Specialist2025 Exam 1 · Q4Tech-free5 marks

Expected value by partial fractions and sample means

The waiting time, $T$ hours, to see a particular doctor at a clinic has a distribution with a probability density function $f$ defined by

$f(t)=\begin{cases}\dfrac{3}{2\log_e(2)}\left(\dfrac{1}{(t+1)(2-t)}\right), & 0<t\le 1\\ 0, & \text{elsewhere}\end{cases}$

Use integration to show that $\operatorname{E}(T)=\frac{1}{2}$ .

3 marks

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For random samples of 25 waiting times, it may be assumed that the sample means are approximately normally distributed.

Find the probability that the average waiting time for a random sample of 25 patients is between 0.44 hours and 0.5 hours.

Use $\sigma = 0.3$ and $\Pr(Z<1)=0.84$

2 marks

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