Question

20 samples, each the size of 7. The center line of a three sigma pair of X bar and R chart, used for monitoring the process, is 55.9 & 12.6, respectively. Assume the process is IN statistical CONTROL. What is the Average Run Length to detect a shift of the process mean to 60. std remains the SAME

Answer 1

X is normally distributed

$\bar x=\mu=55.9$

$\sigma=\epsilon=12.6$

n=7

$\mu_{\bar{x}}=55.9$

$\sigma_{\bar{x}}=\frac{\sigma}{\sqrt{n}}=\frac{12.6}{\sqrt{7}}=4.7624$

$P(20<z<60)=P(\frac{20-55.9}{4.7624}<\frac{x-\mu_{\bar{x}}}{\sigma_{\bar{x}}}<\frac{60-55.9}{4.7624})$

$=P(-7.5382<z<0.8609)$

$=P(z<0.8609)-P(z<-7.5382)$

$=1.613$