Can someone please explain this step by step? Tiger Tools, a division of Drillmore Industries, was about to launch a new product. Production Manager Michelle York asked her assistant, Jim Peterson, to check the capability of the oven used in the process. Jim obtained $18$ random samples of $20$ pieces each. The results of those samples are shown in the following table. After he analyzed the data, he concluded that the process was not capable based on a specification width of $1\mathrm{.}44$ cm$.$

Michelle was quite disappointed when she heard this. She had hoped that with the introduction of the new product her operation could run close to full capacity and regain some of its lost luster. The company had a freeze on capital expenditures of more than $$10,000,$ and a replacement oven would cost many times that amount. Jim Peterson worked with the oven crew to see if perhaps different settings could produce the desired results, but they were unable to achieve any meaningful improvements.

Still not ready to concede, Michelle contacted one of her former professors and explained the problem. The professor suggested obtaining another set of samples, this time using a smaller sample size and taking more samples. Michelle then conferred with Jim and they agreed that he would take $27$ samples of five observations each. The results are shown in the following table.

Questions

Consider the following questions, and then write a brief report to Michelle summarizing your findings. Be sure to include your calculations for supporting your findings. To guide your report and expectations see the Case Study rubric in the syllabus or in the course information area. Submit your report in your week $8$ drop$-$box labeled Final Exam Tiger Tools Case Study.

$1.$ How did Jim conclude that the process was not capable based on his first set of samples? $($Hint: Estimate the process standard deviation, $\backslash$sigma $,$ using $)$

$2.$ Does the second set of samples show anything that the first set didn't? Explain what and why.

$3.$ Assuming the problem can be found and corrected, what impact do you think this would have on the capability of the process? Compute the potential process capability using the second data set.

$4.$ If small samples can reveal something that large samples might not, why not just take small samples in every situation?