A pizza restaurant monitors the size (measured by the diameter)of the 10-inch pizzas that it prepares. Pizza crusts are made fromdoughsthat are prepared and prepackaged in boxes of 15 by asupplier. Doughsare thawed and pressed in a pressing machine. Thetoppings are added, and the pizzas are baked. The wetness of thedoughsvaries from box to box, and if the dough is too wet orgreasy, it is difficult to press, resulting in a crust that is toosmall. The first shift of workers begins work at 4 P.M., and a newshift takes over at 9 P.M. and works until closing. The pressingmachine is readjusted at the beginning of each shift. Therestaurant takes five consecutive pizzas prepared at the beginningof each hour from opening to closing on a particular day. Thediameter of each baked pizza in the subgroups is measured, and thepizza crust diameters obtained are given in Table.

Subgroup	Pizza crust diameter in inches
	Time	1	2	3	4	5
1	4 pm	9.8	9.0	9.0	9.2	9.2
2	5 pm	9.5	10.3	10.2	10.0	10.0
3	6 pm	10.3	10.5	9.8	10.0	10.3
4	7 pm	10.7	9.5	9.8	10.0	10.0
5	8 pm	10.0	10.5	10.0	10.5	10.3
6	9 pm	10.0	9.0	9.0	9.2	9.3
7	10 pm	11.0	10.0	10.3	10.3	10.0
8	11 pm	10.0	10.2	10.1	10.3	11.0
9	12 am	10.0	10.4	10.4	10.5	10.0
10	1 am	11.0	10.5	10.1	10.2	10.2

Use the pizza crust diameter data to do the following:

a. Show that X bar =10.028 and R bar = 0.84.

b. Find the center lines and control limits for the X bar andRcharts for the pizza crust data.

c. Set up the X bar and Rcharts for the pizza crust data.

d. Is the R chart for the pizza crust data in statisticalcontrol? Explain.

e. Is the X bar chart for the pizza crust data in statisticalcontrol? If not, use the X bar chart and the information given withthe data to try to identify any assignable causes that mightexist.

f. Suppose that, based on the X bar chart, the manager of therestaurant decides that the employees do not know how to properlyadjust the dough pressing machine. Because of this, the managerthoroughly trains the employees in the use of this equipment.Because an assignable cause (incorrect adjustment of the pressingmachine) has been found and eliminated, we can remove the subgroupsaffected by this unusual process variation from the data set. Wetherefore drop subgroups 1 and 6 from the data. Use the remainingeight subgroups to show that we obtain revised center lines of Xbar = 10.2225 and R bar = 0.825.

g. Use the revised values of and to compute revised and R chartcontrol limits for the pizza crust diameter data. Set up X bar andR charts using these revised limits. Be sure to omit subgroup meansand ranges for subgroups 1 and 6 when setting up these charts.

h. Has removing the assignable cause brought the process intostatistical control? Explain.