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Part $1$

Aspen Plastics produces plastic bottles to customer order. The quality inspector randomly selects four bottles from the bottle machine and measures the outside diameter of the bottle$$ neck, a critical quality dimension that determines whether the bottle cap will fit properly. The dimensions$($in$.)$ from the last six samples are

Bottle

Sample

$1$

$2$

$3$

$4$

$1$

$0\mathrm{.}596$

$0\mathrm{.}616$

$0\mathrm{.}575$

$0\mathrm{.}609$

$2$

$0\mathrm{.}598$

$0\mathrm{.}620$

$0\mathrm{.}611$

$0\mathrm{.}579$

$3$

$0\mathrm{.}594$

$0\mathrm{.}580$

$0\mathrm{.}615$

$0\mathrm{.}603$

$4$

$0\mathrm{.}573$

$0\mathrm{.}622$

$0\mathrm{.}577$

$0\mathrm{.}612$

$5$

$0\mathrm{.}610$

$0\mathrm{.}631$

$0\mathrm{.}575$

$0\mathrm{.}604$

$6$

$0\mathrm{.}588$

$0\mathrm{.}620$

$0\mathrm{.}591$

$0\mathrm{.}609$

LOADING... Click the icon to view the table of factors for calculating$$ three$-$sigma limits for the x overbar$-$chart and$$ R$-$chart.

Part $2$

Assume that only these six samples are$$ sufficient, and use the data to determine control limits for an$$ R$-$ and an x overbar$-$chart.

To answer this$$ question, first compute the range R and the sample mean x overbar for each sample.

Part $3$

Note that the$$ range, R$,$ is the difference between the largest observation in a sample and the smallest. The sample mean x overbar is the sum of the observations divided by the total number of observations.

Part $4$

Compute the range R and the sample mean x overbar for each sample.

Part $5$Part $6$Part $7$Part $8$Part $9$

Bottle

Sample

$1$

$2$

$3$

$4$

R

x overbar

$1$

$0\mathrm{.}596$

$0\mathrm{.}616$

$0\mathrm{.}575$

$0\mathrm{.}609$

$0\mathrm{.}041$

$0\mathrm{.}599$

$2$

$0\mathrm{.}598$

$0\mathrm{.}620$

$0\mathrm{.}611$

$0\mathrm{.}579$

$0\mathrm{.}041$

$0\mathrm{.}602$

$3$

$0\mathrm{.}594$

$0\mathrm{.}580$

$0\mathrm{.}615$

$0\mathrm{.}603$

$0\mathrm{.}035$

$0\mathrm{.}598$

$4$

$0\mathrm{.}573$

$0\mathrm{.}622$

$0\mathrm{.}577$

$0\mathrm{.}612$

$0\mathrm{.}049$

$0\mathrm{.}596$

$5$

$0\mathrm{.}610$

$0\mathrm{.}631$

$0\mathrm{.}575$

$0\mathrm{.}604$

$0\mathrm{.}056$

$0\mathrm{.}605$

$6$

$0\mathrm{.}588$

$0\mathrm{.}620$

$0\mathrm{.}591$

$0\mathrm{.}609$

$0\mathrm{.}032$

$0\mathrm{.}602$

Part $10$

A range$$ chart, or$$ R$-$chart, is used to monitor process variability. The control limits for the$$ R$-$chart are given by the formulas$$ below, where Upper R overbar is the average of several past R values and the central line of the control$$ chart, and Upper D $3$ and Upper D $4$ are the constants that provide three standard deviation$($three$-$sigma$)$ limits for a given sample size.