- A machine makes bolts such that the length, \(L \mathrm {~cm}\), of a bolt has distribution \(L \sim \mathrm {~N} \left( 4.1,0.125 ^ { 2 } \right)\)
A bolt is selected at random.
- Find the probability that the length of this bolt is more than 4.3 cm .
- Show that \(\mathrm { P } ( 3.9 < L < 4.3 )\) is 0.890 correct to 3 decimal places.
The machine makes 500 bolts.
The cost to make each bolt is 5 pence.
Only bolts with length between 3.9 cm and 4.3 cm can be used. These are sold for 9 pence each. All the bolts that cannot be used are recycled with a scrap value of 1 pence each. - Calculate an estimate for the profit made on these 500 bolts.
Following adjustments to the machine, the length of a bolt, \(B \mathrm {~cm}\), made by the machine is such that \(B \sim \mathrm {~N} \left( \mu , \sigma ^ { 2 } \right)\)
Given that \(\mathrm { P } ( B > 4.198 ) = 0.025\) and \(\mathrm { P } ( B < 4.065 ) = 0.242\)
- find the value of \(\mu\) and the value of \(\sigma\)
- State, giving a reason, whether the adjustments to the machine will result in a decrease or an increase in the profit made on 500 bolts.