Questions S4 (317 questions)

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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 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)
Edexcel S4 2004 June Q4
9 marks Standard +0.3
4. A doctor believes that the span of a person's dominant hand is greater than that of the weaker hand. To test this theory, the doctor measures the spans of the dominant and weaker hands of a random sample of 8 people. He subtracts the span of the weaker hand from that of the dominant hand. The spans, in mm , are summarised in the table below.
\(A\)\(B\)\(C\)\(D\)\(E\)\(F\)\(G\)\(H\)
Dominant hand202251215235210195191230
Weaker hand195249218234211197181225
Test, at the 5\% significance level, the doctor's belief.
(9)
Edexcel S4 2004 June Q5
15 marks Standard +0.3
5. (a) Explain briefly what you understand by
  1. an unbiased estimator,
  2. a consistent estimator.
    of an unknown population parameter \(\theta\). From a binomial population, in which the proportion of successes is \(p , 3\) samples of size \(n\) are taken. The number of successes \(X _ { 1 } , X _ { 2 }\), and \(X _ { 3 }\) are recorded and used to estimate \(p\).
    (b) Determine the bias, if any, of each of the following estimators of \(p\). $$\begin{aligned} & \hat { p } _ { 1 } = \frac { X _ { 1 } + X _ { 2 } + X _ { 3 } } { 3 n } \\ & \hat { p } _ { 2 } = \frac { X _ { 1 } + 3 X _ { 2 } + X _ { 3 } } { 6 n } \\ & \hat { p } _ { 3 } = \frac { 2 X _ { 1 } + 3 X _ { 2 } + X _ { 3 } } { 6 n } \end{aligned}$$ (c) Find the variance of each of these estimators.
    (d) State, giving a reason, which of the three estimators for \(p\) is
  3. the best estimator,
  4. the worst estimator.
Edexcel S4 2004 June Q6
16 marks Standard +0.3
6. A supervisor wishes to cheek the typing speed of a new typist. On 10 randomly selected occasions, the supervisor records the time taken for the new typist to type 100 words. The results, in seconds, are given below. $$110,125,130,126,128,127,118,120,122,125$$ The supervisor assumes that the time taken to type 100 words is normally distributed.
  1. Calculate a 95\% confidence interval for
    1. the mean,
    2. the variance
      of the population of times taken by this typist to type 100 words. The supervisor requires the average time needed to type 100 words to be no more than 130 seconds and the standard deviation to be no more than 4 seconds.
  2. Comment on whether or not the supervisor should be concerned about the speed of the new typist.
Edexcel S4 2004 June Q7
16 marks Standard +0.8
7. A grocer receives deliveries of cauliflowers from two different growers, \(A\) and \(B\). The grocer takes random samples of cauliflowers from those supplied by each grower. He measures the weight \(x\), in grams, of each cauliflower. The results are summarised in the table below.
Sample size\(\Sigma x\)\(\Sigma x ^ { 2 }\)
\(A\)1166003960540
\(B\)1398157410579
  1. Show, at the \(10 \%\) significance level, that the variances of the populations from which the samples are drawn can be assumed to be equal by testing the hypothesis \(\mathrm { H } _ { 0 } : \sigma _ { A } ^ { 2 } = \sigma _ { B } ^ { 2 }\) against hypothesis \(\mathrm { H } _ { 1 } : \sigma _ { A } ^ { 2 } \neq \sigma _ { B } ^ { 2 }\).
    (You may assume that the two samples come from normal populations.)
    (6) The grocer believes that the mean weight of cauliflowers provided by \(B\) is at least 150 g more than the mean weight of cauliflowers provided by \(A\).
  2. Use a \(5 \%\) significance level to test the grocer's belief.
  3. Justify your choice of test.
Edexcel S4 2005 June Q1
6 marks Moderate -0.5
  1. The random variable \(X\) has a \(\chi ^ { 2 }\)-distribution with 9 degrees of freedom.
    1. Find \(\mathrm { P } ( 2.088 < X < 19.023 )\).
    The random variable \(Y\) follows an \(F\)-distribution with 12 and 5 degrees of freedom.
  2. Find the upper and lower \(5 \%\) critical values for \(Y\).
    (3)
    (Total 6 marks)
Edexcel S4 2005 June Q2
6 marks Standard +0.3
2. The standard deviation of the length of a random sample of 8 fence posts produced by a timber yard was 8 mm . A second timber yard produced a random sample of 13 fence posts with a standard deviation of 14 mm .
