AQA S2 (Statistics 2) 2010 June

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. \(88 \quad 81 \quad 84 \quad 89 \quad 90 \quad 77 \quad 72 \quad 80 \quad 82 \quad 81 \quad 75 \quad 85\) Stating a necessary distributional assumption, test Judith's belief at the \(5\%\) level of significance. [9 marks]

It is claimed that a new drug is effective in the prevention of sickness in holiday-makers. A sample of \(100\) holiday-makers was surveyed, with the following results. Assuming that the \(100\) holiday-makers are a random sample, use a \(\chi^2\) test, at the \(5\%\) level of significance, to investigate the claim. [8 marks]

The continuous random variable \(X\) has a rectangular distribution defined by $$f(x) = \begin{cases} k & -3k \leqslant x \leqslant k \\ 0 & \text{otherwise} \end{cases}$$

1. 1. Sketch the graph of f. [2 marks]
   2. Hence show that \(k = \frac{1}{2}\). [2 marks]
2. Find the exact numerical values for the mean and the standard deviation of \(X\). [3 marks]
3. 1. Find \(\mathrm{P}\left(X \geqslant -\frac{1}{4}\right)\). [2 marks]
   2. Write down the value of \(\mathrm{P}\left(X \neq -\frac{1}{4}\right)\). [1 mark]

The error, \(X\) °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, \(x\) °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]

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 \(5\); [1 mark]
   2. exactly \(7\); [2 marks]
   3. at least \(5\) but fewer than \(10\). [3 marks]
2. Write down the distribution for the number of telephone calls received each hour that request an urgent visit by an engineer. [1 mark]
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 marks]
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]

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.

   \(r\)	\(\leqslant 2\)	\(3\)	\(4\)	\(5\)	\(6\)	\(7\)	\(8\)	\(\geqslant 9\)
   \(\mathrm{P}(R = r)\)	\(0\)	\(0.1\)	\(0.2\)	\(0.3\)	\(0.25\)	\(0.1\)	\(0.05\)	\(0\)

   1. Determine the probability that a member, selected at random, takes at least \(5\) strokes to complete the first hole. [1 mark]
   2. Calculate \(\mathrm{E}(R)\). [2 marks]
   3. Show that \(\mathrm{Var}(R) = 1.66\). [4 marks]
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.

   \(s\)	\(\leqslant 2\)	\(3\)	\(4\)	\(5\)	\(6\)	\(7\)	\(8\)	\(\geqslant 9\)
   \(\mathrm{P}(S = s)\)	\(0\)	\(0.15\)	\(0.4\)	\(0.3\)	\(0.1\)	\(0.03\)	\(0.02\)	\(0\)

   Assuming that \(R\) and \(S\) are independent:
   1. show that \(\mathrm{P}(R + S \leqslant 8) = 0.24\); [5 marks]
   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 marks]
   3. calculate \(\mathrm{P}(R = 4 \mid R + S \leqslant 8)\). [3 marks]

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 marks]
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 \(\int (x - 4)^2 \, dx = \frac{1}{3}(x - 4)^3 + c\).) [4 marks]
3. Determine \(\mathrm{P}(2 \leqslant X \leqslant 3)\). [2 marks]
4. 1. Show that \(q\), the upper quartile of \(X\), satisfies the equation \((q - 4)^3 = -13.5\). [3 marks]
   2. Hence evaluate \(q\) to three decimal places. [1 mark]