Questions S2 (1597 questions)

Browse by module
All questions AEA AS Paper 1 AS Paper 2 AS Pure C1 C12 C2 C3 C34 C4 CP AS CP1 CP2 D1 D2 F1 F2 F3 FD1 FD1 AS FD2 FD2 AS FM1 FM1 AS FM2 FM2 AS FP1 FP1 AS FP2 FP2 AS FP3 FS1 FS1 AS FS2 FS2 AS Further AS Paper 1 Further AS Paper 2 Discrete Further AS Paper 2 Mechanics Further AS Paper 2 Statistics Further Additional Pure Further Additional Pure AS Further Discrete Further Discrete AS Further Extra Pure Further Mechanics Further Mechanics A AS Further Mechanics AS Further Mechanics B AS Further Mechanics Major Further Mechanics Minor Further Numerical Methods Further Paper 1 Further Paper 2 Further Paper 3 Further Paper 3 Discrete Further Paper 3 Mechanics Further Paper 3 Statistics Further Paper 4 Further Pure Core Further Pure Core 1 Further Pure Core 2 Further Pure Core AS Further Pure with Technology Further Statistics Further Statistics A AS Further Statistics AS Further Statistics B AS Further Statistics Major Further Statistics Minor Further Unit 1 Further Unit 2 Further Unit 3 Further Unit 4 Further Unit 5 Further Unit 6 H240/01 H240/02 H240/03 M1 M2 M3 M4 M5 Mechanics 1 P1 P2 P3 P4 PMT Mocks PURE Paper 1 Paper 2 Paper 3 Pure 1 S1 S2 S3 S4 SPS ASFM SPS ASFM Mechanics SPS ASFM Pure SPS ASFM Statistics SPS FM SPS FM Mechanics SPS FM Pure SPS FM Statistics SPS SM SPS SM Mechanics SPS SM Pure SPS SM Statistics Stats 1 Unit 1 Unit 2 Unit 3 Unit 4
Browse by board
AQA AS Paper 1 AS Paper 2 C1 C2 C3 C4 D1 D2 FP1 FP2 FP3 Further AS Paper 1 Further AS Paper 2 Discrete Further AS Paper 2 Mechanics Further AS Paper 2 Statistics Further Paper 1 Further Paper 2 Further Paper 3 Discrete Further Paper 3 Mechanics Further Paper 3 Statistics M1 M2 M3 Paper 1 Paper 2 Paper 3 S1 S2 S3 CAIE FP1 FP2 Further Paper 1 Further Paper 2 Further Paper 3 Further Paper 4 M1 M2 P1 P2 P3 S1 S2 Edexcel AEA AS Paper 1 AS Paper 2 C1 C12 C2 C3 C34 C4 CP AS CP1 CP2 D1 D2 F1 F2 F3 FD1 FD1 AS FD2 FD2 AS FM1 FM1 AS FM2 FM2 AS FP1 FP1 AS FP2 FP2 AS FP3 FS1 FS1 AS FS2 FS2 AS M1 M2 M3 M4 M5 P1 P2 P3 P4 PMT Mocks Paper 1 Paper 2 Paper 3 S1 S2 S3 S4 OCR AS Pure C1 C2 C3 C4 D1 D2 FD1 AS FM1 AS FP1 FP1 AS FP2 FP3 FS1 AS Further Additional Pure Further Additional Pure AS Further Discrete Further Discrete AS Further Mechanics Further Mechanics AS Further Pure Core 1 Further Pure Core 2 Further Pure Core AS Further Statistics Further Statistics AS H240/01 H240/02 H240/03 M1 M2 M3 M4 Mechanics 1 PURE Pure 1 S1 S2 S3 S4 Stats 1 OCR MEI AS Paper 1 AS Paper 2 C1 C2 C3 C4 D1 D2 FP1 FP2 FP3 Further Extra Pure Further Mechanics A AS Further Mechanics B AS Further Mechanics Major Further Mechanics Minor Further Numerical Methods Further Pure Core Further Pure Core AS Further Pure with Technology Further Statistics A AS Further Statistics B AS Further Statistics Major Further Statistics Minor M1 M2 M3 M4 Paper 1 Paper 2 Paper 3 S1 S2 S3 S4 SPS SPS ASFM SPS ASFM Mechanics SPS ASFM Pure SPS ASFM Statistics SPS FM SPS FM Mechanics SPS FM Pure SPS FM Statistics SPS SM SPS SM Mechanics SPS SM Pure SPS SM Statistics WJEC Further Unit 1 Further Unit 2 Further Unit 3 Further Unit 4 Further Unit 5 Further Unit 6 Unit 1 Unit 2 Unit 3 Unit 4
AQA S2 2006 June Q5
5
  1. The continuous random variable \(X\) follows a rectangular distribution with probability density function defined by $$f ( x ) = \begin{cases} \frac { 1 } { b } & 0 \leqslant x \leqslant b
