Question

Consider a normally distributed population with unknown variance ơ2. To test the null hypothesis that σ2 is equal to σ. we collect a sample of size n form the population and then calculate the sample variance S2. It is well known that we would fail-to-reject the null hypothesis if the following holds: 1-2n-1n-1 Note: χ2.v is such that P(x:Xp.v), and χ is a chi-squared random variable with v degrees of freedom. Use the above result to develop a hypothesis testing procedure for the potential process capability index Cp

Answer 1

1) Process capability index (Cpk) is a statistical tool, to measure the ability of a process to produce output within customer’s specification limits. In simple words, it measures producer’s capability to produce a product within customer’s tolerance range. Cpk is used to estimate how close you are to a given target and how consistent you are to around your average performance. Cpk gives you the best-case scenario for the existing process. It can also estimate future process performance, assuming performance is consistent over time.

2) Cpk is a standard index to state the capability of one process, the higher the Cpk value the better the process is.

3) Cpk = or >1.33 indicates that the process is capable and meets specification limits. Any value less than this may mean variation is too wide compared to the specification or the process average is away from the target.

4) C_pk is an index (a simple number) which measures how close a process is running to its specification limits, relative to the natural variability of the process. The larger the index, the less likely it is that any item will be outside the process.”

5) C_p should always be greater than 2.0 for a good process which is under statistical control. For a good process under statistical control, C_pk should be greater than 1.5.”

${\hat {C}}_{p}={\frac {USL-LSL}{6{\hat {\sigma }}}}$	Estimates what the process is capable of producing if the process mean were to be centered between the specification limits. Assumes process output is approximately normally distributed.
${\hat {C}}_{{p,lower}}={{\hat {\mu }}-LSL \over 3{\hat {\sigma }}}$	Estimates process capability for specifications that consist of a lower limit only (for example, strength). Assumes process output is approximately normally distributed.
${\hat {C}}_{{p,upper}}={USL-{\hat {\mu }} \over 3{\hat {\sigma }}}$	Estimates process capability for specifications that consist of an upper limit only (for example, concentration). Assumes process output is approximately normally distributed.