2021 VCE Maths Methods Mini Test 8
Number of marks: 14
Reading time: 2 minutes
Writing time: 21 minutes
Section B – Calculator Allowed
Instructions
• Answer all questions in the spaces provided.
• Write your responses in English.
• In questions where a numerical answer is required, an exact value must be given unless otherwise specified.
• In questions where more than one mark is available, appropriate working must be shown.
• Unless otherwise indicated, the diagrams in this book are not drawn to scale.
A teacher coaches their school's table tennis team. The teacher has an adjustable ball machine that they use to help the players practise. The speed, measured in metres per second, of the balls shot by the ball machine is a normally distributed random variable \(W\). The teacher sets the ball machine with a mean speed of 10 metres per second and a standard deviation of 0.8 metres per second.
a. Determine \(\Pr(W \ge 11)\), correct to three decimal places. 1 mark
b. Find the value of \(k\), in metres per second, which 80% of ball speeds are below. Give your answer in metres per second, correct to one decimal place. 1 mark
The teacher adjusts the height setting for the ball machine. The machine now shoots balls high above the table tennis table. Unfortunately, with the new height setting, 8% of balls do not land on the table. Let \(\hat{P}\) be the random variable representing the sample proportion of balls that do not land on the table in random samples of 25 balls.
c. Find the mean and the standard deviation of \(\hat{P}\). 2 marks
d. Use the binomial distribution to find \(\Pr(\hat{P} > 0.1)\), correct to three decimal places. 2 marks
The teacher can also adjust the spin setting on the ball machine. The spin, measured in revolutions per second, is a continuous random variable \(X\) with the probability density function \[ f(x) = \begin{cases} \frac{x}{500} & 0 \le x < 20 \\ \frac{50-x}{750} & 20 \le x \le 50 \\ 0 & \text{elsewhere} \end{cases} \]
e. Find the maximum possible spin applied by the ball machine, in revolutions per second. 1 mark
f. Find the median spin, in revolutions per second, correct to one decimal place. 2 marks
g. Find the standard deviation of the spin, in revolutions per second, correct to one decimal place. 3 marks
h. The teacher adjusts the spin setting so that the median spin becomes 30 revolutions per second. This will transform the original probability density function \(f\) to a new probability density function \(g\), where \(g(x) = af(\frac{x}{b})\). Find the values of \(a\) and \(b\) for which the new median spin is 30 revolutions per second, giving your answer correct to two decimal places. 2 marks
End of examination questions
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