The duration, T T\,T hours, for which a specific type of industrial lithium battery maintains an optimal charge under standard operating conditions is modelled by a normal distribution with mean 1500 hours and standard deviation 120 hours.
State the probability that the duration for which a randomly selected battery maintains its optimal charge is not exactly 1500 hours.
A production manager suspects that a recent update to the chemical composition of the electrolyte has caused the mean duration to change. To investigate this, a random sample of 25 updated batteries is tested, and the total duration for the sample is found to be 39,250 39,250\,39,250 hours.
Assuming that the distribution of durations remains normal with a standard deviation of 120 hours, perform a hypothesis test at the 1% significance level to determine whether the mean duration has changed.
A technician asserts that if another random sample of 25 batteries had been tested instead, the outcome of the hypothesis test in part (b) would certainly remain identical. Comment on the validity of this assertion.
