1 Andy makes repeated attempts to thread a needle. The number of attempts up to and including his first success is denoted by \(X\).
- State two conditions necessary for \(X\) to have a geometric distribution.
- Assuming that \(X\) has the distribution \(\operatorname { Geo } ( 0.3 )\), find
(a) \(\mathrm { P } ( X = 5 )\),
(b) \(\mathrm { P } ( X > 5 )\). - Suggest a reason why one of the conditions you have given in part (i) might not be satisfied in this context.
240 people were asked to guess the length of a certain road. Each person gave their guess, \(l \mathrm {~km}\), correct to the nearest kilometre. The results are summarised below.
| \(l\) | \(10 - 12\) | \(13 - 15\) | \(16 - 20\) | \(21 - 30\) |
| Frequency | 1 | 13 | 20 | 6 |
- (a) Use appropriate formulae to calculate estimates of the mean and standard deviation of \(l\).
(b) Explain why your answers are only estimates. - A histogram is to be drawn to illustrate the data. Calculate the frequency density of the block for the 16-20 class.
- Explain which class contains the median value of \(l\).
- Later, the person whose guess was between 10 km and 12 km changed his guess to between 13 km and 15 km . Without calculation state whether the following will increase, decrease or remain the same:
(a) the mean of \(l\),
(b) the standard deviation of \(l\).