Interquartile range and percentiles

Questions asking to find quartiles, percentiles, or the interquartile range of a uniform distribution.

3 questions

CAIE S2 2012 November Q1
1
\includegraphics[max width=\textwidth, alt={}, center]{0cd5fc36-486d-4c24-b809-907b3e87cfd7-2_371_531_255_806} The diagram shows the graph of the probability density function, f , of a random variable \(X\). Find the median of \(X\).
Edexcel S2 2003 June Q5
5. A drinks machine dispenses lemonade into cups. It is electronically controlled to cut off the flow of lemonade randomly between 180 ml and 200 ml . The random variable \(X\) is the volume of lemonade dispensed into a cup.
  1. Specify the probability density function of \(X\) and sketch its graph.
  2. Find the probability that the machine dispenses
    1. less than 183 ml ,
    2. exactly 183 ml .
  3. Calculate the inter-quartile range of \(X\).
  4. Determine the value of \(x\) such that \(\mathrm { P } ( X \geq x ) = 2 \mathrm { P } ( X \leq x )\).
  5. Interpret in words your value of \(x\).
Edexcel S2 Q1
  1. A golfer believes that the distance, in metres, that she hits a ball with a 5 iron, follows a continuous uniform distribution over the interval [100, 150].
    1. Find the median and interquartile range of the distance she hits a ball, that would be predicted by this model.
    2. Explain why the continuous uniform distribution may not be a suitable model.
      (2 marks)
    3. The continuous random variable \(X\) has the following cumulative distribution function:
    $$\mathrm { F } ( x ) = \begin{cases} 0 , & x < 0
    \frac { 1 } { 64 } \left( 16 x - x ^ { 2 } \right) , & 0 \leq x \leq 8
    1 , & x > 8 \end{cases}$$
  2. Find \(\mathrm { P } ( X > 5 )\).
  3. Find and specify fully the probability density function \(\mathrm { f } ( x )\) of \(X\).
  4. Sketch \(\mathrm { f } ( x )\) for all values of \(x\).