Note: The 10 samples referred to in the question are in the table contained in the file attached.
Answer:
Upper Control Limit for mean chart = 11.62
Lower Control Limit for mean chart = 7.98
Upper Control Limit for range chart = 8.88
Lower Control Limit for range chart = 0
The process is in control
Step-by-step explanation:
Step 1: Calculate the mean [tex]\bar{X}[/tex] and the range R for each of the 10 samples.
The [tex]\bar{X}[/tex] and R of each sample are in each column of the table in the second file attached.
e.g. For sample 1,
[tex]\bar{X} = (11+9+9+10+12)/5 = 10.2\\R = highest no. - smallest no. = 12-9 = 3[/tex]
Step 2: Calculate [tex]\bar{R}[/tex] and [tex]\bar{\bar{X}}[/tex]
[tex]\bar{R} = (3+3+6+2+3+5+4+5+6+5)/10 = 4.2[/tex][tex]\bar{\bar{X}} = (10.2+9.4+9.8+9.6+10.6+10.6+9+9.4+9.4+10)/10 = 9.80[/tex]Step 3: Calculate Standard Deviation
If z = 3, [tex]\sigma_{x} = 1.36[/tex], n = 5
Standard deviation of [tex]\bar{X}[/tex], [tex]\sigma_{\bar{x}} = \frac{\sigma_{x} }{\sqrt{n} }[/tex]
[tex]\sigma_{\bar{x}} = \frac{1.36 }{\sqrt{5} }[/tex] = 0.6082
Step 4: Calculate the Control limits for the mean chart:
Upper Control Limit(UCL)
[tex]UCL = \bar{\bar{X}} + z* \sigma_{\bar{x}}\\UCL = 9.8+(3*0.6082) = 11.62[/tex]
Lower Control Limit (LCL)
[tex]LCL = \bar{\bar{X}} + z* \sigma_{\bar{x}}\\LCL = 9.8-(3*0.6082) = 7.98[/tex]
All the [tex]\bar{X}[/tex] for the 10 samples fall within the LCL and UCL for the mean chart
Step 5: Calculate the Control limits for the range chart:
Upper control limit (UCL)
[tex]UCL = \bar{R} * D_{4}[/tex]
[tex]D_4 = 2.114[/tex] for n = 5 (from the control chart constant table)
UCL = 2.114 * 4.2 = 8.88
Lower control limit (LCL)
[tex]LCL = \bar{R} * D_3[/tex]
[tex]D_3 = 0[/tex] for n = 5 (from the control chart constant table)
LCL = 4.2*0 = 0
All the for the 10 samples fall within the LCL and UCL for the range chart
Since the average for the 10 samples fall within the LCL and UCL for both the mean and range charts, the process is in control.
