QCAA Continuous Random Variables Mini Test 6
External Assessment Paper 2 — Technology-active
Number of marks: 10
Perusal time: 1 minute
Writing time: 15 minutes
Section 2
Instructions
• Write using black or blue pen.
• Questions worth more than one mark require mathematical reasoning and/or working to be shown to support answers.
• If you need more space for a response, use the additional pages at the back of this book.
– On the additional pages, write the question number you are responding to.
– Cancel any incorrect response by ruling a single diagonal line through your work.
– Write the page number of your alternative/additional response, i.e. See page …
– If you do not do this, your original response will be marked.
• This section has nine questions and is worth 45 marks.
Let \(X\) denote the time in minutes between the arrival of trains at a station. The cumulative distribution function of \(X\) is defined by
\[ F(x) = \begin{cases} 2 - \frac{10}{x}, & 5 \le x \le 10 \\ 0, & \text{otherwise} \end{cases} \]a) Determine the probability density function of \(X\). [3 marks]
b) Determine the probability that \(5 < X < 7\). [1 mark]
c) Determine the mean time between the arrival of trains at the station. [2 marks]
Bottles of soft drink should contain a volume with a mean of 591 mL, but some variation is expected. Any bottle at or below the 20th percentile of the volume distribution is rejected. A percentile is a measure in statistics that shows the values below which a given percentage of observations occur.
Thirty-five per cent of the bottles contain 593 mL or more of soft drink.
Assuming the volumes are normally distributed, determine the smallest volume (in mL) that will be accepted.
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