Questions S4 (317 questions)

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Edexcel S4 2017 June Q1
14 marks Challenging +1.2
  1. The times taken by children to run 150 m are normally distributed. The times taken, \(x\) seconds, by a random sample of 9 boys and an independent random sample of 6 girls are recorded. The following statistics are obtained.
Number of childrenSample mean \(\bar { x }\)\(\sum x ^ { 2 }\)
Boys922.84693.60
Girls629.55236.12
  1. Test, at the \(10 \%\) level of significance, whether or not the variances of the two distributions are equal. State your hypotheses clearly. The Headteacher claims that the mean time taken for the girls is more than 5 seconds greater than the mean time taken for the boys.
  2. Stating your hypotheses clearly, test the Headteacher's claim. Use a \(1 \%\) level of significance and show your working clearly.
Edexcel S4 2017 June Q2
8 marks Standard +0.8
  1. The number of accidents per year in Daftstown follows a Poisson distribution with mean \(\lambda\). The value of \(\lambda\) has previously been 6 but Jonty claims that since the Council increased the speed limit, the value of \(\lambda\) has increased.
Jonty records the number of accidents in Daftstown in the first year after the speed limit was increased. He plans to test, at the \(5 \%\) significance level, whether or not there is evidence of an increase in the mean number of accidents in Daftstown per year.
  1. Stating your hypotheses clearly, calculate the probability of a Type I error for this test. Given that there were 9 accidents in the first year after the speed limit was increased,
  2. state, giving a reason, whether or not there is evidence to support Jonty's claim.
  3. Given that the value of \(\lambda\) has actually increased to 8, calculate the probability of drawing the conclusion, using this test, that the number of accidents per year in Daftstown has not increased.
Edexcel S4 2017 June Q3
11 marks Standard +0.3
3. The lengths, \(X \mathrm {~mm}\), of the wings of adult blackbirds follow a normal distribution. A random sample of 5 adult blackbirds is taken and the lengths of the wings are measured. The results are summarised below $$\sum x = 655 \text { and } \sum x ^ { 2 } = 85845$$
  1. Test, at the \(10 \%\) level of significance, whether or not the mean length of an adult blackbird's wing is less than 135 mm . State your hypotheses clearly.
  2. Find the \(90 \%\) confidence interval for the variance of the lengths of adult blackbirds' wings. Show your working clearly.
Edexcel S4 2017 June Q4
12 marks Standard +0.3
4. A coach believes that the average score in the final round of a golf tournament is more than one point below the average score in the first round. To test this belief, the scores of 8 randomly selected players are recorded. The results are given in the table below.
Player\(A\)\(B\)\(C\)\(D\)\(E\)\(F\)\(G\)\(H\)
First round7680727883888172
Final round7078757579848369
    1. State why a paired \(t\)-test is suitable for use with these data.
    2. State an assumption that needs to be made in order to carry out a paired \(t\)-test in this case.
  2. Test, at the \(5 \%\) level of significance, whether or not there is evidence to support the coach's belief. Show your working clearly.
  3. Explain, in the context of the coach's belief, what a Type II error would be in this case.
Edexcel S4 2017 June Q5
11 marks Challenging +1.2
  1. Jamland and Goodjam are two suppliers of jars of jam. The weights of the jars of jam produced by each supplier can be assumed to be normally distributed with unknown, but equal, variances. A random sample of 20 jars of jam is taken from those supplied by Jamland.
Based on this sample, the 95\% confidence interval for the mean weight of a jar of Jamland jam, in grams, is
[0pt] [ 492, 507 ] A random sample of 10 jars of jam is selected from those supplied by Goodjam. The weight of each jar of Goodjam jam, \(y\) grams, is recorded. The results are summarised as follows $$\bar { y } = 480 \quad s _ { y } ^ { 2 } = 280$$ Find a 90\% confidence interval for the value by which the mean weight of a jar of jam supplied by Jamland exceeds the mean weight of a jar of jam supplied by Goodjam.
Edexcel S4 2017 June Q6
19 marks Challenging +1.2
6. The independent random variables \(X _ { 1 }\) and \(X _ { 2 }\) are each distributed \(\mathrm { B } ( n , p )\), where \(n > 1\) An unbiased estimator for \(p\) is given by $$\hat { p } = \frac { a X _ { 1 } + b X _ { 2 } } { n }$$ where \(a\) and \(b\) are constants.
