The quality control department of a precision engineering firm monitors the manufacturing of high-tensile bolts. Historical data indicates that the mean length of the bolts is 145.0 mm145.0\text{ mm}145.0 mm with a standard deviation of 0.8 mm0.8\text{ mm}0.8 mm.
Following a recalibration of the cutting machines, a technician suspects that the mean length of the bolts has changed. To investigate this suspicion, a random sample of 256256256 bolts is measured.
The total length of these 256256256 bolts is found to be 37148.16 mm37148.16\text{ mm}37148.16 mm.
Assuming the lengths are normally distributed, test the technician's suspicion at the 5%5\%5% significance level.
