A manufacturer produces lithium-ion cells for high-performance drones. The energy capacity C C\,C of a cell, measured in milliampere-hours (mAh), is modeled as a normal distribution with mean μ\muμ. To maintain strict quality control, the cells must be consistent enough such that the probability of a cell's capacity exceeding the mean by more than 15 mAh is exactly 0.005.
Show that this requirement implies a standard deviation of σ=5.823 mAh\sigma = 5.823 \text{ mAh}σ=5.823 mAh to 3 decimal places.
A technician suspects that a recent calibration error has caused the mean capacity of the production line to increase above the intended 4200 mAh. They test a random sample of 10 cells, obtaining the following capacities:
4205.2,4191.8,4209.5,4207.1,4197.6,4201.4,4214.9,4206.3,4189.7,4203.5 4205.2, 4191.8, 4209.5, 4207.1, 4197.6, 4201.4, 4214.9, 4206.3, 4189.7, 4203.5 4205.2,4191.8,4209.5,4207.1,4197.6,4201.4,4214.9,4206.3,4189.7,4203.5Assuming the population standard deviation remains 5.823 mAh, conduct a hypothesis test at the 1% significance level to determine if there is evidence that the mean capacity is greater than 4200 mAh. State your hypotheses clearly.