[0pt] [You may assume that if \(X _ { 1 }\) and \(X _ { 2 }\) are independent then \(\mathrm { E } \left( X _ { 1 } X _ { 2 } \right) = \mathrm { E } \left( X _ { 1 } \right) \mathrm { E } \left( X _ { 2 } \right)\) ]
  1. Show that \(a + b = 1\)
  2. Show that \(\operatorname { Var } ( \hat { p } ) = \frac { \left( 2 a ^ { 2 } - 2 a + 1 \right) p ( 1 - p ) } { n }\)
  3. Hence, justifying your answer, determine the value of \(a\) and the value of \(b\) for which \(\hat { p }\) has minimum variance.
    1. Show that \(\hat { p } ^ { 2 }\) is a biased estimator for \(p ^ { 2 }\)
    2. Show that the bias \(\rightarrow 0\) as \(n \rightarrow \infty\)
  5. By considering \(\mathrm { E } \left[ X _ { 1 } \left( X _ { 1 } - 1 \right) \right]\) find an unbiased estimator for \(p ^ { 2 }\)
Edexcel S4 2018 June Q1
5 marks Moderate -0.3
  1. A machine fills packets with almonds. The weight, in grams, of almonds in a packet is modelled by \(\mathrm { N } \left( \mu , \sigma ^ { 2 } \right)\). To check that the machine is working properly, a random sample of 10 packets is selected and unbiased estimates for \(\mu\) and \(\sigma ^ { 2 }\) are
$$\bar { x } = 202 \quad \text { and } \quad s ^ { 2 } = 3.6$$ Stating your hypotheses clearly, test, at the \(1 \%\) level of significance, whether or not the mean weight of almonds in a packet is more than 200 g .
Edexcel S4 2018 June Q2
13 marks Standard +0.3
  1. Jeremiah currently uses a Fruity model of juicer. He agrees to trial a new model of juicer, Zesty. The amounts of juice extracted, \(x \mathrm { ml }\), from each of 9 randomly selected oranges, using the Zesty are summarised as
$$\sum x = 468 \quad \sum x ^ { 2 } = 24560$$ Given that the amounts of juice extracted follow a normal distribution,
  1. calculate a 95\% confidence interval for
    1. the mean amount of juice extracted from an orange using the Zesty,
    2. the standard deviation of the amount of juice extracted from an orange using the Zesty. Jeremiah knows that, for his Fruity, the mean amount of juice extracted from an orange is 38 ml and the standard deviation of juice extracted from an orange is 5 ml . He decides that he will replace his Fruity with a Zesty if both
      • the mean for the Zesty is more than \(20 \%\) higher than the mean for his Fruity and
  2. the standard deviation for the Zesty is less than 5.5 ml .
  3. Using your answers to part (a), explain whether or not Jeremiah should replace his Fruity with the Zesty.
Edexcel S4 2018 June Q3
10 marks Challenging +1.2
  1. A random sample of 8 students is selected from a school database.
Each student's reaction time is measured at the start of the school day and again at the end of the school day. The reaction times, in milliseconds, are recorded below.
StudentA\(B\)CD\(E\)\(F\)G\(H\)
Reaction time at the start of the school day10.87.28.76.89.410.911.17.6
Reaction time at the end of the school day106.18.85.78.78.19.86.8
  1. State one assumption that needs to be made in order to carry out a paired \(t\)-test.
    (1) The random variable \(R\) is the reaction time at the start of the school day minus the reaction time at the end of the school day. The mean of \(R\) is \(\mu\). John uses a paired \(t\)-test to test the hypotheses $$\mathrm { H } _ { 0 } : \mu = m \quad \mathrm { H } _ { 1 } : \mu \neq m$$ Given that \(\mathrm { H } _ { 0 }\) is rejected at the 5\% level of significance but accepted at the 1\% level of significance,
  2. find the ranges of possible values for \(m\).
Edexcel S4 2018 June Q4
17 marks Challenging +1.2
  1. A glue supplier claims that Goglue is stronger than Tackfast. A company is presently using Tackfast but agrees to change to Goglue if, at the 5\% significance level,
  • the standard deviation of the force required for Goglue to fail is not greater than the standard deviation of the force required for Tackfast to fail and
  • the mean force required for Goglue to fail is more than 4 newtons greater than the mean force for Tackfast to fail.
