Question

A pizza restaurant monitors the size (measured by the diameter) of the 10-inch pizzas that it prepares. Pizza crusts are made from doughsthat are prepared and prepackaged in boxes of 15 by a supplier. Doughsare thawed and pressed in a pressing machine. The toppings are added, and the pizzas are baked. The wetness of the doughsvaries from box to box, and if the dough is too wet or greasy, it is difficult to press, resulting in a crust that is too small. The first shift of workers begins work at 4 P.M., and a new shift takes over at 9 P.M. and works until closing. The pressing machine is readjusted at the beginning of each shift. The restaurant takes five consecutive pizzas prepared at the beginning of each hour from opening to closing on a particular day. The diameter of each baked pizza in the subgroups is measured, and the pizza 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 and Rcharts 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 statistical control? Explain.

e. Is the X bar chart for the pizza crust data in statistical control? If not, use the X bar chart and the information given with the data to try to identify any assignable causes that might exist.

f. Suppose that, based on the X bar chart, the manager of the restaurant decides that the employees do not know how to properly adjust the dough pressing machine. Because of this, the manager thoroughly trains the employees in the use of this equipment. Because an assignable cause (incorrect adjustment of the pressing machine) has been found and eliminated, we can remove the subgroups affected by this unusual process variation from the data set. We therefore drop subgroups 1 and 6 from the data. Use the remaining eight subgroups to show that we obtain revised center lines of X bar = 10.2225 and R bar = 0.825.

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

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

Answer 1

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

1. First we need to find average and range of the all 5 observations

Obervations					Sample mean	Range
9.8	9	9	9.2	9.2	9.24	0.8
9.5	10.3	10.2	10	10	10	0.8
10.3	10.5	9.8	10	10.3	10.18	0.7
10.7	9.5	9.8	10	10	10	1.2
10	10.5	10	10.5	10.3	10.26	0.5
10	9	9	9.2	9.3	9.3	1
11	10	10.3	10.3	10	10.32	1
10	10.2	10.1	10.3	11	10.32	1
10	10.4	10.4	10.5	10	10.26	0.5
11	10.5	10.1	10.2	10.2	10.4	0.9

sample mean formula : =AVERAGE(I6:M6) "in each observation row"

range formula : "=MAX(I6:M6)-MIN(I6:M6)"

2. We have 10 rows of observations so we have to refer table of control chart of constants and update A2,D3 and D4 to prepare chart

3.

4. Formula reference:

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

Refer the above snap and answer..

X-bar chart:

center lie : Mean = 10.03

UCL : 10.29

LCL: 9.77

Rchart:

Center line: R-bar : 0,84

UCL: 1.49

LCL: 0.187

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

yes ,R chart for the pizza crust data is fully  in statistical control

e. Is the X bar chart for the pizza crust data in statistical control? If not, use the X bar chart and the information given with the data to try to identify any assignable causes that might exist.

No X-bar chart is not in control.

Refer the x-bar chart.. observation 10 is above the upper control limit

f. Suppose that, based on the X bar chart, the manager of the restaurant decides that the employees do not know how to properly adjust the dough pressing machine. Because of this, the manager thoroughly trains the employees in the use of this equipment. Because an assignable cause (incorrect adjustment of the pressing machine) has been found and eliminated, we can remove the subgroups affected by this unusual process variation from the data set. We therefore drop subgroups 1 and 6 from the data. Use the remaining eight subgroups to show that we obtain revised center lines of X bar = 10.2225 and R bar = 0.825.

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

yes both x-bar and R-chart is control