spen 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

$1$

$2$

$3$

$4$

$1$

$0\mathrm{.}580$

$0\mathrm{.}580$

$0\mathrm{.}586$

$0\mathrm{.}589$

$2$

$0\mathrm{.}611$

$0\mathrm{.}570$

$0\mathrm{.}623$

$0\mathrm{.}575$

$3$

$0\mathrm{.}580$

$0\mathrm{.}576$

$0\mathrm{.}621$

$0\mathrm{.}625$

$4$

$0\mathrm{.}592$

$0\mathrm{.}588$

$0\mathrm{.}605$

$0\mathrm{.}582$

$5$

$0\mathrm{.}591$

$0\mathrm{.}593$

$0\mathrm{.}615$

$0\mathrm{.}608$

$6$

$0\mathrm{.}603$

$0\mathrm{.}619$

$0\mathrm{.}614$

$0\mathrm{.}587$

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.

Compute the range R and the sample mean x overbar for each sample. $($Enter your responses rounded to three decimal places$.)$

Bottle

Sample

$1$

$2$

$3$

$4$

R

x overbar

$1$

$0\mathrm{.}580$

$0\mathrm{.}580$

$0\mathrm{.}586$

$0\mathrm{.}589$

$0\mathrm{.}009$

$0\mathrm{.}584$

$2$

$0\mathrm{.}611$

$0\mathrm{.}570$

$0\mathrm{.}623$

$0\mathrm{.}575$

$0\mathrm{.}053$

$0\mathrm{.}595$

$3$

$0\mathrm{.}580$

$0\mathrm{.}576$

$0\mathrm{.}621$

$0\mathrm{.}625$

$0\mathrm{.}049$

$0\mathrm{.}601$

$4$

$0\mathrm{.}592$

$0\mathrm{.}588$

$0\mathrm{.}605$

$0\mathrm{.}582$

$0\mathrm{.}023$

$0\mathrm{.}592$

$5$

$0\mathrm{.}591$

$0\mathrm{.}593$

$0\mathrm{.}615$

$0\mathrm{.}608$

$$

$0\mathrm{.}024$

$$

$0\mathrm{.}60175$

$6$

$0\mathrm{.}603$

$0\mathrm{.}619$

$0\mathrm{.}614$

$0\mathrm{.}587$

$$

$0\mathrm{.}032$

$$

$0\mathrm{.}60575$

Part $3$

The UCL Subscript Upper R equals

$$

$0\mathrm{.}070$ inches and the LCL Subscript Upper R equals

$$

$0$ inches. $($Enter your responses rounded to three decimal places. If your answer for LCL Subscript Upper R is$$ negative, enter this value as $0.)$

Part $4$

The UCL Subscript x overbar equals

$$

$0\mathrm{.}630$ inches and the LCL Subscript x overbar equals

$$

$0\mathrm{.}570$ inches. $($Enter your responses rounded to three decimal places$.)$