Questions Further Paper 3 Statistics (56 questions)

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AQA Further Paper 3 Statistics Specimen Q2
2 The continuous random variable \(Y\) has cumulative distribution function defined by $$\mathrm { F } ( y ) = \left\{ \begin{array} { c c } 0 & y < 0
\frac { y ^ { 2 } } { 36 } & 0 \leq y \leq 6
1 & y > 6 \end{array} \right.$$ Find the value of \(\mathrm { P } ( Y > 4 )\)
Circle your answer.
\(\frac { 4 } { 9 }\)
\(\frac { 5 } { 9 }\)
\(\frac { 16 } { 27 }\)
\(\frac { 11 } { 27 }\)
AQA Further Paper 3 Statistics Specimen Q3
4 marks
3 The continuous random variable \(R\) follows a rectangular distribution with probability density function given by $$f ( r ) = \begin{cases} k & - a \leq r \leq b
0 & \text { otherwise } \end{cases}$$ Prove, using integration, that \(\mathrm { E } ( R ) = \frac { 1 } { 2 } ( b - a )\)
[0pt] [4 marks]
AQA Further Paper 3 Statistics Specimen Q4
3 marks
4 David, a zoologist, is investigating a particular species of monitor lizard. He measures the lengths, in centimetres, of a random sample of this particular species of lizard. His measured lengths are $$\begin{array} { l l l l l l l l l l } 53.2 & 57.8 & 55.3 & 58.9 & 59.0 & 60.2 & 61.8 & 62.3 & 65.4 & 66.5 \end{array}$$ The lengths may be assumed to be normally distributed.
David correctly constructed a 90\% confidence interval for the mean length of lizard using the measured lengths given and the formula \(\bar { x } \pm \left( b \times \frac { s } { \sqrt { n } } \right)\) This interval had limits of 57.63 and 62.45, correct to two decimal places.
4
  1. State the value for \(b\) used in David's formula. 4
  2. David interprets his interval and states,
    "My confidence interval indicates that exactly 90\% of the population of lizard lengths for this particular species lies between 57.63 cm and \(62.45 \mathrm {~cm} ^ { \prime \prime }\). Do you think David's statement is true? Explain your reasoning. 4
  3. David's assistant, Amina, correctly constructs a \(\beta \%\) confidence interval from David's random sample of measured lengths. Amina informs David that the width of her confidence interval is 8.54 .
    Find the value of \(\beta\).
    [0pt] [3 marks]
    Turn over for the next question
AQA Further Paper 3 Statistics Specimen Q5
8 marks
5 Students at a science department of a university are offered the opportunity to study an optional language module, either German or Mandarin, during their second year of study. From a sample of 50 students who opted to study a language module, 31 were female. Of those who opted to study Mandarin, 8 were female and 12 were male. Test, using the \(5 \%\) level of significance, whether choice of language is independent of gender. The sample of students may be regarded as random.
[0pt] [8 marks] Turn over for the next question
AQA Further Paper 3 Statistics Specimen Q6
9 marks
6 The random variable \(T\) can take the value \(T = - 2\) or any value in the range \(0 \leq T < 12\) The distribution of \(T\) is given by \(\mathrm { P } ( T = - 2 ) = c , \mathrm { P } ( 0 \leq T \leq t ) = 225 k - k ( 15 - t ) ^ { 2 }\) 6
    1. Show that \(1 - c = 216 k\)
      [0pt] [3 marks] 6
  2. (ii) Given that \(c = 0.1\), find the value of \(\mathrm { E } ( T )\)
    [0pt] [3 marks]
    6
  3. Show that \(\mathrm { E } ( \sqrt { | T | } ) = \frac { 5 \sqrt { 2 } + 52 \sqrt { 3 } } { 50 }\)
    [0pt] [3 marks]
AQA Further Paper 3 Statistics Specimen Q7
9 marks
7 Petroxide Industries produces a chemical used in the production of mobile phone covers for a mobile phone company. The chemical becomes less effective when the mean level of impurity is greater than 3 per cent.
Sunita is the Quality Control manager at Petroxide Industries. After a complaint from the mobile phone company, Sunita obtains a random sample of this chemical from 9 batches. She measures the level of impurity, \(X\) per cent, in each sample.
The summarised results are as follows. $$\sum x = 28.8 \quad \sum ( x - \bar { x } ) ^ { 2 } = 0.6$$ 7
    1. Investigate using the \(5 \%\) level of significance whether the mean level of impurity in the chemical is greater than 3 per cent.
