Question

or a process with a mean of 100, a standard deviation of 10 and an upper specification of 120, what is the probability that a randomly selected item is defective(or beyond the upper specification limit)?

Answer 1

$X\sim N(\mu, \sigma^2)$

Then $Z=\dfrac{X-\mu}{\sigma/}\sim N(0, 1)$

Given $\mu=100, \sigma=10$

$$P(X>120)=1-P(X\leq 120)$$

$$=1-P(Z\leq\dfrac{120-100}{10})=1-P(Z\leq2)$$

$$\approx1-0.97725\approx0.02275$$

$\dfrac{120-100}{10}=2$

95% of data fall with 2 standard deviations of the mean

$P(X>120)\approx\dfrac{1-0.95}{2}=0.025$