    0 & \text { otherwise } \end{cases}$$
    1. Write down \(\mathrm { E } ( X )\).
    2. Prove, using integration, that $$\operatorname { Var } ( X ) = \frac { 1 } { 12 } b ^ { 2 }$$
  2. At an athletics meeting, the error, in seconds, made in recording the time taken to complete the 10000 metres race may be modelled by the random variable \(T\), having the probability density function $$f ( t ) = \left\{ \begin{array} { c c } 5 & - 0.1 \leqslant t \leqslant 0.1
    0 & \text { otherwise } \end{array} \right.$$ Calculate \(\mathrm { P } ( | T | > 0.02 )\).
AQA S2 2006 June Q6
6 The lifetime, \(X\) hours, of Everwhite camera batteries is normally distributed. The manufacturer claims that the mean lifetime of these batteries is 100 hours.
  1. The members of a photography club suspect that the batteries do not last as long as is claimed by the manufacturer. In order to investigate their suspicion, the members test a random sample of five of these batteries and find the lifetimes, in hours, to be as follows: $$\begin{array} { l l l l l } 85 & 92 & 100 & 95 & 99 \end{array}$$ Test the members' suspicion at the \(5 \%\) level of significance.
  2. The manufacturer, believing that the mean lifetime of these batteries has not changed from 100 hours, decides to determine the lifetime, \(x\) hours, of each of a random sample of 80 Everwhite camera batteries. The manufacturer obtains the following results, where \(\bar { x }\) denotes the sample mean: $$\sum x = 8080 \quad \text { and } \quad \sum ( x - \bar { x } ) ^ { 2 } = 6399$$ Test the manufacturer's belief at the \(5 \%\) level of significance.
AQA S2 2006 June Q7
7 The continuous random variable \(X\) has probability density function defined by $$f ( x ) = \begin{cases} \frac { 1 } { 5 } ( 2 x + 1 ) & 0 \leqslant x \leqslant 1
\frac { 1 } { 15 } ( 4 - x ) ^ { 2 } & 1 < x \leqslant 4
0 & \text { otherwise } \end{cases}$$
  1. Sketch the graph of f.
    1. Show that the cumulative distribution function, \(\mathrm { F } ( x )\), for \(0 \leqslant x \leqslant 1\) is $$\mathrm { F } ( x ) = \frac { 1 } { 5 } x ( x + 1 )$$
    2. Hence write down the value of \(\mathrm { P } ( X \leqslant 1 )\).
    3. Find the value of \(x\) for which \(\mathrm { P } ( X \geqslant x ) = \frac { 17 } { 20 }\).
    4. Find the lower quartile of the distribution.
AQA S2 2008 June Q1
1 It is thought that the incidence of asthma in children is associated with the volume of traffic in the area where they live. Two surveys of children were conducted: one in an area where the volume of traffic was heavy and the other in an area where the volume of traffic was light. For each area, the table shows the number of children in the survey who had asthma and the number who did not have asthma.
\cline { 2 - 4 } \multicolumn{1}{c|}{}AsthmaNo asthmaTotal
Heavy traffic5258110
Light traffic286290
Total80120200
  1. Use a \(\chi ^ { 2 }\) test, at the \(5 \%\) level of significance, to determine whether the incidence of asthma in children is associated with the volume of traffic in the area where they live.
  2. Comment on the number of children in the survey who had asthma, given that they lived in an area where the volume of traffic was heavy.
AQA S2 2008 June Q2
2
  1. The number of telephone calls, \(X\), received per hour for Dr Able may be modelled by a Poisson distribution with mean 6 . Determine \(\mathrm { P } ( X = 8 )\).
  2. The number of telephone calls, \(Y\), received per hour for Dr Bracken may be modelled by a Poisson distribution with mean \(\lambda\) and standard deviation 3 .