      [0pt] [7 marks]
      7
  2. (ii) State the assumption that it was necessary for you to make in order for the test in part (a)(i) to be valid.
    7
  3. State the changes that would be required to your test in part (a) if you were told that the standard deviation of the level of impurity is known to be 0.25 per cent.
    [0pt] [2 marks]
    Turn over for the next question
AQA Further Paper 3 Statistics Specimen Q8
9 marks
8 The time in hours to failure of a component may be modelled by an exponential distribution with parameter \(\lambda = 0.025\) In a manufacturing process, the machine involved uses one of these components continuously until it fails. The component is then immediately replaced.
8
  1. Write down the mean time to failure for a component. 8
  2. Find the probability that a component will fail during a 12-hour shift. 8
  3. A component has not failed for 30 hours. Find the probability that this component lasts for at least another 30 hours.
    [0pt] [2 marks] 8
  4. Find the probability that a component does not fail during 4 consecutive 12-hour shifts.
    [0pt] [3 marks]
    8
    1. State the distribution that can be used to model the number of components that fail during one hour of the manufacturing process.
      [0pt] [2 marks]
      8
  6. (ii) Hence, or otherwise, find the probability that no components fail during 5 consecutive 12-hour shifts.
    [0pt] [2 marks]
AQA Further Paper 3 Statistics 2019 June Q1
1 The discrete random variable \(X\) has \(\operatorname { Var } ( X ) = 5\)
Find \(\operatorname { Var } ( 4 X - 3 )\) Circle your answer.
17207780
AQA Further Paper 3 Statistics 2019 June Q2
2 Amy takes a sample of size 50 from a normal distribution with mean \(\mu\) and variance 16 She conducts a hypothesis test with hypotheses: $$\begin{aligned} & \mathrm { H } _ { 0 } : \mu = 52
& \mathrm { H } _ { 1 } : \mu > 52 \end{aligned}$$ She rejects the null hypothesis if her sample has a mean greater than 53
The actual population mean is 53.5
Find the probability that Amy makes a Type II error.
Circle your answer. \(0.4 \% 3.9 \% 18.9 \% 15.0 \%\)
AQA Further Paper 3 Statistics 2019 June Q3
3 Alan's journey time to work can be modelled by a normal distribution with standard deviation 6 minutes. Alan measures the journey time to work for a random sample of 5 journeys. The mean of the 5 journey times is 36 minutes. 3
  1. Construct a 95\% confidence interval for Alan's mean journey time to work, giving your values to one decimal place.
    3
  2. Alan claims that his mean journey time to work is 30 minutes.
    State, with a reason, whether or not the confidence interval found in part (a) supports Alan's claim.
    3
  3. Suppose that the standard deviation is not known but a sample standard deviation is found from Alan's sample and calculated to be 6 Explain how the working in part (a) would change.
AQA Further Paper 3 Statistics 2019 June Q4
4 A random variable \(X\) has a rectangular distribution. The mean of \(X\) is 3 and the variance of \(X\) is 3
4
  1. Determine the probability density function of \(X\).
    Fully justify your answer. 4
  2. A 6 metre clothes line is connected between the point \(P\) on one building and the point \(Q\) on a second building. Roy is concerned the clothes line may break. He uses the random variable \(X\) to model the distance in metres from \(P\) where the clothes line breaks. 4
    1. State a criticism of Roy's model. 4
  4. (ii) On the axes below, sketch the probability density function for an alternative model for the clothes line.
    \includegraphics[max width=\textwidth, alt={}, center]{3219e2fe-7757-469a-9d0d-654b3e180e8d-05_584_1162_1210_438}
AQA Further Paper 3 Statistics 2019 June Q5
5 An insurance company models the claims it pays out in pounds \(( \pounds )\) with a random variable \(X\) which has probability density function $$f ( x ) = \begin{cases} \frac { k } { x } & 1 < x < a
0 & \text { otherwise } \end{cases}$$ 5
  1. The median claim is \(\pounds 200\)
    Show that \(k = \frac { 1 } { 2 \ln 200 }\)
    5
  2. Find \(\mathrm { P } ( X < 2000 )\), giving your answer to three significant figures.
    5
  3. The insurance company finds that the maximum possible claim is \(\pounds 2000\) and they decide to refine their probability density function. Suggest how this could be done.