A series of trials is carried out, using Goglue and Tackfast, and the glues are tested to destruction. The force, \(x\) newtons, at which each glue fails is recorded.
Sample size \(( n )\)Sample mean \(( \bar { x } )\)Standard deviation \(( s )\)
Tackfast \(( T )\)65.270.31
Goglue \(( G )\)510.120.66
It can be assumed that the force at which each glue fails is normally distributed.
  1. Test, at the \(5 \%\) level of significance, whether or not there is evidence that the standard deviation of the force required for Goglue to fail is greater than the standard deviation of the force required for the Tackfast to fail. State your hypotheses clearly. The supplier claims that the mean force required for its Goglue to fail is more than 4 newtons greater than the mean force required for Tackfast to fail.
  2. Stating your hypotheses clearly and using a \(5 \%\) level of significance, test the supplier's claim.
  3. Show that, at the \(5 \%\) level of significance, the supplier's claim will be accepted if \(\bar { X } _ { G } - \bar { X } _ { T } > 4.55\), where \(\bar { X } _ { G }\) and \(\bar { X } _ { T }\) are the mean forces required for Goglue to fail and Tackfast to fail respectively. Later, it was found that an error had been made when recording the results for Goglue. This resulted in all the forces recorded for Goglue being 0.5 newtons more than they should have been. The results for Tackfast were correct.
  4. Explain whether or not this information affects the decision about which glue the supplier decides to use.
Edexcel S4 2018 June Q5
11 marks Challenging +1.2
  1. A machine makes posts. The length of a post is normally distributed with unknown mean \(\mu\) and standard deviation 4 cm .
A random sample of size \(n\) is taken to test, at the \(5 \%\) significance level, the hypotheses $$\mathrm { H } _ { 0 } : \mu = 150 \quad \mathrm { H } _ { 1 } : \mu > 150$$
  1. State the probability of a Type I error for this test. The manufacturer requires the probability of a Type II error to be less than 0.1 when the actual value of \(\mu\) is 152
  2. Calculate the minimum value of \(n\).
Edexcel S4 2018 June Q6
19 marks Challenging +1.2
  1. The continuous random variable \(X\) has probability density function \(\mathrm { f } ( x )\)
$$f ( x ) = \left\{ \begin{array} { c c } \frac { x } { 2 \theta ^ { 2 } } & 0 \leqslant x \leqslant 2 \theta \\ 0 & \text { otherwise } \end{array} \right.$$ where \(\theta\) is a constant.
  1. Use integration to show that \(\mathrm { E } \left( X ^ { N } \right) = \frac { 2 ^ { N + 1 } } { N + 2 } \theta ^ { N }\)
  2. Hence
    1. write down an expression for \(\mathrm { E } ( X )\) in terms of \(\theta\)
    2. find \(\operatorname { Var } ( X )\) in terms of \(\theta\) A random sample \(X _ { 1 } , X _ { 2 } , \ldots , X _ { n }\) where \(n \geqslant 2\) is taken to estimate the value of \(\theta\) The random variable \(S _ { 1 } = q \bar { X }\) is an unbiased estimator of \(\theta\)
  3. Write down the value of \(q\) and show that \(S _ { 1 }\) is a consistent estimator of \(\theta\) The continuous random variable \(Y\) is independent of \(X\) and is uniformly distributed over the interval \(\left[ 0 , \frac { 2 \theta } { 3 } \right]\), where \(\theta\) is the same unknown constant as in \(\mathrm { f } ( x )\). The random variable \(S _ { 2 } = a X + b Y\) is an unbiased estimator of \(\theta\) and is based on one observation of \(X\) and one observation of \(Y\).
  4. Find the value of \(a\) and the value of \(b\) for which \(S _ { 2 }\) has minimum variance.
  5. Show that the minimum variance of \(S _ { 2 }\) is \(\frac { \theta ^ { 2 } } { 11 }\)
  6. Explain which of \(S _ { 1 }\) or \(S _ { 2 }\) is the better estimator for \(\theta\)
Edexcel S4 Q1
8 marks Standard +0.3
  1. The weights of the contents of jars of jam are normally distributed with a stated mean of 100 g . A random sample of 7 jars was taken and the contents of each jar, \(x\) grams, was weighed. The results are summarised by the following statistics.
$$\sum x = 710.9 , \sum x ^ { 2 } = 72219.45 .$$ Test at the \(5 \%\) level of significance whether or not there is evidence that the mean weight of the contents of the jars is greater than 100 g . State your hypotheses clearly.