    1. Write down the value of \(\lambda\).
    2. Determine \(\mathrm { P } ( Y > \lambda )\).
    1. Assuming that \(X\) and \(Y\) are independent Poisson variables, write down the distribution of the total number of telephone calls received per hour for Dr Able and Dr Bracken.
    2. Determine the probability that a total of at most 20 telephone calls will be received during any one-hour period.
    3. The total number of telephone calls received during each of 6 one-hour periods is to be recorded. Calculate the probability that a total of at least 21 telephone calls will be received during exactly 4 of these one-hour periods.
AQA S2 2008 June Q3
3 Alan's company produces packets of crisps. The standard deviation of the weight of a packet of crisps is known to be 2.5 grams. Alan believes that, due to the extra demand on the production line at a busy time of the year, the mean weight of packets of crisps is not equal to the target weight of 34.5 grams. In an experiment set up to investigate Alan's belief, the weights of a random sample of 50 packets of crisps were recorded. The mean weight of this sample is 35.1 grams. Investigate Alan's belief at the \(5 \%\) level of significance.
AQA S2 2008 June Q4
4 The delay, in hours, of certain flights from Australia may be modelled by the continuous random variable \(T\), with probability density function $$\mathrm { f } ( t ) = \left\{ \begin{array} { c c } \frac { 2 } { 15 } t & 0 \leqslant t \leqslant 3
1 - \frac { 1 } { 5 } t & 3 \leqslant t \leqslant 5
0 & \text { otherwise } \end{array} \right.$$
  1. Sketch the graph of f.
  2. Calculate:
    1. \(\mathrm { P } ( T \leqslant 2 )\);
    2. \(\mathrm { P } ( 2 < T < 4 )\).
  3. Determine \(\mathrm { E } ( T )\).
AQA S2 2008 June Q5
5 The weight of fat in a digestive biscuit is known to be normally distributed.
Pat conducted an experiment in which she measured the weight of fat, \(x\) grams, in each of a random sample of 10 digestive biscuits, with the following results: $$\sum x = 31.9 \quad \text { and } \quad \sum ( x - \bar { x } ) ^ { 2 } = 1.849$$
    1. Construct a \(99 \%\) confidence interval for the mean weight of fat in digestive biscuits.
    2. Comment on a claim that the mean weight of fat in digestive biscuits is 3.5 grams.
  2. If 200 such \(99 \%\) confidence intervals were constructed, how many would you expect not to contain the population mean?
AQA S2 2008 June Q6
6 The management of the Wellfit gym claims that the mean cholesterol level of those members who have held membership of the gym for more than one year is 3.8 . A local doctor believes that the management's claim is too low and investigates by measuring the cholesterol levels of a random sample of 7 such members of the Wellfit gym, with the following results: $$\begin{array} { l l l l l l l } 4.2 & 4.3 & 3.9 & 3.8 & 3.6 & 4.8 & 4.1 \end{array}$$ Is there evidence, at the \(5 \%\) level of significance, to justify the doctor's belief that the mean cholesterol level is greater than the management's claim? State any assumption that you make.
AQA S2 2008 June Q7
7
  1. The number of text messages, \(N\), sent by Peter each month on his mobile phone never exceeds 40. When \(0 \leqslant N \leqslant 10\), he is charged for 5 messages.
    When \(10 < N \leqslant 20\), he is charged for 15 messages.
    When \(20 < N \leqslant 30\), he is charged for 25 messages.
    When \(30 < N \leqslant 40\), he is charged for 35 messages.
    The number of text messages, \(Y\), that Peter is charged for each month has the following probability distribution:
    \(\boldsymbol { y }\)5152535
    \(\mathbf { P } ( \boldsymbol { Y } = \boldsymbol { y } )\)0.10.20.30.4
    1. Calculate the mean and the standard deviation of \(Y\).
    2. The Goodtime phone company makes a total charge for text messages, \(C\) pence, each month given by $$C = 10 Y + 5$$ Calculate \(\mathrm { E } ( C )\).
  2. The number of text messages, \(X\), sent by Joanne each month on her mobile phone is such that $$\mathrm { E } ( X ) = 8.35 \quad \text { and } \quad \mathrm { E } \left( X ^ { 2 } \right) = 75.25$$ The Newtime phone company makes a total charge for text messages, \(T\) pence, each month given by $$T = 0.4 X + 250$$ Calculate \(\operatorname { Var } ( T )\).
