Task 5 The lengths of 40 steel pins (6 mm diameter) were measured (to the nearest tenth of a millimetre), with the following results. (a) A firm uses a machine to produce steel pins with lengths that are normally distributed with mean 50.00 mm and standard deviation 0.24 mm. A pin must have a length between 49.60 mm and 50.35 mm to be considered acceptable. Pins with lengths below this range are scrapped and pins with lengths above the range require grinding to an acceptable length. Calculate, showing your working: (i) the proportion of pins that are scrapped; (ii) the proportion of pins that require grinding; (iii) the proportion of pins that are acceptable. (b) In a manufacturing company a group of nine employees are chosen for a new project. The company is made up of manufacturing staff and administrative support staff. (i) Give the full binomial expansion for (m + a). The company is made up of manufacturing staff and administrative support staff in the ratio of m: a (see Engineering Maths Unique Student Values for your ratio). (ii) State the probability an employee chosen at random is from the manufacturing department (P(m)) and the probability an employee is from the administrative support department (P(a)). Find the probability that the new project team consists of: (iii) All administrative staff; (iv) All manufacturing staff; (v) (vi) Exactly four manufacturing staff and five administrative staff; At least seven administrative staff?