Software upgrade times $($in minutes$)$ are being evaluated. Five samples of six observations $($n$=6)$ each have been taken, and the results are as listed.

SAMPLE DATA

Sample $1$ Sample $2$ Sample $3$ Sample $4$ Sample $5$

$77\mathrm{.}080\mathrm{.}579\mathrm{.}978\mathrm{.}980\mathrm{.}8$

$78\mathrm{.}178\mathrm{.}779\mathrm{.}679\mathrm{.}479\mathrm{.}6$

$80\mathrm{.}081\mathrm{.}080\mathrm{.}479\mathrm{.}780\mathrm{.}4$

$78\mathrm{.}480\mathrm{.}480\mathrm{.}379\mathrm{.}480\mathrm{.}8$

$79\mathrm{.}580\mathrm{.}180\mathrm{.}880\mathrm{.}678\mathrm{.}8$

$78\mathrm{.}180\mathrm{.}681\mathrm{.}081\mathrm{.}182\mathrm{.}3$

A$.$ Calculate the mean for each sample $($x$-$bar$)$ and the grand mean $($the average of all the means, x$-$double$-$bar$).($Round your sample means and grand mean to $2$ decimal places.$)$

Sample $1$ mean:

Sample $2$ mean:

Sample $3$ mean:

Sample $4$ mean:

Sample $5$ mean:

Grand mean:

B$.$ Calculate the range for each sample $($R$)$ and the average range of the samples $($R$-$bar$).($Round the R$-$bar value to $2$ decimals.$)$

Sample $1$ R:

Sample $2$ R:

Sample $3$ R:

Sample $4$ R:

Sample $5$ R:

R$-$bar:

Factors for three$-$sigma control limits for x$$

and R charts.

FACTORS FOR R CHARTS

Number of Observations in Subgroup,

n Factor for

x$$

Chart,

A$2$ Lower

Control Limit,

D$3$ Upper

Control Limit,

D$4$

$21\mathrm{.}8803\mathrm{.}27$

$31\mathrm{.}0202\mathrm{.}57$

$40\mathrm{.}7302\mathrm{.}28$

$50\mathrm{.}5802\mathrm{.}11$

$60\mathrm{.}4802\mathrm{.}00$

$70\mathrm{.}420\mathrm{.}081\mathrm{.}92$

$80\mathrm{.}370\mathrm{.}141\mathrm{.}86$

$90\mathrm{.}340\mathrm{.}181\mathrm{.}82$

$100\mathrm{.}310\mathrm{.}221\mathrm{.}78$

$110\mathrm{.}290\mathrm{.}261\mathrm{.}74$

$120\mathrm{.}270\mathrm{.}281\mathrm{.}72$

$130\mathrm{.}250\mathrm{.}311\mathrm{.}69$

$140\mathrm{.}240\mathrm{.}331\mathrm{.}67$

$150\mathrm{.}220\mathrm{.}351\mathrm{.}65$

$160\mathrm{.}210\mathrm{.}361\mathrm{.}64$

$170\mathrm{.}200\mathrm{.}381\mathrm{.}62$

$180\mathrm{.}190\mathrm{.}391\mathrm{.}61$

$190\mathrm{.}190\mathrm{.}401\mathrm{.}60$

$200\mathrm{.}180\mathrm{.}411\mathrm{.}59$

C$.$ Using the A$2$ factors from the above table, and the grand mean and R$-$bar from parts A and B$,$ determine upper and lower control limits for the mean chart $($also called the X$-$bar chart$).($Use your rounded answers from A and B$.$ Round your final answers to $2$ decimal places.$)$

Lower Control Limit:

Upper Control Limit:

D$.$ Using the D$3$ and D$4$ factors from the above table, and the R$-$bar from part B$,$ determine upper and lower control limits for the range chart $($also called the R$-$chart$).($Use your rounded answer from B$,$ and round your final answers to $2$ decimal places.$)$

Lower Control Limit:

Upper Control Limit:

E$.$ Build the x$-$bar chart and plot the $5$ sample means on it$.$ Build the R$-$chart and plot the $5$ sample ranges. Based on your charts, is this process in control? Why or why not?

multiple choice

The process is not in control because the Range chart is not in control.

The process is not in control because the x$-$bar chart is not in control.

The process is in control because both the x$-$bar and range chart are in control

The process is not in control because both the x$-$bar and Range chart are not in control.

The process is in control because at least one chart is in control.