AQA S2 2008 June Q8
8 The continuous random variable \(X\) has cumulative distribution function $$\mathrm { F } ( x ) = \left\{ \begin{array} { c c } 0 & x < - 1
\frac { x + 1 } { k + 1 } & - 1 \leqslant x \leqslant k
1 & x > k \end{array} \right.$$ where \(k\) is a positive constant.
  1. Find, in terms of \(k\), an expression for \(\mathrm { P } ( X < 0 )\).
  2. Determine an expression, in terms of \(k\), for the lower quartile, \(q _ { 1 }\).
  3. Show that the probability density function of \(X\) is defined by $$\mathrm { f } ( x ) = \left\{ \begin{array} { c c } \frac { 1 } { k + 1 } & - 1 \leqslant x \leqslant k
    0 & \text { otherwise } \end{array} \right.$$
  4. Given that \(k = 11\) :
    1. sketch the graph of f;
    2. determine \(\mathrm { E } ( X )\) and \(\operatorname { Var } ( X )\);
    3. show that \(\mathrm { P } \left( q _ { 1 } < X < \mathrm { E } ( X ) \right) = 0.25\).
AQA S2 2010 June Q1
1 Judith, the village postmistress, believes that, since moving the post office counter into the local pharmacy, the mean daily number of customers that she serves has increased from 79. In order to investigate her belief, she counts the number of customers that she serves on 12 randomly selected days, with the following results. $$\begin{array} { l l l l l l l l l l l l } 88 & 81 & 84 & 89 & 90 & 77 & 72 & 80 & 82 & 81 & 75 & 85 \end{array}$$ Stating a necessary distributional assumption, test Judith's belief at the 5\% level of significance. \begin{verbatim} QUESTION PART REFERENCE \end{verbatim}
\includegraphics[max width=\textwidth, alt={}]{c31c5c67-834e-42ce-b4af-555890c393d5-03_2484_1709_223_153}
\includegraphics[max width=\textwidth, alt={}]{c31c5c67-834e-42ce-b4af-555890c393d5-04_2496_1724_214_143}
\includegraphics[max width=\textwidth, alt={}]{c31c5c67-834e-42ce-b4af-555890c393d5-05_2484_1709_223_153}
AQA S2 2010 June Q3
3 The continuous random variable \(X\) has a rectangular distribution defined by $$\mathrm { f } ( x ) = \begin{cases} k & - 3 k \leqslant x \leqslant k
0 & \text { otherwise } \end{cases}$$
    1. Sketch the graph of f.
    2. Hence show that \(k = \frac { 1 } { 2 }\).
  2. Find the exact numerical values for the mean and the standard deviation of \(X\).
    1. Find \(\mathrm { P } \left( X \geqslant - \frac { 1 } { 4 } \right)\).
    2. Write down the value of \(\mathrm { P } \left( X \neq - \frac { 1 } { 4 } \right)\).
      \includegraphics[max width=\textwidth, alt={}]{c31c5c67-834e-42ce-b4af-555890c393d5-07_2484_1709_223_153}
AQA S2 2010 June Q4
4 The error, \(X ^ { \circ } \mathrm { C }\), made in measuring a patient's temperature at a local doctors' surgery may be modelled by a normal distribution with mean \(\mu\) and standard deviation \(\sigma\). The errors, \(x ^ { \circ } \mathrm { C }\), made in measuring the temperature of each of a random sample of 10 patients are summarised below. $$\sum x = 0.35 \quad \text { and } \quad \sum ( x - \bar { x } ) ^ { 2 } = 0.12705$$ Construct a \(99 \%\) confidence interval for \(\mu\), giving the limits to three decimal places.
(5 marks)
\includegraphics[max width=\textwidth, alt={}]{c31c5c67-834e-42ce-b4af-555890c393d5-09_2484_1709_223_153}
AQA S2 2010 June Q5
5 The number of telephone calls received, during an 8-hour period, by an IT company that request an urgent visit by an engineer may be modelled by a Poisson distribution with a mean of 7 .
  1. Determine the probability that, during a given 8 -hour period, the number of telephone calls received that request an urgent visit by an engineer is:
    1. at most 5 ;
    2. exactly 7 ;
    3. at least 5 but fewer than 10 .
  2. Write down the distribution for the number of telephone calls received each hour that request an urgent visit by an engineer.
