A biomedical engineering company manufacture artificial hip joints from titanium alloy. The femoral head for the joint needs to be precision machined to fit the corresponding socket in the pelvis. To...

A biomedical engineering company manufacture artificial hip joints from titanium alloy. The femoral head for the joint needs to be precision machined to fit the corresponding socket in the pelvis. To test their manufacturing process, the company have produced 25 femoral heads with an intended diameter of 45mm. The femoral heads were measured, and their mean and standard deviation were found to be 45.1 mm and 0.5 mm, respectively.

(i) If another 25 femoral heads were produced, what is the probability that their mean diameter would be within 0.1 mm of the target value of 45 mm? Work out the answer by hand.