Thirty cards, marked with the even numbers from 2 to 60 inclusive, are shuffled and one card is withdrawn at random and then replaced. The random variable \(X\) takes the value of the number on the card each time the experiment is repeated.
What must be assumed about the cards if the distribution of \(X\) is modelled by a discrete uniform distribution?
Making this modelling assumption, find the expectation and the variance of \(X\).
(a) Explain briefly why, for data grouped in unequal classes, the class with the highest frequency may not be the modal class.
In a histogram drawn to represent the annual incomes (in thousands of pounds) of 1000 families, the modal class was \(15 - 20\) (i.e. \(\mathrm { f } x\), where \(15000 \leq x < 20000\) ), with frequency 300 . The highest frequency in a class was 400 , for the class \(30 - 40\), and the bar representing this class was 8 cm high. The total area under the histogram was \(50 \mathrm {~cm} ^ { 2 }\).
Find the height and the width of the bar representing the modal class.