QCAA Continuous Random Variables Mini Test 5
External Assessment Paper 2 — Technology-active
Number of marks: 10
Perusal time: 1 minute
Writing time: 15 minutes
Section 1
Instructions
• This section has 10 questions and is worth 10 marks.
• Use a 2B pencil to fill in the A, B, C or D answer bubble completely.
• Choose the best answer for Questions 1 10.
• If you change your mind or make a mistake, use an eraser to remove your response and fill in the new answer bubble completely.
In a certain normal distribution curve, 95% of the area lies between the values 50.32 and 113.68. The mean of this distribution is 82.
Determine the standard deviation.
- (A) 16.16
- (B) 21.12
- (C) 31.68
- (D) 63.36
The continuous random variable \(X\) has the probability density function
\[ f(x) = \begin{cases} \frac{\cos(x)}{2}, & -\frac{\pi}{2} \le x \le \frac{\pi}{2} \\ 0, & \text{otherwise} \end{cases} \]The standard deviation of \(X\) is
- (A) 0.467
- (B) 0.684
- (C) 1.211
- (D) 1.467
A stall at the school fete sells cups of lemonade. Assuming the amount of lemonade in a cup is normally distributed with a mean of 60 mL and a standard deviation of 3 mL, 80% of the cups contain more than
- (A) 52.4 mL
- (B) 57.5 mL
- (C) 61.6 mL
- (D) 62.5 mL
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.
Suppose that the distance travelled by vehicles in a year can be modelled by a normal distribution. In 2021, vehicles travelled a mean of 13 700 km with a standard deviation of 3400 km.
a) Determine the probability that a vehicle chosen at random travelled less than 12 000 km in 2021. [2 marks]
b) Determine the value of \(x\) where 60% of vehicles travelled less than \(x\) km in 2021. [2 marks]
The intelligence quotient (IQ) of individuals in a population is normally distributed, with a mean of 100 and a standard deviation of 16.
Nine individuals are chosen at random from the population.
Determine the probability that no more than two of the individuals have an IQ of at least 120.
END OF PAPER
