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AQA S2 2007 June Q4
7 marks Moderate -0.8
4 Students are each asked to measure the distance between two points to the nearest tenth of a metre.
  1. Given that the rounding error, \(X\) metres, in these measurements has a rectangular distribution, explain why its probability density function is $$f ( x ) = \left\{ \begin{array} { c c } 10 & - 0.05 < x \leqslant 0.05 \\ 0 & \text { otherwise } \end{array} \right.$$
  2. Calculate \(\mathrm { P } ( - 0.01 < X < 0.02 )\).
  3. Find the mean and the standard deviation of \(X\).
AQA S2 2007 June Q5
10 marks Standard +0.3
5 Members of a residents' association are concerned about the speeds of cars travelling through their village. They decide to record the speed, in mph , of each of a random sample of 10 cars travelling through their village, with the following results: $$\begin{array} { l l l l l l l l l l } 33 & 27 & 34 & 30 & 48 & 35 & 34 & 33 & 43 & 39 \end{array}$$
  1. Construct a \(99 \%\) confidence interval for \(\mu\), the mean speed of cars travelling through the village, stating any assumption that you make.
  2. Comment on the claim that a 30 mph speed limit is being adhered to by most motorists.
    (3 marks)
AQA S2 2007 June Q6
12 marks Standard +0.8
6 The continuous random variable \(X\) has the probability density function given by $$f ( x ) = \left\{ \begin{array} { c c } 3 x ^ { 2 } & 0 < x \leqslant 1 \\ 0 & \text { otherwise } \end{array} \right.$$
  1. Determine:
    1. \(\mathrm { E } \left( \frac { 1 } { X } \right)\);
      (3 marks)
    2. \(\operatorname { Var } \left( \frac { 1 } { X } \right)\).
  2. Hence, or otherwise, find the mean and the variance of \(\left( \frac { 5 + 2 X } { X } \right)\).
AQA S2 2007 June Q7
7 marks Moderate -0.8
7 On a multiple choice examination paper, each question has five alternative answers given, only one of which is correct. For each question, candidates gain 4 marks for a correct answer but lose 1 mark for an incorrect answer.
  1. James guesses the answer to each question.
    1. Copy and complete the following table for the probability distribution of \(X\), the number of marks obtained by James for each question.
      \(\boldsymbol { x }\)4- 1
      \(\mathbf { P } ( \boldsymbol { X } = \boldsymbol { x } )\)
    2. Hence find \(\mathrm { E } ( X )\).
  2. Karen is able to eliminate two of the incorrect answers from the five alternative answers given for each question before guessing the answer from those remaining. Given that the examination paper contains 24 questions, calculate Karen's expected total mark.
AQA S2 2007 June Q8
11 marks Moderate -0.3
8 A jam producer claims that the mean weight of jam in a jar is 230 grams.
  1. A random sample of 8 jars is selected and the weight of jam in each jar is determined. The results, in grams, are $$\begin{array} { l l l l l l l l } 220 & 228 & 232 & 219 & 221 & 223 & 230 & 229 \end{array}$$ Assuming that the weight of jam in a jar is normally distributed, test, at the \(5 \%\) level of significance, the jam producer's claim.
  2. It is later discovered that the mean weight of jam in a jar is indeed 230 grams. Indicate whether a Type I error, a Type II error or neither has occurred in carrying out the hypothesis test in part (a). Give a reason for your answer.
AQA S2 2009 June Q1
6 marks Moderate -0.3
1 A machine fills bottles with bleach. The volume, in millilitres, of bleach dispensed by the machine into a bottle may be modelled by a normal distribution with mean \(\mu\) and standard deviation 8 . A recent inspection indicated that the value of \(\mu\) was 768 . Yvonne, the machine's operator, claims that this value has not subsequently changed. Zara, the quality control supervisor, records the volume of bleach in each of a random sample of 18 bottles filled by the machine and calculates their mean to be 764.8 ml . Test, at the \(5 \%\) level of significance, Yvonne's claim that the mean volume of bleach dispensed by the machine has not changed from 768 ml .
AQA S2 2009 June Q2
14 marks Moderate -0.3
2 John works from home. The number of business letters, \(X\), that he receives on a weekday may be modelled by a Poisson distribution with mean 5.0. The number of private letters, \(Y\), that he receives on a weekday may be modelled by a Poisson distribution with mean 1.5.
  1. Find, for a given weekday:
    1. \(\mathrm { P } ( X < 4 )\);
    2. \(\quad \mathrm { P } ( Y = 4 )\).