(8 marks)
Edexcel S4 Q2
8 marks Standard +0.3
2. An engineer decided to investigate whether or not the strength of rope was affected by water. A random sample of 9 pieces of rope was taken and each piece was cut in half. One half of each piece was soaked in water over night, and then each piece of rope was tested to find its strength. The results, in coded units, are given in the table below
Rope no.123456789
Dry rope9.78.56.38.37.25.46.88.15.9
Wet rope9.19.58.29.78.54.98.48.77.7
Assuming that the strength of rope follows a normal distribution, test whether or not there is any difference between the mean strengths of dry and wet rope. State your hypotheses clearly and use a \(1 \%\) level of significance.
(8 marks)
Edexcel S4 Q3
13 marks Standard +0.8
3. A certain vaccine is known to be only \(70 \%\) effective against a particular virus; thus \(30 \%\) of those vaccinated will actually catch the virus. In order to test whether or not a new and more expensive vaccine provides better protection against the same virus, a random sample of 30 people were chosen and given the new vaccine. If fewer than 6 people contracted the virus the new vaccine would be considered more effective than the current one.
  1. Write down suitable hypotheses for this test.
  2. Find the probability of making a Type I error.
  3. Find the power of this test if the new vaccine is
    1. \(80 \%\) effective,
    2. \(90 \%\) effective. An independent research organisation decided to test the new vaccine on a random sample of 50 people to see if it could be considered more than \(70 \%\) effective. They required the probability of a Type I error to be as close as possible to 0.05 .
  4. Find the critical region for this test.
  5. State the size of this critical region.
  6. Find the power of this test if the new vaccine is
    1. \(80 \%\) effective,
    2. \(90 \%\) effective.
  7. Give one advantage and one disadvantage of the second test.
Edexcel S4 Q4
14 marks Standard +0.8
4. Gill, a member of the accounts department in a large company, is studying the expenses claims of company employees. She assumes that the claims, in \(\pounds\), follow a normal distribution with mean \(\mu\) and variance \(\sigma ^ { 2 }\). As a first stage in her investigation she took the following random sample of 10 claims. $$30.85,99.75,142.73,223.16,75.43,28.57,53.90,81.43,68.62,43.45 .$$
  1. Find a 95\% confidence interval for \(\mu\). The chief accountant would like a \(95 \%\) confidence interval where the difference between the upper confidence limit and the lower confidence limit is less than 20 .
  2. Show that \(\frac { \sigma ^ { 2 } } { n } < 26.03\) (to 2 decimal places), where \(n\) is the size of the sample required to achieve this. Gill decides to use her original sample of 10 to obtain a value for \(\sigma ^ { 2 }\) so that the chance of her value being an underestimate is 0.01 .
  3. Find such a value for \(\sigma ^ { 2 }\).
  4. Use this value for \(\sigma ^ { 2 }\) to estimate the size of sample the chief accountant requires.
Edexcel S4 Q5
16 marks Standard +0.8
5. An educational researcher is testing the effectiveness of a new method of teaching a topic in mathematics. A random sample of 10 children were taught by the new method and a second random sample of 9 children, of similar age and ability, were taught by the conventional method. At the end of the teaching, the same test was given to both groups of children. The marks obtained by the two groups are summarised in the table below.
New methodConventional method
Mean \(( \bar { x } )\)82.378.2
Standard deviation \(( s )\)3.55.7
Number of students \(( n )\)109
  1. Stating your hypotheses clearly and using a \(5 \%\) level of significance, investigate whether or not
    1. the variance of the marks of children taught by the conventional method is greater than that of children taught by the new method,
    2. the mean score of children taught by the conventional method is lower than the mean score of those taught by the new method.
      [0pt] [In each case you should give full details of the calculation of the test statistics.]
  2. State any assumptions you made in order to carry out these tests.
  3. Find a 95\% confidence interval for the common variance of the marks of the two groups.
Edexcel S4 Q6
18 marks Standard +0.3
6. A statistics student is trying to estimate the probability, \(p\), of rolling a 6 with a particular die. The die is rolled 10 times and the random variable \(X _ { 1 }\) represents the number of sixes obtained. The random variable \(R _ { 1 } = \frac { X _ { 1 } } { 10 }\) is proposed as an estimator of \(p\).
