1.The weights, in grams, of the contents of tins of mackerel fillets are normally distributed with mean and standard deviation 2.5. The value of may be adjusted as required

(a)Find the proportion of tins with contents weighing between 125.0 grams and 130.0 grams when =129.0

(b)(i) State, without proof, the value of which would maximize the proportion of tins with contents weighing between 125.0 grams and 130.0 grams.

(ii)Find the proportion of tins contents weighing between 125.0 grams and 130.0 grams when is equal to the value you have specified in part (b) (i).

(c ) Find, to one decimal place, the value such that 99% of the tins have contents weighing more than 125.0 grams.

(d ) The normal distribution provides a good model for many continuous distribution which arise in production processes or in nature. Explain why the central Limit theorem provides another reason for the importance of the normal distribution.

10. The White Hot Peppers is traditional jazz band. The length, in minutes, of each piece of music played by the band may be modeled by a normal distribution with mean 5 and standard deviation 1.5, and may be assumed to be independent of the lengths of all other pieces.

(a)Find the probability that a particular piece will last between 3.5 minutes and 7.25 minutes.

(b)Find the probability that the mean of the length of the next six pieces to be played by the band will be less than 4 minutes

(c)There is an interval of exactly one minute between the band finishing on piece and starting the next. The band starts to play its last six pieces at 10.31 pm. Using your answer to part (b), state whether you think it likely that the band will still be playing after 11.00 pm. Justify your answer.

(d)Give a reason why the assumption that the lengths of all pieces are independent may be unrealistic.