    1. Assuming that \(X\) and \(Y\) are independent random variables, determine the probability that, on a given weekday, John receives a total of more than 5 business and private letters.
    2. Hence calculate the probability that John receives a total of more than 5 business and private letters on at least 7 out of 8 given weekdays.
  3. The numbers of letters received by John's neighbour, Brenda, on 10 consecutive weekdays are $$\begin{array} { l l l l l l l l l l } 15 & 8 & 14 & 7 & 6 & 8 & 2 & 8 & 9 & 3 \end{array}$$
    1. Calculate the mean and the variance of these data.
    2. State, giving a reason based on your answers to part (c)(i), whether or not a Poisson distribution might provide a suitable model for the number of letters received by Brenda on a weekday.
AQA S2 2009 June Q3
12 marks Standard +0.3
3 A sample survey, conducted to determine the attitudes of residents to a proposed reorganisation of local schools, gave the following results.
Against reorganisationNot against reorganisation
\multirow{5}{*}{Age of resident}16-1792
18-211710
22-4911590
50-654134
Over 6534
Use a \(\chi ^ { 2 }\) test, at the \(5 \%\) level of significance, to determine whether there is an association between the ages of residents and their attitudes to the proposed reorganisation of local schools.
AQA S2 2009 June Q4
12 marks Standard +0.3
4 The continuous random variable \(X\) has probability density function given by $$f ( x ) = \left\{ \begin{array} { c c } \frac { 1 } { 2 } & 0 \leqslant x \leqslant 1 \\ \frac { 3 - x } { 4 } & 1 \leqslant x \leqslant 3 \\ 0 & \text { otherwise } \end{array} \right.$$
  1. Sketch the graph of f.
  2. Explain why the value of \(\eta\), the median of \(X\), is 1 .
  3. Show that the value of \(\mu\), the mean of \(X\), is \(\frac { 13 } { 12 }\).
  4. Find \(\mathrm { P } ( X < 3 \mu - \eta )\).
AQA S2 2009 June Q5
15 marks Moderate -0.3
5 Joanne has 10 identically-shaped discs, of which 1 is blue, 2 are green, 3 are yellow and 4 are red. She places the 10 discs in a bag and asks her friend David to play a game by selecting, at random and without replacement, two discs from the bag.
  1. Show that:
    1. the probability that the two discs selected are the same colour is \(\frac { 2 } { 9 }\);
    2. the probability that exactly one of the two discs selected is blue is \(\frac { 1 } { 5 }\).
  2. Using the discs, Joanne plays the game with David, under the following conditions: If the two discs selected by David are the same colour, she will pay him 135p. If exactly one of the two discs selected by David is blue, she will pay him 145p. Otherwise David will pay Joanne 45p.
    1. When a game is played, \(X\) is the amount, in pence, won by David. Construct the probability distribution for \(X\), in the form of a table.
    2. Show that \(\mathrm { E } ( X ) = 33\).
  3. Joanne modifies the game so that the amount per game, \(Y\) pence, that she wins may be modelled by $$Y = 104 - 3 X$$
    1. Determine how much Joanne would expect to win if the game is played 100 times.
    2. Calculate the standard deviation of \(Y\), giving your answer to the nearest 1 p .
AQA S2 2009 June Q6
16 marks Standard +0.3
6 Bishen believes that the mean weight of boxes of black peppercorns is 45 grams. Abi, thinking that this is not the case, weighs, in grams, a random sample of 8 boxes of black peppercorns, with the following results. $$\begin{array} { l l l l l l l l } 44 & 44 & 43 & 46 & 42 & 40 & 43 & 46 \end{array}$$
    1. Construct a \(95 \%\) confidence interval for the mean weight of boxes of black peppercorns, stating any assumption that you make.
    2. Comment on Bishen's belief.
    1. Abi claims that the mean weight of boxes of black peppercorns is less than 45 grams. Test this claim at the \(5 \%\) level of significance.
    2. If Bishen's belief is true, state, with a reason, what type of error, if any, may have occurred when conclusions to the test in part (b)(i) were drawn.
      (2 marks)
OCR S2 Q1
7 marks Moderate -0.3
1 In a study of urban foxes it is found that on average there are 2 foxes in every 3 acres.
  1. Use a Poisson distribution to find the probability that, at a given moment,
    1. in a randomly chosen area of 3 acres there are at least 4 foxes,
    2. in a randomly chosen area of 1 acre there are exactly 2 foxes.
