Question

Integrated circuits consist of electric channels that are etched onto silicon wafers. A certain proportion of circuits are defective because of "undercutting," which occurs when too much material is etched away so that the channels, which consist of the unetched portions of the wafers, are too narrow. A redesigned process, involving lower pressure in the etching chamber, is being investigated. The goal is to reduce the rate of undercutting to less than 5%. Out of the first 1000 circuits manufactured by the new process, only 35 show evidence of undercutting. Can you conclude that the goal has been met?

Answer 1

We can conclude that the rate of undercutting is less than 5% and the goal has been met.

Explanation:

We have a goal of a rate of 5% or lower of defects. We took a sample and now we have to perform an hypothesis test of the proportions to conclude (or not) that the rate of defects is below 5%.

The null and alternative hypothesis are:

`H_0: \pi\geq0.05\\\\H_1: \pi<0.05`

The significance level is defined as 0.05.

The sample proportion is

`p=35/1000=0.035`

The standard deviation is

`\sigma=\sqrt{(\pi(1-\pi))/(N) }= \sqrt{(0.05(1-0.05))/(1000) }=0.007`

We can calculate the statistic z-value as

`z=(p-\pi+0.5/N)/(\sigma) =(0.035-0.05+0.5/1000)/(0.007)= (-0.0145)/(0.007)= -2.07`

The p-value for this test statistic is

`P(z<-2.07)=0.019`

The P-value (0.02) is smaller than the significance level (0.05), so the effect is significant. There is enugh evidence to reject the null hypothesis.

We can conclude that the rate of undercutting is less than 5% and the goal has been met.