Answer all Parts $($A$,$ B$,$ C and D$)$ independently.

Smart Widget Inc. $($SWI$)$ manufactures and distributes precision systems for pharmaceutical

production lines, touchscreen displays, hospital needs and other gadgets. Answer the following

questions regarding their selected products. Note that each part is independent of each other.

$($A$)$ SWI developed a new advanced precision system for the pharmaceutical production line.

Extensive testing has revealed that the mean dosage delivered by this system is $32\mathrm{.}0370$

milligrams with a standard deviation of $0\mathrm{.}25$ milligrams. The typical dosage requirement for this

product is $32+-0\mathrm{.}45$ milligrams. Calculate and interpret the process capability index. What

percentage of the current production is expected to be outside the tolerance limits given?

$($B$)$ SWI is trying to improve its production process. They set a quality control policy for their newly

developed advanced precision system for the pharmaceutical production line that dosage of

the product outside $32+-0\mathrm{.}45$ milligrams will be considered defective. A sample of $135$

dosages was picked daily over the past ten days. The daily number of defective dosages is as

follows:

With this kind of data, which type of process control chart would be appropriate? Specify the

process control limits for which one would expect that only $2\%$ of the production will fall outside

these control limits$.$

$($C$)$Smart Widget Inc. produces touchscreen displays and sources glass panels. A specific circuit

board KZ$323$ used in this product is purchased from an outside vendor. The vendor makes the

KZ$323$ circuit board to an acceptable quality limit of $2\mathrm{.}10\%$ defectives and is willing to run a $5$

percent risk $($a$)$ of having lots of this level or fewer defectives rejected. SWI considers a lot of

$8\%$ or more defectives unacceptable and wants to ensure it will accept such poor quality lots no

more than $10\%$ of the time $($ beta $).$ A large shipment has just been delivered. What value of sample

size $($ n $)$ and acceptance number $($c$)$ should be selected to determine the quality of this lot?

$($D$)$SWI$\text{'}$s quality department is monitoring the quality of its non$-$rebreather masks. These masks

are used in emergencies to deliver high concentrations of oxygen quickly. These masks have a

bag, called a reservoir bag, that is filled with oxygen and attached to the mask with a one$-$way

valve. When the patient inhales, they breathe in oxygen from the reservoir bag. They are used

in situations like smoke inhalation, carbon monoxide poisoning and severe lung injuries. To

statistically control this product's production process, $13$ masks are randomly picked from the

production line every hour. The measurements of flow rate $($liters of oxygen per minute$)$ are

recorded sequentially for the samples over the $12$ hours of operation. The overall average of

the $12$ sample means is $14\mathrm{.}35$ liters per minute. The overall average of the $12$ sample ranges

is $0\mathrm{.}6950$ liters per minute. Three sigma control limits are applied in the statistical process

control of this item. Determine the upper and lower control limits of the X$-$bar chart and R$-$

chart.

Show all calculations$/$results to at$-$least $4$places after the decimal point$($expect when all succeeding digits are $0)$