A games developer claims that the average amount of time a player of a new MMORPG (Massively Multiplayer Online Roleplaying Game) spends in the online game world during a single session is 40 minutes. It has been suggested by the developer that the time spent by players in a single session of the game follows an exponential distribution.

(a) On the basis of the information provided by the games developer, write down the cumulative probability function for the time spent by players in a single session of the game.

(b) What is the probability that a player will spend less than 25 minutes in the game world during a single session of the game?

(c) What is the probability that a player will spend between 50 and 75 minutes in the game world during a single session of the game?

(d) A computer games website have decided to investigate the developer's suggestion that single session playing durations follow an exponential distribution. Data have been collected on the duration of a single gaming session from a random sample of 100 players of the game and are summarised below:

Duration of a single session /Number of observed players

Less than 25 minutes/ 31

Between 25 and 50 minutes/ 32

Between 50 and 75 minutes /19

Between 75 and 100 minutes/ 11

Between 100 and 125 minutes/ 4

More than 125 minutes /3

Using the information provided, test the developer's suggestion that single sessionplaying durations follow an exponential distribution with a mean of 40 minutes at the 5% level of significance.