$1.(10$ points$)$ Ten samples of $15$ parts each were taken from an ongoing process to establish a p$-$chart for control. The samples and the number of defects in each are shown below $($and in the Homework $6$ data file $$ tab Problem $6\mathrm{.}1).$

Sample n # Defects Sample n # Defects

$11536152$

$21517150$

$31508153$

$41509151$

$515010150$

a$)$ Compute the lower and upper control limits for a $95\%$ confidence interval $($that is$,1\mathrm{.}96$ standard deviations$)$

b$)$ Based on the actual data versus the control limits$,$ what can you say about the process?

$2.(15$ points$)$ You are the new production manager at Buff Soda, Inc. and you$$ve been assigned to monitor the performance of the company$$s $12-$ounce bottle filling line. The bottles must be filled within the specification limits set for this product, which are $12$ ounces $\backslash$pm $0\mathrm{.}1$ ounces $($In other words, the lower specification limit is $11\mathrm{.}9$ ounces, and the upper specification limit is $12\mathrm{.}1$ ounces$).$ You start by measuring the actual amount of soda dispensed by the filling line in four bottles sampled each hour. The data is shown in the Homework $6$ data file $($see tab Problem $6\mathrm{.}2).$

a$)$ Use this data to determine the upper and lower control limits for an R$-$ and X$-$bar chart.

b$)$ Is the process under control?

c$)$ What is the process capability index, Cpk$?$ Does it meet the company$$s target of $1\mathrm{.}33?$

Hint: Use the Table in lecture slides to estimate the standard deviation of the process.

$3.(9$ points$)$ A large national retail chain uses $$mystery shoppers$$ to visit their stores and evaluate the quality of service, cleanliness of the stores, and the maintenance and orderliness of the merchandise layout. The mystery shopper evaluates the store with a checklist and indicates whether the item is acceptable or not. The mystery shopper has visited the Broomfield store $20$ times over the last year, and the average number of unacceptable findings over that period is $10.$

a$)$ Which type of control chart could be constructed based on this data alone?

b$)$ What are the upper and lower control limits for this control chart?

c$)$ When the mystery shopper visited the store recently, they noted $20$ unacceptable findings, which seems like a significant increase. Is this sufficient evidence to justify a claim that something has changed at the store and there are special causes or is this simply common cause variation? Note: Obtain your answer using the control limits$.$

$4.(6$ points$)$ There is a $3$ percent error rate at a specific point in a production process. If an inspector is placed at this point, $85\%$ of the errors can be detected and eliminated. However, the inspector is paid $$80/$hr and can inspect the units at a rate of $30/$hr$.$ If no inspector is used, there is a cost of $$100/$unit to correct the defects later on$.$ Should an inspector be hired?

$5.(5$ points$)$ The purpose of a Sampling Plan $($with parameters n and c$)$ is:

a$.$ To correctly account for desired Type I and Type II error levels

b$.$ To ensure zero$-$defects

c$.$ To minimize consumer risk

d$.$ To minimize consumer $+$ producer risk

e$.$ To compute an AQL

$6.(5$ points$)$ Identify the true statements below:

a$.$ ISO $9000$ is focused on product standards.

b$.$ Deming, Nash, and Juran were giants in establishing quality principles.

c$.$ The Six Sigma program was originally developed by Toyota.

d$.$ Quality improvements always pay for themselves.

e$.$ The Malcolm Baldridge award does not include service industries.