Thickness of a circuit board

Topic: Functions of random variables and limit theorems · more questions on this topic
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Boards should have thickness between 0.146 cm and 0.154 cm. A machine produces thickness X\sim\operatorname{N}(0.151, 0.003^2).

1. Find the probability that one board meets the specification. Use \Phi(1)\approx 0.8413 and \Phi(1.67)\approx 0.9525.

2. Let \overline X be the mean thickness of 25 independent boards from this machine. Find the probability that the batch mean meets the same specification.

Worked answer (click to show)

1. Standardise the limits 0.146 and 0.154 with mean 0.151 and standard deviation 0.003:

\mathbb{P}(0.146\le X\le 0.154)=\Phi(1)-\bigl(1-\Phi(1.667)\bigr)\approx 0.8413-(1-0.9525)=0.794.

2. The batch mean is \operatorname{N}(0.151, 0.003^2/25). The standardised limits are -8.33 and 5. Both are so far into the tails that the probability is 1 for any ordinary purpose.