Sample size for hypothesis testing for the Cp and Cpk index

The BIS.Net Team BIS.Net Team

The CP and Cpk index remain one of the most popular Quality Indexes used by manufacturing.

The baseline is considered 1.0 being the borderline case. A lower value means that more than a trivial amount of non-conforming items are produced. Although confidence intervals can provide information on whether a batch or process is such that these indexes are above 1 or higher, hypothesis testing has one major advantage over the use of confidence intervals. Sample size can be tailored to focus on the amount of concern, i.e. the difference about the base line that we wish to ensure has been met. The norm is to target Cp and Cpk to be above 1.33 and hence the amount of concern relative to the baseline (reference value) is 1.33 – 1 = .333

To determine the sample size the analyst will need to specify a reference value for the null hypothesis which can be considered the status quo. This does not have to be 1.0. If the process was previously stable at around 1.333 and it is desired to now test whether it has improved to 1.5, the reference value would be set at 1.33

The amount of concern, i.e. the minimum change from the reference value that is important to detect must also be specified.

Additionally, the analyst must specify two types of risk. The first is the risk or probability of falsely concluding there is a change at least equal to the specified amount of concern when there isn’t one. This is called the alpha risk. The second is the risk or probability of concluding there has been no change when in fact there was a change greeter or equal to the amount of concern. This is called the beta risk. Both risks are due to ‘the way the numbers fall’ when sampling. By chance, sampling may have selected samples with measurements larger or smaller than expected. Finally, the type of change must be specified, which can be > or < or <>

It is not possible to directly calculate the required sample size using numerical methods. Instead a machine learning algorithm is used which learns to discard incorrect solution paths, ultimately finding the required sample size.

Once the sample size is determined it is possible to calculate critical values. An amount beyond the critical value implies that there is reasonable evidence that there has been a change beyond the alternative hypothesis. For a > than hypothesis an upper critical value is calculated. For a < than hypothesis a lower critical value is calculated. For a <> equal hypothesis both a lower and upper critical value is calculated.

A preferred and recommended option to using critical values is to perform a hypothesis test after sampling with the calculated sample size and make decisions based on the p value.

The BIS.Net Sample Size app also displays an OC and Power curve such as shown in the image below.

Hypothesis testing on proportions

The analyst can use this chart to answer what if questions, such as what is the power of detecting a change different to the amount of concern.

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