## Answers

is the mean of all the sample means. Hence we have as below

= (9.402+9.404+9.391+9.408+9.399+9.401+9.403+9.403+9.393+9.401+9.401+9.406)/12 = 112.812/12 = **9.401**

**Hence = 9.401 inches**

Also, is the mean of all the sample ranges. Hence is as below

= (0.033+0.041+0.034+0.051+0.031+0.036+0.041+0.034+0.027+0.029+0.039+0.047)/12 = 0.443/12 = **0.0369**

**Hence is 0.0369 inches**

Now for the x-bar chart the control limits are given as per below formula

Control Limits = () A_{2}*()

It is given that the sample size is 5. From the given table, for sample size of 5, A_{2} is 0.577

Hence we have the below control limits

UCL_{x} = 9.401 + 0.577*0.0369 = 9.401 + 0.0213 = **9.4223 inches**

LCL_{x} = 9.401 - 0.577*0.0369 = 9.401 - 0.0213 = **9.3797 inches**

We can see that all the sample means lie within the control limits. Hence we have the below

**Based on x-chart, is one or more samples beyond the control limits : No**

For the given data, is **0.0369** as calculated above

The control limits for R-chart are calculated below

LCL = D_{3}*

UCL = D_{4}*

From the given table for sample size of 5, D_{3} = 0 and D_{4} = 2.115

Hence we have the control limits as below

LCL = 0*0.0369 = 0 inches

UCL = 2.115*0.0369 = 0.0780 inches

**Hence UCL _{R} = 0.0780 inches and LCL_{R} = 0 inches**

We can see that all the sample ranges lie within the control limits.

Hence we have the below

**Based on R-chart, is one or more samples beyond the control limits : No**

is the mean of all the sample means. Hence we have as below

= (9.402+9.404+9.391+9.408+9.399+9.401+9.403+9.403+9.393+9.401+9.401+9.406)/12 = 112.812/12 = **9.401**

**Hence = 9.401 inches**

Also, is the mean of all the sample ranges. Hence is as below

= (0.033+0.041+0.034+0.051+0.031+0.036+0.041+0.034+0.027+0.029+0.039+0.047)/12 = 0.443/12 = **0.0369**

**Hence is 0.0369 inches**

Now for the x-bar chart the control limits are given as per below formula

Control Limits = () A_{2}*()

It is given that the sample size is 5. From the given table, for sample size of 5, A_{2} is 0.577

Hence we have the below control limits

UCL_{x} = 9.401 + 0.577*0.0369 = 9.401 + 0.0213 = **9.4223 inches**

LCL_{x} = 9.401 - 0.577*0.0369 = 9.401 - 0.0213 = **9.3797 inches**

We can see that all the sample means lie within the control limits. Hence we have the below

**Based on x-chart, is one or more samples beyond the control limits : No**

For the given data, is **0.0369** as calculated above

The control limits for R-chart are calculated below

LCL = D_{3}*

UCL = D_{4}*

From the given table for sample size of 5, D_{3} = 0 and D_{4} = 2.115

Hence we have the control limits as below

LCL = 0*0.0369 = 0 inches

UCL = 2.115*0.0369 = 0.0780 inches

**Hence UCL _{R} = 0.0780 inches and LCL_{R} = 0 inches**

We can see that all the sample ranges lie within the control limits.

Hence we have the below

**Based on R-chart, is one or more samples beyond the control limits : No**

is the mean of all the sample means. Hence we have as below

= (9.402+9.404+9.391+9.408+9.399+9.401+9.403+9.403+9.393+9.401+9.401+9.406)/12 = 112.812/12 = **9.401**

**Hence = 9.401 inches**

Also, is the mean of all the sample ranges. Hence is as below

= (0.033+0.041+0.034+0.051+0.031+0.036+0.041+0.034+0.027+0.029+0.039+0.047)/12 = 0.443/12 = **0.0369**

**Hence is 0.0369 inches**

Now for the x-bar chart the control limits are given as per below formula

Control Limits = () A_{2}*()

It is given that the sample size is 5. From the given table, for sample size of 5, A_{2} is 0.577

Hence we have the below control limits

UCL_{x} = 9.401 + 0.577*0.0369 = 9.401 + 0.0213 = **9.4223 inches**

LCL_{x} = 9.401 - 0.577*0.0369 = 9.401 - 0.0213 = **9.3797 inches**

We can see that all the sample means lie within the control limits. Hence we have the below

**Based on x-chart, is one or more samples beyond the control limits : No**

For the given data, is **0.0369** as calculated above

The control limits for R-chart are calculated below

LCL = D_{3}*

UCL = D_{4}*

From the given table for sample size of 5, D_{3} = 0 and D_{4} = 2.115

Hence we have the control limits as below

LCL = 0*0.0369 = 0 inches

UCL = 2.115*0.0369 = 0.0780 inches

**Hence UCL _{R} = 0.0780 inches and LCL_{R} = 0 inches**

We can see that all the sample ranges lie within the control limits.

Hence we have the below

**Based on R-chart, is one or more samples beyond the control limits : No**