  3. The IT company has 4 engineers available for urgent visits and it may be assumed that each of these engineers takes exactly 1 hour for each such visit. At 10 am on a particular day, all 4 engineers are available for urgent visits.
    1. State the maximum possible number of telephone calls received between 10 am and 11 am that request an urgent visit and for which an engineer is immediately available.
      (1 mark)
    2. Calculate the probability that at 11 am an engineer will not be immediately available to make an urgent visit.
  4. Give a reason why a Poisson distribution may not be a suitable model for the number of telephone calls per hour received by the IT company that request an urgent visit by an engineer.
    (1 mark)
    \includegraphics[max width=\textwidth, alt={}]{c31c5c67-834e-42ce-b4af-555890c393d5-11_2484_1709_223_153}
    \includegraphics[max width=\textwidth, alt={}]{c31c5c67-834e-42ce-b4af-555890c393d5-12_2484_1712_223_153}
    \includegraphics[max width=\textwidth, alt={}]{c31c5c67-834e-42ce-b4af-555890c393d5-13_2484_1709_223_153}
AQA S2 2010 June Q6
6
  1. The number of strokes, \(R\), taken by the members of Duffers Golf Club to complete the first hole may be modelled by the following discrete probability distribution.
    \(\boldsymbol { r }\)\(\leqslant 2\)345678\(\geqslant 9\)
    \(\mathbf { P } ( \boldsymbol { R } = \boldsymbol { r } )\)00.10.20.30.250.10.050
    1. Determine the probability that a member, selected at random, takes at least 5 strokes to complete the first hole.
    2. Calculate \(\mathrm { E } ( R )\).
    3. Show that \(\operatorname { Var } ( R ) = 1.66\).
  2. The number of strokes, \(S\), taken by the members of Duffers Golf Club to complete the second hole may be modelled by the following discrete probability distribution.
    \(\boldsymbol { s }\)\(\leqslant 2\)345678\(\geqslant 9\)
    \(\mathbf { P } ( \boldsymbol { S } = \boldsymbol { s } )\)00.150.40.30.10.030.020
    Assuming that \(R\) and \(S\) are independent:
    1. show that \(\mathrm { P } ( R + S \leqslant 8 ) = 0.24\);
    2. calculate the probability that, when 5 members are selected at random, at least 4 of them complete the first two holes in fewer than 9 strokes;
    3. calculate \(\mathrm { P } ( R = 4 \mid R + S \leqslant 8 )\).
      \includegraphics[max width=\textwidth, alt={}]{c31c5c67-834e-42ce-b4af-555890c393d5-15_2484_1709_223_153}
      \includegraphics[max width=\textwidth, alt={}]{c31c5c67-834e-42ce-b4af-555890c393d5-16_2484_1712_223_153}
      \includegraphics[max width=\textwidth, alt={}]{c31c5c67-834e-42ce-b4af-555890c393d5-17_2484_1709_223_153}
AQA S2 2010 June Q7
7 The random variable \(X\) has probability density function defined by $$f ( x ) = \begin{cases} \frac { 1 } { 2 } & 0 \leqslant x \leqslant 1
\frac { 1 } { 18 } ( x - 4 ) ^ { 2 } & 1 \leqslant x \leqslant 4
0 & \text { otherwise } \end{cases}$$
  1. State values for the median and the lower quartile of \(X\).
  2. Show that, for \(1 \leqslant x \leqslant 4\), the cumulative distribution function, \(\mathrm { F } ( x )\), of \(X\) is given by $$\mathrm { F } ( x ) = 1 + \frac { 1 } { 54 } ( x - 4 ) ^ { 3 }$$ (You may assume that \(\int ( x - 4 ) ^ { 2 } \mathrm {~d} x = \frac { 1 } { 3 } ( x - 4 ) ^ { 3 } + c\).)
  3. Determine \(\mathrm { P } ( 2 \leqslant X \leqslant 3 )\).
    1. Show that \(q\), the upper quartile of \(X\), satisfies the equation \(( q - 4 ) ^ { 3 } = - 13.5\).
    2. Hence evaluate \(q\) to three decimal places.
      \includegraphics[max width=\textwidth, alt={}]{c31c5c67-834e-42ce-b4af-555890c393d5-19_2484_1709_223_153}
AQA S2 2011 June Q1
1 The number of cars passing a speed camera on a main road between 9.30 am and 11.30 am may be modelled by a Poisson distribution with a mean rate of 2.6 per minute.
