Question

Question 3 

The time to replace vehicle wiper blades at a service center was monitored using a mean and a range chart. Six samples of n = 20 observations were obtained, and the sample means and ranges computed: 

Sample Mean Range Sample Mean Range 3.06 .42 4 3.13 46 3.15 .50 3.06 46 3.11 41 3.09 45 

Using the factors in the provided table, determine upper and lower limits for mean and range charts. Is the process in control?

Answer 1

Control Charts are graphs that show if a sample is within statistical control limits. Here, control limits are the upper and lower bands of a control chart. Control charts have two basic purposes: 1. To establish the control limits for a process 2. To monitor the process to indicate when it is out of control. X-bar charts are used to check the productivity of the process with respect to its average performances and predetermined targets. R-charts are used to check and monitor the variability in the process.

To prepare y chart and R chart, begin by calculating the average mean and average range of the samples, as shown below: Α. B C 1 Sample Mean Range 2 1 3.06 0.42 3 2 3.15 0.5 4 3 3.11 0.41 5 4 3.13 0.46 6 5 3.06 0.46 76 3.09 0.45 8 Average =AVERAGE(B2:B7) =AVERAGE(C2:07)

A B C Sample Mean Range 1 3.06 0.42 3 2 3.15 0.50 3.11 0.41 5 4 3.13 0.46 3.06 0.46 6 3.09 0.45 8 Average 3.1000.450 Thus, * = 3.100 and R = 0.450

The factors for 3 Sigma control limits for and R chart for n = 20 are 4, 0.18 D, = 0.41 D. = 1.59 The upper and lower controllimits for x chart are calculated using the following formulae Upper Control Limit = x + A R - 3.100+0.18x(0.450) - 3.100+0,081 UCL = 3.181 Lower Control Limit = x - AR = 3.100-0.18x(0.450) -- 3.100-0,081 Cl = 3.019) The upper and lower control limits of x chart are 3.181 and 3.019 respectively.

The upper and lower control limits for R chart are calculated using the following formulae Upper Control Limit = D.R = 1.590.45 UCL = 0.7155 Lower Control Limit = D,R = 0.410.45 LCL = 0.1845 The upper and lower control limits of R chart are 0.7155 and 0.1845 respectively.

To determine whether the process is in control or not, construct the X and R chart in Excel by following the below steps: Enter the control limits in Excel as shown below I E F I J M Sample G H X-bar chart UCL LCL 3.181 3.019 3.181 3.019 3.181 3.019 3.181 3.019 3.1813.019 3.181 3.019 Mean 3.06 3.15 3.11 3.13 3.06 3.09 5 6 7 K L R- chart UCL LCL 0.7155 0.1845 0.7155 0.1845 0.7155 0.1845 0.7155 0.1845 0.7155 0.1845 0.7155 0.1845 3 4 5 Range 3.100 0.42 3.100 0.50 3.100 0.41 3.100 0.46 3.1000 .46 3.100 0.45 R 0.450 0.450 0.450 0.450 0.450 0.450 Select the cell range F3:18. Go to the Insert Tab and choose the line chart from the options

The chart is as follows: - UCL - 3.181 1-3.100 LCL - 3.019 Figure 1: X-bar Chart One can see that all the data points are within the control limits x chart Thus, the process is in control as per the x chart.

Now, to draw R chart, select the cell range 13. M8. Go to the Insert Tab and choose the line chart from the options The R chart is as follows: 0.90 -- UCL -0.7155 R-0.450 LCL = 0.1845 1 2 5 6 3 4 Figure 2: R Chart One can see that all the points are within the control limits of R-chart.

Hence, one can say that the process is under control both in terms of variability in mean and range of the data.