Question $2$

Beaver Brothers, Inc., is conducting a study to assess the capability of its $150-$gram bar soap production line. A critical quality measure is the weight of the soap bars after stamping. The lower and upper specification limits are $162$ and $170$ grams, respectively. As a part of an initial capability study, $25$ samples of size $5$ were collected by the quality assurance group and the observations in Table $3\mathrm{.}7$ were recorded.

After analyzing the data by using statistical control charts, the quality assurance group calculated the process capability ratio, $Cp,$ and the process capability index, $Cpk.$ It then decided to improve the stamping process, especially the feeder mechanism. After making all the changes that were deemed necessary, $18$ additional samples were collected. The summary data for these samples are

$1$

$\frac{?}{\text{b}\text{a}\text{r}}(\stackrel{}{x})=163$ grams$\frac{}{b}$

$ar(R)=2\mathrm{.}326$grams

$s=1$gram

All sample observations were within the control chart limits$.$ With the new data, the quality assurance group recalculated the process capability measures. It was pleased with the improved $Cp$ but felt that the process should be centered at $166$ grams to ensure that everything was in order. Its decision concluded the study.

a$.$ Draw the control charts for the data obtained in the initial study and verify that the process was in statistical control.

b$.$ What were the values obtained by the group for $Cp$ and $Cpk$ for the initial capability study? Comment on your findings and explain why further improvements were necessary.

c$.$ What are the $Cp$ and $Cpk$ after the improvements? Comment on your findings, indicating why the group decided to change the centering of the process.

d$.$ What are the $Cp$ and $Cpk$ if the process were centered at $166?$ Comment on your findings.