    1. Write down the distribution of \(X\), the number of cars passing the speed camera during a 5-minute interval between 9.30 am and 11.30 am .
    2. Determine \(\mathrm { P } ( X = 20 )\).
    3. Determine \(\mathrm { P } ( 6 \leqslant X \leqslant 18 )\).
  2. Give two reasons why a Poisson distribution with mean 2.6 may not be a suitable model for the number of cars passing the speed camera during a 1 -minute interval between 8.00 am and 9.30 am on weekdays.
  3. When \(n\) cars pass the speed camera, the number of cars, \(Y\), that exceed 60 mph may be modelled by the distribution \(\mathrm { B } ( n , 0.2 )\). Given that \(n = 20\), determine \(\mathrm { P } ( Y \geqslant 5 )\).
  4. Stating a necessary assumption, calculate the probability that, during a given 5-minute interval between 9.30 am and 11.30 am , exactly 20 cars pass the speed camera of which at least 5 are exceeding 60 mph .
AQA S2 2011 June Q2
2
  1. The continuous random variable \(X\) has a rectangular distribution defined by the probability density function $$f ( x ) = \begin{cases} 0.01 \pi & u \leqslant x \leqslant 11 u
    0 & \text { otherwise } \end{cases}$$ where \(u\) is a constant.
    1. Show that \(u = \frac { 10 } { \pi }\).
    2. Using the formulae for the mean and the variance of a rectangular distribution, find, in terms of \(\pi\), values for \(\mathrm { E } ( X )\) and \(\operatorname { Var } ( X )\).
    3. Calculate exact values for the mean and the variance of the circumferences of circles having diameters of length \(\left( X + \frac { 10 } { \pi } \right)\).
  2. A machine produces circular discs which have an area of \(Y \mathrm {~cm} ^ { 2 }\). The distribution of \(Y\) has mean \(\mu\) and variance 25 . A random sample of 100 such discs is selected. The mean area of the discs in this sample is calculated to be \(40.5 \mathrm {~cm} ^ { 2 }\). Calculate a 95\% confidence interval for \(\mu\). Emily believed that the performances of 16-year-old students in their GCSEs are associated with the schools that they attend. To investigate her belief, Emily collected data on the GCSE results for 2010 from four schools in her area. The table shows Emily's collected data, denoted by \(O _ { i }\), together with the corresponding expected frequencies, \(E _ { i }\), necessary for a \(\chi ^ { 2 }\) test.
    \multirow{2}{*}{}\(\boldsymbol { \geq } \mathbf { 5 }\) GCSEs\(\mathbf { 1 } \boldsymbol { \leqslant }\) GCSEs < \(\mathbf { 5 }\)No GCSEs
    \(O _ { i }\)\(E _ { i }\)\(O _ { i }\)\(E _ { i }\)\(O _ { i }\)\(E _ { i }\)
    Jolliffe College for the Arts187193.159390.623026.23
    Volpe Science Academy175184.439786.522425.05
    Radok Music School183183.817886.233424.96
    Bailey Language School265248.61112116.632233.76
    Emily used these values to correctly conduct a \(\chi ^ { 2 }\) test at the \(1 \%\) level of significance.
AQA S2 2011 June Q4
4 A discrete random variable \(X\) has the probability distribution $$\mathrm { P } ( X = x ) = \left\{ \begin{array} { c l } \frac { 3 x } { 40 } & x = 1,2,3,4
\frac { x } { 20 } & x = 5
0 & \text { otherwise } \end{array} \right.$$
  1. Calculate \(\mathrm { E } ( X )\).
  2. Show that:
    1. \(\quad \mathrm { E } \left( \frac { 1 } { X } \right) = \frac { 7 } { 20 }\);
      (2 marks)
    2. \(\operatorname { Var } \left( \frac { 1 } { X } \right) = \frac { 7 } { 160 }\).
  3. The discrete random variable \(Y\) is such that \(Y = \frac { 40 } { X }\). Calculate:
    1. \(\mathrm { P } ( Y < 20 )\);
    2. \(\mathrm { P } ( X < 4 \mid Y < 20 )\).
AQA S2 2011 June Q5
5
  1. The lifetime of a new 16-watt energy-saving light bulb may be modelled by a normal random variable with standard deviation 640 hours. A random sample of 25 bulbs, taken by the manufacturer from this distribution, has a mean lifetime of 19700 hours. Carry out a hypothesis test, at the \(1 \%\) level of significance, to determine whether the mean lifetime has changed from 20000 hours.
