- A manufacturer uses a machine to make metal rods.
The length of a metal rod, \(L \mathrm {~cm}\), is normally distributed with
- a mean of 8 cm
- a standard deviation of \(x \mathrm {~cm}\)
Given that the proportion of metal rods less than 7.902 cm in length is \(2.5 \%\)
- show that \(x = 0.05\) to 2 decimal places.
- Calculate the proportion of metal rods that are between 7.94 cm and 8.09 cm in length.
The cost of producing a single metal rod is 20p
A metal rod
- where \(L < 7.94\) is sold for scrap for 5 p
- where \(7.94 \leqslant L \leqslant 8.09\) is sold for 50 p
- where \(L > 8.09\) is shortened for an extra cost of 10 p and then sold for 50 p
- Calculate the expected profit per 500 of the metal rods.
Give your answer to the nearest pound.
The same manufacturer makes metal hinges in large batches.
The hinges each have a probability of 0.015 of having a fault.
A random sample of 200 hinges is taken from each batch and the batch is accepted if fewer than 6 hinges are faulty.
The manufacturer's aim is for 95\% of batches to be accepted. - Explain whether the manufacturer is likely to achieve its aim.