AQA Further Paper 3 Statistics (Further Paper 3 Statistics) Specimen

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.
\(\frac { 4 } { 9 }\)
\(\frac { 5 } { 9 }\)
\(\frac { 16 } { 27 }\)
\(\frac { 11 } { 27 }\)

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]

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

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

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. 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]

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. 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

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
5. 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]