QCAA Continuous Random Variables Mini Test 2
External Assessment Paper 2 — Technology-active
Number of marks: 11
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.
The term extremely tall is used to describe any person whose height is three standard deviations or more above the mean height of the population.
A person who just qualifies as extremely tall in a country where heights are normally distributed with a mean height of 180 cm and a standard deviation of 10 cm travels to another country. The person discovers they are taller than exactly 90% of the destination country's population.
Assuming that the standard deviation of both countries is the same, determine the minimum height required to be considered extremely tall in the destination country.
The amount of gravel (in tonnes) sold by a construction company in a given week is a continuous random variable \(X\) and has a probability density function defined by: \[ f(x) = \begin{cases} c(1-x^2), & 0 \le x \le 1 \\ 0, & \text{otherwise} \end{cases} \]
a) Show that \( c = \frac{3}{2} \). [1 mark]
b) Determine \( P(X < 0.25) \). [2 marks]
c) Determine the variance of \(X\). [4 marks]
END OF PAPER