    3. Explain briefly why a Poisson distribution might not be a suitable model.
OCR S2 Q2
7 marks Moderate -0.8
2 The random variable \(W\) has the distribution \(B \left( 40 , \frac { 2 } { 7 } \right)\). Use an appropriate approximation to find \(\mathrm { P } ( W > 13 )\).
OCR S2 Q3
7 marks Moderate -0.3
3 The manufacturers of a brand of chocolates claim that, on average, \(30 \%\) of their chocolates have hard centres. In a random sample of 8 chocolates from this manufacturer, 5 had hard centres. Test, at the \(5 \%\) significance level, whether there is evidence that the population proportion of chocolates with hard centres is not \(30 \%\), stating your hypotheses clearly. Show the values of any relevant probabilities.
OCR S2 Q4
7 marks Moderate -0.8
4 DVD players are tested after manufacture. The probability that a randomly chosen DVD player is defective is 0.02 . The number of defective players in a random sample of size 80 is denoted by \(R\).
  1. Use an appropriate approximation to find \(\mathrm { P } ( R \geqslant 2 )\).
  2. Find the smallest value of \(r\) for which \(\mathrm { P } ( R \geqslant r ) < 0.01\).
OCR S2 Q5
9 marks Challenging +1.2
5 In an investment model the increase, \(Y \%\), in the value of an investment in one year is modelled as a continuous random variable with the distribution \(\mathrm { N } \left( \mu , \frac { 1 } { 4 } \mu ^ { 2 } \right)\). The value of \(\mu\) depends on the type of investment chosen.
  1. Find \(\mathrm { P } ( Y < 0 )\), showing that it is independent of the value of \(\mu\).
  2. Given that \(\mu = 6\), find the probability that \(Y < 9\) in each of three randomly chosen years.
  3. Explain why the calculation in part (ii) might not be valid if applied to three consecutive years.
OCR S2 Q6
10 marks Standard +0.3
6 Alex obtained the actual waist measurements, \(w\) inches, of a random sample of 50 pairs of jeans, each of which was labelled as having a 32 -inch waist. The results are summarised by $$n = 50 , \quad \Sigma w = 1615.0 , \quad \Sigma w ^ { 2 } = 52214.50$$ Test, at the \(0.1 \%\) significance level, whether this sample provides evidence that the mean waist measurement of jeans labelled as having 32 -inch waists is in fact greater than 32 inches. State your hypotheses clearly. \section*{Jan 2006}
OCR S2 Q7
10 marks Standard +0.3
7 The random variable \(X\) has the distribution \(\mathrm { N } \left( \mu , 8 ^ { 2 } \right)\). The mean of a random sample of 12 observations of \(X\) is denoted by \(\bar { X }\). A test is carried out at the \(1 \%\) significance level of the null hypothesis \(\mathrm { H } _ { 0 } : \mu = 80\) against the alternative hypothesis \(\mathrm { H } _ { 1 } : \mu < 80\). The test is summarised as follows: 'Reject \(\mathrm { H } _ { 0 }\) if \(\bar { X } < c\); otherwise do not reject \(\mathrm { H } _ { 0 } { } ^ { \prime }\).
  1. Calculate the value of \(c\).
  2. Assuming that \(\mu = 80\), state whether the conclusion of the test is correct, results in a Type I error, or results in a Type II error if:
    1. \(\bar { X } = 74.0\),
    2. \(\bar { X } = 75.0\).
    3. Independent repetitions of the above test, using the value of \(c\) found in part (i), suggest that in fact the probability of rejecting the null hypothesis is 0.06 . Use this information to calculate the value of \(\mu\).
OCR S2 Q8
15 marks Moderate -0.3
8 A continuous random variable \(X\) has probability density function given by $$f ( x ) = \begin{cases} k x ^ { n } & 0 \leqslant x \leqslant 1 \\ 0 & \text { otherwise } \end{cases}$$ where \(n\) and \(k\) are positive constants.
  1. Find \(k\) in terms of \(n\).
  2. Show that \(\mathrm { E } ( X ) = \frac { n + 1 } { n + 2 }\). It is given that \(n = 3\).
  3. Find the variance of \(X\).
  4. One hundred observations of \(X\) are taken, and the mean of the observations is denoted by \(\bar { X }\). Write down the approximate distribution of \(\bar { X }\), giving the values of any parameters.