  1. Test, at the \(10 \%\) significance level, whether or not there is evidence that the lengths of fence posts produced by these timber yards differ in variability. State your hypotheses clearly.
  2. State an assumption you have made in order to carry out the test in part (a).
Edexcel S4 2005 June Q3
8 marks Standard +0.3
3. A machine is set to fill bags with flour such that the mean weight is 1010 grams. To check that the machine is working properly, a random sample of 8 bags is selected. The weight of flour, in grams, in each bag is as follows. $$\begin{array} { l l l l l l l l } 1010 & 1015 & 1005 & 1000 & 998 & 1008 & 1012 & 1007 \end{array}$$ Carry out a suitable test, at the \(5 \%\) significance level, to test whether or not the mean weight of flour in the bags is less than 1010 grams. (You may assume that the weight of flour delivered by the machine is normally distributed.)
(Total 8 marks)
Edexcel S4 2005 June Q4
13 marks Standard +0.3
4. A farmer set up a trial to assess the effect of two different diets on the increase in the weight of his lambs. He randomly selected 20 lambs. Ten of the lambs were given \(\operatorname { diet } A\) and the other 10 lambs were given diet \(B\). The gain in weight, in kg , of each lamb over the period of the trial was recorded.
  1. State why a paired \(t\)-test is not suitable for use with these data.
  2. Suggest an alternative method for selecting the sample which would make the use of a paired \(t\)-test valid.
  3. Suggest two other factors that the farmer might consider when selecting the sample. The following paired data were collected.
    Diet \(A\)5674.66.15.76.27.453
    Diet \(B\)77.286.45.17.98.26.26.15.8
  4. Using a paired \(t\)-test, at the \(5 \%\) significance level, test whether or not there is evidence of a difference in the weight gained by the lambs using \(\operatorname { diet } A\) compared with those using \(\operatorname { diet } B\).
  5. State, giving a reason, which diet you would recommend the farmer to use for his lambs.
    (Total 13 marks)
Edexcel S4 2005 June Q5
12 marks Standard +0.3
5. Define
  1. a Type I error,
  2. the size of a test. Jane claims that she can read Alan's mind. To test this claim Alan randomly chooses a card with one of 4 symbols on it. He then concentrates on the symbol. Jane then attempts to read Alan's mind by stating what symbol she thinks is on the card. The experiment is carried out 8 times and the number of times \(X\) that Jane is correct is recorded. The probability of Jane stating the correct symbol is denoted by \(p\).
    To test the hypothesis \(\mathrm { H } _ { 0 } : p = 0.25\) against \(\mathrm { H } _ { 1 } : p > 0.25\), a critical region of \(X > 6\) is used.
  3. Find the size of this test.
  4. Show that the power function of this test is \(8 p ^ { 7 } - 7 p ^ { 8 }\). Given that \(p = 0.3\), calculate
  5. the power of this test,
  6. the probability of a Type II error.
  7. Suggest two ways in which you might reduce the probability of a Type II error.
Edexcel S4 2005 June Q6
12 marks Standard +0.3
6. Brickland and Goodbrick are two manufacturers of bricks. The lengths of the bricks produced by each manufacturer can be assumed to be normally distributed. A random sample of 20 bricks is taken from Brickland and the length, \(x \mathrm {~mm}\), of each brick is recorded. The mean of this sample is 207.1 mm and the variance is \(3.2 \mathrm {~mm} ^ { 2 }\).
  1. Calculate the \(98 \%\) confidence interval for the mean length of brick from Brickland. A random sample of 10 bricks is selected from those manufactured by Goodbrick. The length of each brick, \(y \mathrm {~mm}\), is recorded. The results are summarised as follows. $$\sum y = 2046.2 \quad \sum y ^ { 2 } = 418785.4$$ The variances of the length of brick for each manufacturer are assumed to be the same.
  2. Find a \(90 \%\) confidence interval for the value by which the mean length of brick made by Brickland exceeds the mean length of brick made by Goodbrick.
