Question

A manufacturer makes integrated circuits that each have a resistance layer with a target thickness of 200 units. A circuit won't work well if this thickness varies too much from the target value. These thickness measurements are approximately normally distributed with a mean of 200 units and a standard deviation of 12 units. A random sample of 16 measurements is selected for a quality inspection. We can assume that the measurements in the sample are independent.
What is the probability that the mean thickness in these 16 measurements xˉ is farther than 3 units away from the target value?

Answer 1

Answer:

0.3174

Step-by-step explanation:

Given that a manufacturer makes integrated circuits that each have a resistance layer with a target thickness of 200 units

These thickness measurements, X,  are approximately normally distributed with a mean of 200 units and a standard deviation of 12 units.

Sample size n = 16

Sample mean will follow N(mu = 200, std error = 12/sqrt 16)

i.e. $\bar X$: N(200, 3)

Required probability

= probability that the mean thickness in these 16 measurements xˉ is farther than 3 units away from the target value

= $P(|bar x-200|>3)\\=P(|z|>1)\\= 1-0.6826\\= 0.3174$