  2. The lifetime of a new 11-watt energy-saving light bulb may be modelled by a normal random variable with mean \(\mu\) hours and standard deviation \(\sigma\) hours. The manufacturer claims that the mean lifetime of these energy-saving bulbs is 10000 hours. Christine, from a consumer organisation, believes that this is an overestimate. To investigate her belief, she carries out a hypothesis test at the \(5 \%\) level of significance based on the null hypothesis \(\mathrm { H } _ { 0 } : \mu = 10000\).
    1. State the alternative hypothesis that should be used by Christine in this test.
    2. From the lifetimes of a random sample of 16 bulbs, Christine finds that \(s = 500\) hours. Determine the range of values for the sample mean which would lead Christine not to reject her null hypothesis.
    3. It was later revealed that \(\mu = 10000\). State which type of error, if any, was made by Christine if she concluded that her null hypothesis should not be rejected.
      (l mark)
AQA S2 2011 June Q6
6 The continuous random variable \(X\) has the probability density function defined by $$f ( x ) = \begin{cases} \frac { 3 } { 8 } \left( x ^ { 2 } + 1 \right) & 0 \leqslant x \leqslant 1
\frac { 1 } { 4 } ( 5 - 2 x ) & 1 \leqslant x \leqslant 2
0 & \text { otherwise } \end{cases}$$
  1. The cumulative distribution function of \(X\) is denoted by \(\mathrm { F } ( x )\). Show that, for \(0 \leqslant x \leqslant 1\), $$\mathrm { F } ( x ) = \frac { 1 } { 8 } x \left( x ^ { 2 } + 3 \right)$$
  2. Hence, or otherwise, verify that the median value of \(X\) is 1 .
  3. Show that the upper quartile, \(q\), satisfies the equation \(q ^ { 2 } - 5 q + 5 = 0\) and hence that \(q = \frac { 1 } { 2 } ( 5 - \sqrt { 5 } )\).
  4. Calculate the exact value of \(\mathrm { P } ( q < X < 1.5 )\).
AQA S2 2012 June Q1
1 At the start of the 2012 season, the ages of the members of the Warwickshire Acorns Cricket Club could be modelled by a normal random variable, \(X\) years, with mean \(\mu\) and standard deviation \(\sigma\). The ages, \(x\) years, of a random sample of 15 such members are summarised below. $$\sum x = 546 \quad \text { and } \quad \sum ( x - \bar { x } ) ^ { 2 } = 1407.6$$
  1. Construct a \(98 \%\) confidence interval for \(\mu\), giving the limits to one decimal place.
    (6 marks)
  2. At the start of the 2005 season, the mean age of the members was 40.0 years. Use your confidence interval constructed in part (a) to indicate, with a reason, whether the mean age had changed.
AQA S2 2012 June Q2
2 The times taken to complete a round of golf at Slowpace Golf Club may be modelled by a random variable with mean \(\mu\) hours and standard deviation 1.1 hours. Julian claims that, on average, the time taken to complete a round of golf at Slowpace Golf Club is greater than 4 hours. The times of 40 randomly selected completed rounds of golf at Slowpace Golf Club result in a mean of 4.2 hours.
  1. Investigate Julian's claim at the \(5 \%\) level of significance.
  2. If the actual mean time taken to complete a round of golf at Slowpace Golf Club is 4.5 hours, determine whether a Type I error, a Type II error or neither was made in the test conducted in part (a). Give a reason for your answer.
AQA S2 2012 June Q3
3 The continuous random variable \(X\) has a cumulative distribution function defined by $$\mathrm { F } ( x ) = \left\{ \begin{array} { c l } 0 & x < - 5
\frac { x + 5 } { 20 } & - 5 \leqslant x \leqslant 15
1 & x > 15 \end{array} \right.$$
  1. Show that, for \(- 5 \leqslant x \leqslant 15\), the probability density function, \(\mathrm { f } ( x )\), of \(X\) is given by \(\mathrm { f } ( x ) = \frac { 1 } { 20 }\).
    (1 mark)
  2. Find:
    1. \(\mathrm { P } ( X \geqslant 7 )\);
    2. \(\mathrm { P } ( X \neq 7 )\);
    3. \(\mathrm { E } ( X )\);
    4. \(\mathrm { E } \left( 3 X ^ { 2 } \right)\).