Edexcel S4 2005 June Q7
17 marks Standard +0.3
7. A bag contains marbles of which an unknown proportion \(p\) is red. A random sample of \(n\) marbles is drawn, with replacement, from the bag. The number \(X\) of red marbles drawn is noted. A second random sample of \(m\) marbles is drawn, with replacement. The number \(Y\) of red marbles drawn is noted. Given that \(p _ { 1 } = \frac { a X } { n } + \frac { b Y } { m }\) is an unbiased estimator of \(p\),
  1. show that \(a + b = 1\). Given that \(p _ { 2 } = \frac { ( X + Y ) } { n + m }\),
  2. show that \(p _ { 2 }\) is an unbiased estimator for \(p\).
  3. Show that the variance of \(p _ { 1 }\) is \(p ( 1 - p ) \left( \frac { a ^ { 2 } } { n } + \frac { b ^ { 2 } } { m } \right)\).
  4. Find the variance of \(p _ { 2 }\).
  5. Given that \(a = 0.4 , m = 10\) and \(n = 20\), explain which estimator \(p _ { 1 }\) or \(p _ { 2 }\) you should use.
Edexcel S4 2006 June Q1
7 marks Standard +0.3
  1. Historical records from a large colony of squirrels show that the weight of squirrels is normally distributed with a mean of 1012 g . Following a change in the diet of squirrels, a biologist is interested in whether or not the mean weight has changed.
A random sample of 14 squirrels is weighed and their weights \(x\), in grams, recorded. The results are summarised as follows: $$\Sigma x = 13700 , \quad \Sigma x ^ { 2 } = 13448750 .$$ Stating your hypotheses clearly test, at the \(5 \%\) level of significance, whether or not there has been a change in the mean weight of the squirrels.
Edexcel S4 2006 June Q2
12 marks Challenging +1.2
2. The weights, in grams, of apples are assumed to follow a normal distribution. The weights of apples sold by a supermarket have variance \(\sigma _ { s } { } ^ { 2 }\). A random sample of 4 apples from the supermarket had weights $$\text { 114, 100, 119, } 123 .$$
  1. Find a 95\% confidence interval for \(\sigma _ { s } ^ { 2 }\). The weights of apples sold on a market stall have variance \(\sigma _ { M } ^ { 2 }\). A second random sample of 7 apples was taken from the market stall. The sample variance \(s _ { M } ^ { 2 }\) of the apples was 318.8.
  2. Stating your hypotheses clearly test, at the \(1 \%\) levcel of significnace, whether or not there is evidence that \(\sigma _ { M } ^ { 2 } > \sigma _ { s } ^ { 2 }\).
Edexcel S4 2006 June Q3
9 marks Standard +0.3
3. As part of an investigation into the effectiveness of solar heating, a pair of houses was identified where the mean weekly fuel consumption was the same. One of the houses was then fitted with solar heating and the other was not. Following the fitting of the solar heating, a random sample of 9 weeks was taken and the table below shows the weekly fuel consumption for each house.
Week123456789
Without solar heating191918146753143
With solar heating1322111614102038
Units of fuel used per week
  1. Stating your hypotheses clearly, test, at the \(5 \%\) level of significance, whether or not there is evidence that the solar heating reduces the mean weekly fuel consumption.
    (8)
  2. State an assumption about weekly fuel consumption that is required to carry out this test.
Edexcel S4 2006 June Q4
13 marks Standard +0.3
4. Two machines \(A\) and \(B\) produce the same type of component in a factory. The factory manager wishes to know whether the lengths, \(x \mathrm {~cm}\), of the components produced by the two machines have the same mean. The manager took a random sample of components from each machine and the results are summarised in the table below.
Sample sizeMean \(\bar { x }\)
Standard
deviation \(s\)
Machine \(A\)94.830.721
Machine \(B\)104.850.572
The lengths of components produced by the machines can be assumed to follow normal distributions.
  1. Use a two tail test to show, at the \(10 \%\) significance level, that the variances of the lengths of components produced by each machine can be assumed to be equal.
    (4)
  2. Showing your working clearly, find a \(95 \%\) confidence interval for \(\mu _ { B } - \mu _ { A }\), where \(\mu _ { A }\) and \(\mu _ { B }\) are the mean lengths of the populations of components produced by machine \(A\) and machine \(B\) respectively. There are serious consequences for the production at the factory if the difference in mean lengths of the components produced by the two machines is more than 0.7 cm .
  3. State, giving your reason, whether or not the factory manager should be concerned.