  5. Write down the mean and the variance of the random variable \(Y\) with probability density function given by $$g ( y ) = \begin{cases} 4 \left( y + \frac { 4 } { 5 } \right) ^ { 3 } & - \frac { 4 } { 5 } \leqslant y \leqslant \frac { 1 } { 5 } \\ 0 & \text { otherwise } \end{cases}$$
AQA S3 2006 June Q1
8 marks Moderate -0.3
1 A council claims that 80 per cent of households are generally satisfied with the services it provides. A random sample of 250 households shows that 209 are generally satisfied with the council's provision of services.
  1. Construct an approximate \(95 \%\) confidence interval for the proportion of households that are generally satisfied with the council's provision of services.
  2. Hence comment on the council's claim.
AQA S3 2006 June Q2
7 marks Standard +0.3
2 The table below shows the heart rates, \(x\) beats per minute, and the systolic blood pressures, \(y\) milligrams of mercury, of a random sample of 10 patients undergoing kidney dialysis.
Patient\(\mathbf { 1 }\)\(\mathbf { 2 }\)\(\mathbf { 3 }\)\(\mathbf { 4 }\)\(\mathbf { 5 }\)\(\mathbf { 6 }\)\(\mathbf { 7 }\)\(\mathbf { 8 }\)\(\mathbf { 9 }\)\(\mathbf { 1 0 }\)
\(\boldsymbol { x }\)838688929498101111115121
\(\boldsymbol { y }\)157172161154171169179180192182
  1. Calculate the value of the product moment correlation coefficient for these data.
  2. Assuming that these data come from a bivariate normal distribution, investigate, at the \(1 \%\) level of significance, the claim that, for patients undergoing kidney dialysis, there is a positive correlation between heart rate and systolic blood pressure.
AQA S3 2006 June Q3
11 marks Moderate -0.3
3 Each enquiry received by a business support unit is dealt with by Ewan, Fay or Gaby. The probabilities of them dealing with an enquiry are \(0.2,0.3\) and 0.5 respectively. Of enquiries dealt with by Ewan, 60\% are answered immediately, 25\% are answered later the same day and the remainder are answered at a later date. Of enquiries dealt with by Fay, 75\% are answered immediately, 15\% are answered later the same day and the remainder are answered at a later date. Of enquiries dealt with by Gaby, 90\% are answered immediately and the remainder are answered at a later date.
  1. Determine the probability that an enquiry:
    1. is dealt with by Gaby and answered immediately;
    2. is answered immediately;
    3. is dealt with by Gaby, given that it is answered immediately.
  2. Determine the probability that an enquiry is dealt with by Ewan, given that it is answered later the same day.
AQA S3 2006 June Q4
6 marks Moderate -0.3
4 The table below shows the probability distribution for the number of students, \(R\), attending classes for a particular mathematics module.
\(\boldsymbol { r }\)678
\(\mathbf { P } ( \boldsymbol { R } = \boldsymbol { r } )\)0.10.60.3
  1. Find values for \(\mathrm { E } ( R )\) and \(\operatorname { Var } ( R )\).
  2. The number of students, \(S\), attending classes for a different mathematics module is such that $$\mathrm { E } ( S ) = 10.9 , \quad \operatorname { Var } ( S ) = 1.69 \quad \text { and } \quad \rho _ { R S } = \frac { 2 } { 3 }$$ Find values for the mean and variance of:
    1. \(T = R + S\);
    2. \(\quad D = S - R\).
AQA S3 2006 June Q5
12 marks Standard +0.3
5 The number of letters per week received at home by Rosa may be modelled by a Poisson distribution with parameter 12.25.
  1. Using a normal approximation, estimate the probability that, during a 4 -week period, Rosa receives at home at least 42 letters but at most 54 letters.
  2. Rosa also receives letters at work. During a 16-week period, she receives at work a total of 248 letters.
    1. Assuming that the number of letters received at work by Rosa may also be modelled by a Poisson distribution, calculate a \(98 \%\) confidence interval for the average number of letters per week received at work by Rosa.
    2. Hence comment on Rosa's belief that she receives, on average, fewer letters at home than at work.
AQA S3 2006 June Q6
8 marks Challenging +1.2
6 The random variable \(X\) has a Poisson distribution with parameter \(\lambda\).
  1. Prove that \(\mathrm { E } ( X ) = \lambda\).
  2. By first proving that \(\mathrm { E } ( X ( X - 1 ) ) = \lambda ^ { 2 }\), or otherwise, prove that \(\operatorname { Var } ( X ) = \lambda\).