Interquartile range and percentiles

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 \(X\).

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. [4]

Find the probability that the machine dispenses

1. less than 183 ml, [3]
2. exactly 183 ml. [1]
3. Calculate the inter-quartile range of X. [3]
4. Determine the value of s such that P(X ≤ s) = 1 - 2P(X ≤ s). [2]
5. Interpret in words your value of s.

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. [4]
2. Find the probability that the machine dispenses
   1. less than 183 ml,
   2. exactly 183 ml.
   [3]
3. Calculate the inter-quartile range of \(X\). [1]
4. Determine the value of \(x\) such that P(\(X \geq x\)) = 2P(\(X \leq x\)). [3]
5. Interpret in words your value of \(x\). [2]

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. [3 marks]
2. Explain why the continuous uniform distribution may not be a suitable model. [2 marks]