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.
616 exam-style questions on Edexcel A Level Maths The Normal Distribution, covering 3.1 The Normal Distribution, 3.2 Finding Probabilities for Normal Distributions, 3.3 The Inverse Normal Distribution Function, 3.4 The Standard Normal Distribution, 3.5 Finding the mean and standard deviation, 3.6 Approximating a Binomial Distribution, and 3.7 Hypothesis Testing with the Normal Distribution. Each one has a worked solution and a mark scheme showing where the marks go.