6 South Riding Alarms (SRA) maintains household burglar-alarm systems. The company aims to carry out an annual service of a system in a mean time of 20 minutes.
Technicians who carry out an annual service must record the times at which they start and finish the service.

1. Gary is employed as a technician by SRA and his manager, Rajul, calculates the times taken for 8 annual services carried out by Gary. The results, in minutes, are as follows. $$\begin{array} { l l l l l l l l } 24 & 25 & 29 & 16 & 18 & 27 & 19 & 23 \end{array}$$ Assume that these times may be regarded as a random sample from a normal distribution. Carry out a hypothesis test, at the \(10 \%\) significance level, to examine whether the mean time for an annual service carried out by Gary is 20 minutes.
   [0pt] [8 marks]
2. Rajul suspects that Gary may be taking longer than 20 minutes on average to carry out an annual service. Rajul therefore calculates the times taken for 100 annual services carried out by Gary. Assume that these times may also be regarded as a random sample from a normal distribution but with a standard deviation of 4.6 minutes. Find the highest value of the sample mean which would not support Rajul's suspicion at the \(5 \%\) significance level. Give your answer to two decimal places.
   [0pt] [4 marks]
   \(7 \quad\) A continuous random variable \(X\) has the probability density function defined by $$f ( x ) = \begin{cases} \frac { 4 } { 5 } x & 0 \leqslant x \leqslant 1
   \frac { 1 } { 20 } ( x - 3 ) ( 3 x - 11 ) & 1 \leqslant x \leqslant 3
   0 & \text { otherwise } \end{cases}$$
3. Find \(\mathrm { P } ( X < 1 )\).
4. 1. Show that, for \(1 \leqslant x \leqslant 3\), the cumulative distribution function, \(\mathrm { F } ( x )\), is given by $$\mathrm { F } ( x ) = \frac { 1 } { 20 } \left( x ^ { 3 } - 10 x ^ { 2 } + 33 x - 16 \right)$$
   2. Hence verify that the median value of \(X\) lies between 1.13 and 1.14 .
      [0pt] [3 marks] QUESTION
      PART Answer space for question 7
      REFERENCE REFERENCE
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