AQA Further Paper 3 Statistics 2019 June Q6
6 During August, 102 candidates took their driving test at centre \(A\) and 60 passed. During the same month, 110 candidates took their driving test at centre \(B\) and 80 passed. 6
  1. Test whether the driving test result is independent of the driving test centre using the \(5 \%\) level of significance. 6
  2. Rebecca claims that if the result of the test in part (a) is to reject the null hypothesis then it is easier to pass a driving test at centre \(B\) than centre \(A\). State, with a reason, whether or not you agree with Rebecca's claim.
AQA Further Paper 3 Statistics 2019 June Q7
7 A shopkeeper sells chocolate bars which are described by the manufacturer as having an average mass of 45 grams. The shopkeeper claims that the mass of the chocolate bars, \(X\) grams, is getting smaller on average. A random sample of 6 chocolate bars is taken and their masses in grams are measured. The results are $$\sum x = 246 \quad \text { and } \quad \sum x ^ { 2 } = 10198$$ Investigate the shopkeeper's claim using the \(5 \%\) level of significance.
State any assumptions that you make.
AQA Further Paper 3 Statistics 2019 June Q8
8 The number of telephone calls received by an office can be modelled by a Poisson distribution with mean 3 calls per 10 minutes. 8
  1. Find the probability that:
    8
    1. the office receives exactly 2 calls in 10 minutes; 8
  3. (ii) the office receives more than 30 calls in an hour.
    8
  4. The office manager splits an hour into 6 periods of 10 minutes and records the number of telephone calls received in each of the 10 minute periods. Find the probability that the office receives exactly 2 calls in a 10 minute period exactly twice within an hour.
    8
  5. The office has just received a call.
    8
    1. Find the probability that the next call is received more than 10 minutes later.
      8
  7. (ii) Mahah arrives at the office 5 minutes after the last call was received.
    State the probability that the next call received by the office is received more than 10 minutes later. Explain your answer.
    \includegraphics[max width=\textwidth, alt={}, center]{3219e2fe-7757-469a-9d0d-654b3e180e8d-14_2492_1721_217_150} Additional page, if required.
    Write the question numbers in the left-hand margin. Additional page, if required.
    Write the question numbers in the left-hand margin.
AQA Further Paper 3 Statistics 2020 June Q1
1 The continuous random variable \(X\) has probability density function $$f ( x ) = \begin{cases} \frac { 1 } { 5 } & 1 \leq x \leq 6
0 & \text { otherwise } \end{cases}$$ Find \(\mathrm { P } ( X \geq 3 )\)
Circle your answer. \(\frac { 1 } { 5 } \quad \frac { 2 } { 5 } \quad \frac { 3 } { 5 } \quad \frac { 4 } { 5 }\)
AQA Further Paper 3 Statistics 2020 June Q2
1 marks
2 Jamie is conducting a hypothesis test on a random variable which has a normal distribution with standard deviation 1 The hypotheses are $$\begin{aligned} & \mathrm { H } _ { 0 } : \mu = 5
& \mathrm { H } _ { 1 } : \mu > 5 \end{aligned}$$ He takes a random sample of size 4
The mean of his sample is 6
He uses a 5\% level of significance.
Before Jamie conducted the test, what was the probability that he would make a Type I error? Circle your answer.
[0pt] [1 mark] \(0.0228 \quad 0.0456 \quad 0.0500 \quad 0.1587\)
AQA Further Paper 3 Statistics 2020 June Q3
3 The mass of male giraffes is assumed to have a normal distribution. Duncan takes a random sample of 600 male giraffes.
The mean mass of the sample is 1196 kilograms.
The standard deviation of the sample is 98 kilograms.
3
  1. Construct a 94\% confidence interval for the mean mass of male giraffes, giving your values to one decimal place.
    3
  2. Explain whether or not your answer to part (a) would change if a sample of size 5 was taken with the same mean and standard deviation.
AQA Further Paper 3 Statistics 2020 June Q4
4 The discrete random variable \(X\) follows a discrete uniform distribution and takes values \(1,2,3 , \ldots , n\). The discrete random variable \(Y\) is defined by \(Y = 2 X\)
4
  1. Using the standard results for \(\sum n , \sum n ^ { 2 }\) and \(\operatorname { Var } ( a X + b )\), prove that $$\operatorname { Var } ( Y ) = \frac { n ^ { 2 } - 1 } { 3 }$$ 4
  2. A spinning toy can land on one of four values: 2, 4, 6 or 8
    Using a discrete uniform distribution, find the probability that the next value the toy lands on is greater than 2 4
  3. State an assumption that is required for the discrete uniform distribution used in part (b) to be valid.
AQA Further Paper 3 Statistics 2020 June Q5
5 Emily claims that the average number of runners per minute passing a shop during a long distance run is 8 Emily conducts a hypothesis test to investigate her claim.
