You may assume that times taken for each school are normally distributed.)
  2. test at the \(5 \%\) significance level, whether or not the mean time taken to complete the test by students from school \(A\) is greater than the mean time taken to complete the test by students from school \(B\).
  3. Why does the result to part (a) enable you to carry out the test in part (b)?
  4. Give one factor that has not been taken into account in your analysis.
Edexcel S4 2003 June Q1
6 marks Standard +0.3
  1. A beach is divided into two areas \(A\) and \(B\). A random sample of pebbles is taken from each of the two areas and the length of each pebble is measured. A sample of size 26 is taken from area \(A\) and the unbiased estimate for the population variance is \(s _ { A } ^ { 2 } = 0.495 \mathrm {~mm} ^ { 2 }\). A sample of size 25 is taken from area \(B\) and the unbiased estimate for the population variance is \(s _ { B } ^ { 2 } = 1.04 \mathrm {~mm} ^ { 2 }\).
    1. Stating your hypotheses clearly test, at the \(10 \%\) significance level, whether or not there is a difference in variability of pebble length between area \(A\) and area \(B\).
    2. State the assumption you have made about the populations of pebble lengths in order to carry out the test.
    3. A random sample of 10 mustard plants had the following heights, in mm , after 4 days growth.
    $$5.0,4.5,4.8,5.2,4.3,5.1,5.2,4.9,5.1,5.0$$ Those grown previously had a mean height of 5.1 mm after 4 days. Using a \(2.5 \%\) significance level, test whether or not the mean height of these plants is less than that of those grown previously.
    (You may assume that the height of mustard plants after 4 days follows a normal distribution.)
Edexcel S4 2003 June Q3
9 marks Standard +0.8
3. A train company claims that the probability \(p\) of one of its trains arriving late is \(10 \%\). A regular traveller on the company's trains believes that the probability is greater than \(10 \%\) and decides to test this by randomly selecting 12 trains and recording the number \(X\) of trains that were late. The traveller sets up the hypotheses \(\mathrm { H } _ { 0 } : p = 0.1\) and \(\mathrm { H } _ { 1 } : p > 0.1\) and accepts the null hypothesis if \(x \leq 2\).
  1. Find the size of the test.
  2. Show that the power function of the test is $$1 - ( 1 - p ) ^ { 10 } \left( 1 + 10 p + 55 p ^ { 2 } \right) .$$
  3. Calculate the power of the test when
    1. \(p = 0.2\),
    2. \(p = 0.6\).
  4. Comment on your results from part (c).
Edexcel S4 2003 June Q4
9 marks Standard +0.8
4. A random sample of 15 tomatoes is taken and the weight \(x\) grams of each tomato is found. The results are summarised by \(\sum x = 208\) and \(\sum x ^ { 2 } = 2962\).
  1. Assuming that the weights of the tomatoes are normally distributed, calculate the \(90 \%\) confidence interval for the variance \(\sigma ^ { 2 }\) of the weights of the tomatoes.
  2. State with a reason whether or not the confidence interval supports the assertion \(\sigma ^ { 2 } = 3\).
Edexcel S4 2003 June Q5
11 marks Standard +0.3
5. (a) Define
  1. a Type I error,
  2. a Type II error. A small aviary, that leaves the eggs with the parent birds, rears chicks at an average rate of 5 per year. In order to increase the number of chicks reared per year it is decided to remove the eggs from the aviary as soon as they are laid and put them in an incubator. At the end of the first year of using an incubator 7 chicks had been successfully reared.
    (b) Assuming that the number of chicks reared per year follows a Poisson distribution test, at the \(5 \%\) significance level, whether or not there is evidence of an increase in the number of chicks reared per year. State your hypotheses clearly.
    (c) Calculate the probability of the Type I error for this test.
    (d) Given that the true average number of chicks reared per year when the eggs are hatched in an incubator is 8 , calculate the probability of a Type II error.
Edexcel S4 2003 June Q6
14 marks Standard +0.3
6. A random sample of three independent variables \(X _ { 1 } , X _ { 2 }\) and \(X _ { 3 }\) is taken from a distribution with mean \(\mu\) and variance \(\sigma ^ { 2 }\).
  1. Show that \(\frac { 2 } { 3 } X _ { 1 } - \frac { 1 } { 2 } X _ { 2 } + \frac { 5 } { 6 } X _ { 3 }\) is an unbiased estimator for \(\mu\). An unbiased estimator for \(\mu\) is given by \(\hat { \mu } = a X _ { 1 } + b X _ { 2 }\) where \(a\) and \(b\) are constants.
  2. Show that \(\operatorname { Var } ( \hat { \mu } ) = \left( 2 a ^ { 2 } - 2 a + 1 \right) \sigma ^ { 2 }\).
  3. Hence determine the value of \(a\) and the value of \(b\) for which \(\hat { \mu }\) has minimum variance.
Edexcel S4 2003 June Q7
17 marks Standard +0.3
7. Two methods of extracting juice from an orange are to be compared. Eight oranges are halved. One half of each orange is chosen at random and allocated to Method \(A\) and the other half is allocated to Method \(B\). The amounts of juice extracted, in ml , are given in the table.
\cline { 2 - 9 } \multicolumn{1}{c|}{}Orange
\cline { 2 - 9 } \multicolumn{1}{c|}{}12345678
Method \(A\)2930262526222328
Method \(B\)2725282423262225
One statistician suggests performing a two-sample \(t\)-test to investigate whether or not there is a difference between the mean amounts of juice extracted by the two methods.
  1. Stating your hypotheses clearly and using a \(5 \%\) significance level, carry out this test.
    (You may assume \(\bar { x } _ { A } = 26.125 , s _ { A } ^ { 2 } = 7.84 , \bar { x } _ { B } = 25 , s _ { B } ^ { 2 } = 4\) and \(\sigma _ { A } ^ { 2 } = \sigma _ { B } ^ { 2 }\) ) Another statistician suggests analysing these data using a paired \(t\)-test.
  2. Using a \(5 \%\) significance level, carry out this test.
  3. State which of these two tests you consider to be more appropriate. Give a reason for your choice.
Edexcel S4 2004 June Q1
4 marks Standard +0.8
  1. The random variable \(X\) has an \(F\)-distribution with 8 and 12 degrees of freedom.
Find \(\mathrm { P } \left( \frac { 1 } { 5.67 } < X < 2.85 \right)\).
(4)
Edexcel S4 2004 June Q2
6 marks Standard +0.3
2. A mechanic is required to change car tyres. An inspector timed a random sample of 20 tyre changes and calculated the unbiased estimate of the population variance to be 6.25 minutes \({ } ^ { 2 }\). Test, at the \(5 \%\) significance level, whether or not the standard deviation of the population of times taken by the mechanic is greater than 2 minutes. State your hypotheses clearly.
(6)
Edexcel S4 2004 June Q3
9 marks Challenging +1.2
3. It is suggested that a Poisson distribution with parameter \(\lambda\) can model the number of currants in a currant bun. A random bun is selected in order to test the hypotheses \(\mathrm { H } _ { 0 } : \lambda = 8\) against \(\mathrm { H } _ { 1 } : \lambda \neq 8\), using a \(10 \%\) level of significance.
  1. Find the critical region for this test, such that the probability in each tail is as close as possible to \(5 \%\).
  2. Given that \(\lambda = 10\), find
    1. the probability of a type II error,
    2. the power of the test.
      (4)