Question

Processing new accounts at a bank is intended to average 10 minutes each. Five samples of four observations each have been taken. Use the sample data below, construct a X-bar and a R (range) chart.

	Sample 1	Sample 2	Sample 3	Sample 4	Sample 5
	10.2	10.3	9.7	9.9	9.8
	9.9	9.8	9.9	10.3	10.2
	9.8	9.9	9.9	10.1	10.3
	10.1	10.4	10.1	10.5	9.7
Total	40	40.4	39.6	40.8	40
Mean	10	10.1	9.9	10.2	10	X= 10.4
Range	0.4	0.6	0.4	0.6	0.6	R= 0.52

1. Determine the mean and range of each sample in above table.

Mean : 10.0+10.1++9.9+10.2+10/5=10.4

Range : 0.4+0.6+0.4+0.6+0.6+0.6/5=0.52

2. Compute the average mean (X) and average range (R ) in above table.

3. If the standard deviation of population is known as 0.25, compute upper and lower limits for a X chart.

4. If the standard deviation of population is unknown, compute upper and lower limits for a X chart. (Obtain the Factor from the Factor table and use it in calculation)

5. Compute the upper and lower limits for a R chart. (Obtain the Factor from the Factor table and use it in calculation)

6. Construct (Draw) a X chart and R chart for this process with three-sigma limits. Is the process in control?

Answer 1

Q - Compute the average mean (X) and average range (R ) in above table.

Answer –

Average mean (X) = Sum of all sample means / Number of samples

= (10.0+10.1++9.9+10.2+10) / 5 = 10.4

Average Range (R) = Sum of all sample range / Number of samples

= (0.4+0.6+0.4+0.6+0.6+0.6) / 5 = 0.52

Q - If the standard deviation of population is known as 0.25, compute upper and lower limits for a X chart.

Answer –

Upper control limit (UCL) X-bar = Mean + [z (standard deviation / √ n)]

Here, Mean = 10.4

Z = 3 (3-sigma limits)

Standard deviation = 0.25

n = sample size = 4

UCL = 10.4 + [3(0.25/√ 4)]

= 10.4 + [3(0.25/2)]

= 10.4 + 0.375

= 10.775

Lower Control Limit (LCL) X-bar = Mean - [z (standard deviation / √ n)]

LCL = 10.4 - [3(0.25/√ 4)]

= 10.4 - [3(0.25/2)]

= 10.4 - 0.375

= 10.025

Q - If the standard deviation of population is unknown, compute upper and lower limits for a X chart. (Obtain the Factor from the Factor table and use it in calculation)

Answer –

By obtaining factors from control chart constant/factor table for n = 4 we have A2 = 0.73

Upper control limit (UCL) X-bar = Average Mean + (A2 X Average Range)

  = 10.04 + (0.73 ∗ 0.52) = 10.42

Lower Control Limit (LCL) X-bar = Average Mean - (A2 X Average Range)

= 10.04 – (0.73 ∗ 0.52) = 9.66

Q - Compute the upper and lower limits for a R chart. (Obtain the Factor from the Factor table and use it in calculation)

Answer-

By obtaining factors from control chart constant/factor table for n = 4 we have:

D3 = 0

D4 = 2.28

Upper control limit (UCL) R-chart = D4 x Average Range

= 2.28 ∗ 0.52 = 1.19

Lower Control Limit (LCL) R-chart = D3 x Average Range

= 0 ∗ 0.52 = 0

As the sample mean and range are within their control limits, process is under control.