Statistics for Business: Decision Making and Analysis, 3rd Edition

$$\begin{array}{l}\text{... within} \text{on} \text{day} 10)\\ ={0.95}^{10}\approx 0.60\end{array}$$

Hence, the probability that at least one mean lies outside the control limits is 1 − 0.60 = 0.40. The chance of a Type I error during 10 days is larger than the chance on any one day. A 5% chance for a Type I error on any one day implies there is a 40% chance for a Type I error during 10 days. Repeated testing eventually signals a problem. Testing the process for 25 consecutive days increases the chance that a Type I error will occur during this period to 1 − 0.95²⁵ ≈ 0.72.

Smaller values of α reduce the chance of a Type I error but at a cost. Reducing α moves the control limits farther from μ, increasing the chance ...