  1. Show that \(R _ { 1 }\) is an unbiased estimator of \(p\). The student decided to roll the die again \(n\) times ( \(n > 10\) ) and the random variable \(X _ { 2 }\) represents the number of sixes in these \(n\) rolls. The random variable \(R _ { 2 } = \frac { X _ { 2 } } { n }\) and the random variable \(Y = \frac { 1 } { 2 } \left( R _ { 1 } + R _ { 2 } \right)\).
  2. Show that both \(R _ { 2 }\) and \(Y\) are unbiased estimators of \(p\).
  3. Find \(\operatorname { Var } \left( R _ { 2 } \right)\) and \(\operatorname { Var } ( Y )\).
  4. State giving a reason which of the 3 estimators \(R _ { 1 } , R _ { 2 }\) and \(Y\) are consistent estimators of \(p\).
  5. For the case \(n = 20\) state, giving a reason, which of the 3 estimators \(R _ { 1 } , R _ { 2 }\) and \(Y\) you would recommend. The student's teacher pointed out that a better estimator could be found based on the random variable \(X _ { 1 } + X _ { 2 }\).
  6. Find a suitable estimator and explain why it is better than \(R _ { 1 } , R _ { 2 }\) and \(Y\). END
OCR S4 2010 June Q3
7 marks Challenging +1.8
  1. Assuming that all rankings are equally likely, show that \(\mathrm { P } ( R \leqslant 17 ) = \frac { 2 } { 231 }\). The marks of 5 randomly chosen students from School \(A\) and 6 randomly chosen students from School \(B\), who took the same examination, achieving different marks, were ranked. The rankings are shown in the table.
    Rank1234567891011
    School\(A\)\(A\)\(A\)\(B\)\(A\)\(A\)\(B\)\(B\)\(B\)\(B\)\(B\)
  2. For a Wilcoxon rank-sum test, obtain the exact smallest significance level for which there is evidence of a difference in performance at the two schools.
Edexcel S4 Q1
6 marks Standard +0.3
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 \(s_A^2 = 0.495 \text{ mm}^2\). A sample of size 25 is taken from area \(B\) and the unbiased estimate for the population variance is \(s_B^2 = 1.04 \text{ mm}^2\).
  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 \(B\). [5]
  2. State the assumption you have made about the populations of pebble lengths in order to carry out this test. [1]
Edexcel S4 Q2
9 marks Standard +0.3
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.) [9]
Edexcel S4 Q3
9 marks Standard +0.8
A train company claims that the probability \(p\) of one of its trains arriving late is 10\%. A regular traveller sets up the hypothesis \(H_0: p = 0.1\) and decides 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 \(H_0: p = 0.1\) and \(H_1: p > 0.1\) and decides to reject \(H_0\) if \(x \ge 2\).
  1. Find the size of the test. [1]
  2. Show that the power function of the test is $$1 - (1 - p)^{10}(1 + 10p + 55p^2).$$ [4]
  3. Calculate the power of the test when
    1. \(p = 0.2\),
    2. \(p = 0.6\). [3]
  4. Comment on your results from part (c). [1]
Edexcel S4 Q4
9 marks Standard +0.8
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. [7]
  2. State with a reason whether or not the confidence interval supports the assertion \(\sigma^2 = 3\). [2]
Edexcel S4 Q5
11 marks Standard +0.3
  1. Define
    1. a Type I error,
    2. a Type II error. [2]
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.
  1. [(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. [4]
  2. Calculate the probability of the Type I error for this test. [3]
  3. 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. [2]
Edexcel S4 Q6
14 marks Standard +0.3
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 \(\frac{1}{3}X_1 + \frac{1}{3}X_2 + \frac{1}{3}X_3\) is an unbiased estimator for \(\mu\). [3]
An unbiased estimator for \(\mu\) is given by \(\hat{\mu} = aX_1 + bX_2\) where \(a\) and \(b\) are constants.
  1. [(b)] Show that Var(\(\hat{\mu}\)) = \((2a^2 - 2a + 1)\sigma^2\). [6]
  2. Hence determine the value of \(a\) and the value of \(b\) for which \(\hat{\mu}\) has minimum variance. [5]