Answer to Question #279046 in Statistics and Probability for Joe

Question

Suppose that the thickness of a part used in a semiconductor is its critical dimension and that the process of manufacturing these parts is considered to be under control if the true variation among the thicknesses of the parts is given by a standard deviation not greater than σ = 0.60 thousandth of an inch. To keep a check on the process, random samples of size n = 20 are taken periodically, and it is regarded to be “out of control” if the probability that S2 will take on a value greater than or equal to the observed sample value is 0.01 or less (even though σ = 0.60). What can one conclude about the process if the standard deviation of such a periodic random sample is s = 0.84 thousandth of an inch?

Answer 1

In order to make a valid conclusion, we perform the following hypothesis test.

$H_0:\sigma^2=0.36$ $vs$ $H_A:\sigma^2\gt 0.36$

$n=20,\space s^2=0.7056$

The test statistic is,

$\chi^2_c={(n-1)s^2\over\sigma^2}={(20-1)0.7056\over 0.36}=37.24$

Since the process is out of control if probability that $s^2$ will take on a value greater than or equal to the observed sample value is 0.01 or less, we determine the following probability.

$p(\chi^2\gt37.24)$. If $p(\chi^2\gt37.24)\leq0.01$, then the process is out of control. To find this probability, we enter the following command in R.

> pchisq(37.24, df = 19,lower.tail = FALSE)

[1] 0.007404034

The probability, $p(\chi^2\gt37.24)=0.0074\lt 0.01$ and therefore, we reject the null hypothesis and conclude that the process is out of control.