5
  1. State the hypotheses for Emily's test. 5
  2. Emily counts the number of runners, \(X\), passing the shop in a randomly chosen minute. The critical region for Emily's test is \(X \leq 2\) or \(X \geq 14\)
    During a randomly chosen minute, Emily counts 3 runners passing the shop.
    Determine the outcome of Emily's hypothesis test.
    5
  3. The actual average number of runners per minute passing the shop is 7 Find the power of Emily's hypothesis test, giving your answer to three significant figures.
AQA Further Paper 3 Statistics 2020 June Q6
2 marks
6 The distance, \(X\) metres, between successive breaks in a water pipe is modelled by an exponential distribution. The mean of \(X\) is 25 The distance between two successive breaks is measured. A water pipe is given a 'Red' rating if the distance is less than \(d\) metres. The government has introduced a new law changing \(d\) to 2
Before the government introduced the new law, the probability that a water pipe is given a 'Red' rating was 0.05 6
  1. Explain whether or not the probability that a water pipe is given a 'Red' rating has increased as a result of the new law.
    6
  2. Find the probability density function of the random variable \(X\). 6
  3. After investigation, the distances between successive breaks in water pipes are found to have a standard deviation of 5 metres. Explain whether or not the use of an exponential model in parts (a) and (b) is appropriate.
    [0pt] [2 marks]
AQA Further Paper 3 Statistics 2020 June Q7
7 The rainfall per day in February in a particular town has been recorded as having a mean of 1.8 inches. Sienna claims that rainfall in February has increased in the town. She records the rainfall in a random sample of 12 days. Her sample mean is 2 inches and her sample standard deviation is 0.4 inches.
It is assumed that rainfall per day has a normal distribution.
7
  1. Investigate Sienna's claim using the \(5 \%\) level of significance.
    7
  2. For the test carried out in part (a), state in context the meaning of a Type II error. 7
  3. The distribution of rainfall per day in February in the town over 10 years is shown in the histogram.
    \includegraphics[max width=\textwidth, alt={}, center]{443e7f17-a555-41ff-9d91-541cf45aae99-11_508_645_849_699} Explain whether or not the assumption that rainfall per day in February has a normal distribution is appropriate.
AQA Further Paper 3 Statistics 2020 June Q8
8 Ray is conducting a hypothesis test with the hypotheses
\(\mathrm { H } _ { 0 }\) : There is no association between time of day and number of snacks eaten
\(\mathrm { H } _ { 1 }\) : There is an association between time of day and number of snacks eaten
He calculates expected frequencies correct to two decimal places, which are given in the following table.
Number of snacks eaten
\cline { 2 - 5 }\cline { 2 - 4 }012 or more
\cline { 2 - 4 } Time of Day23.6821.055.26
\cline { 2 - 5 }Night21.3218.954.74
\cline { 2 - 5 }
\cline { 2 - 5 }
Ray calculates his test statistic using \(\sum \frac { ( O - E ) ^ { 2 } } { E }\)
8
  1. State, with a reason, the error Ray has made and describe any changes Ray will need to make to his test.
    8
  2. Having made the necessary corrections as described in part (a), the correct value of the test statistic is 8.74 Complete Ray's hypothesis test using a \(1 \%\) level of significance.
AQA Further Paper 3 Statistics 2020 June Q9
9 The continuous random variable \(X\) has the cumulative distribution function shown below. $$\mathrm { F } ( x ) = \left\{ \begin{array} { c c } 0 & x < 0
\frac { 1 } { 62 } \left( 4 x ^ { 3 } + 6 x ^ { 2 } + 3 x \right) & 0 \leq x \leq 2
1 & x > 2 \end{array} \right.$$ The discrete random variable \(Y\) has the probability distribution shown below.
\(y\)271319
\(\mathrm { P } ( Y = y )\)0.50.10.10.3
The random variables \(X\) and \(Y\) are independent.
Find the exact value of \(\mathrm { E } \left( X ^ { 3 } + Y \right)\).
AQA Further Paper 3 Statistics 2021 June Q1
1 The discrete uniform distribution \(X\) can take values \(1,2,3 , \ldots , 10\)
Find \(\mathrm { P } ( X \geq 7 )\) Circle your answer. \(0.3 \quad 0.4 \quad 0.6 \quad 0.7\)