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

$1$

$2$

$3$

$4$

$1$

$0\mathrm{.}586$

$0\mathrm{.}601$

$0\mathrm{.}591$

$0\mathrm{.}585$

$2$

$0\mathrm{.}619$

$0\mathrm{.}624$

$0\mathrm{.}619$

$0\mathrm{.}584$

$3$

$0\mathrm{.}584$

$0\mathrm{.}620$

$0\mathrm{.}618$

$0\mathrm{.}604$

$4$

$0\mathrm{.}574$

$0\mathrm{.}609$

$0\mathrm{.}596$

$0\mathrm{.}570$

$5$

$0\mathrm{.}574$

$0\mathrm{.}593$

$0\mathrm{.}601$

$0\mathrm{.}575$

$6$

$0\mathrm{.}572$

$0\mathrm{.}614$

$0\mathrm{.}597$

$0\mathrm{.}592$

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

Part $2$

Suppose that the specification for the bottle neck diameter is $0\mathrm{.}600\backslash$pm $0\mathrm{.}050$ in$.$ and the population standard deviation is $0\mathrm{.}015$ in$.$