Determination of UCL & LCL for Quality Control Chart

PART-1
ANSWER - A
Given Details
(PR) 1.1
Allowance Percentage 15% of job.
Observation Time (minutes)
1 2.07
2 2.08
3 2.11
4 2.12
5 2.06
6 1.98
7 2.04
8 2.02
9 2.05
10 2.03
11 2.09
Calculations: Formula
Average Observation Time
(OT) 2.059

Normal Time (NT) 2.265 Avg OT * PR Factor
Standard Time (ST) 2.60475 NT (1+ Allowance Factor)
ANSWER - B
Given Details
Times busy Times idle Total observations
1 8 1 9
2 9 2 11
3 10 3 13
4 8 0 8
5 9 2 11
6 11 4 15
7 9 3 12
Calculations:
Sum 64 15 79
Estimated % of Idle Time 0.1899 18.98734177
ANSWER - C
Given Details:
n = required sample size
z = 2.33 for a 98% confidence level
p = estimate of idle proportion 0.1899

h = acceptable error of 5% 0.08
Calculation:
N = (Z^2P(1-P))/H^2
n 130
Excess observation Required 51
ANSWER - D
Given Details:
n = required sample size
z = 2.33 for a 98% confidence level
Observations 100.0000
p = estimate of idle proportion 0.1500
h = acceptable error of 5% 0.05
Calculation:
N = (Z^2P(1-P))/H^2
n 277
Excess observation Required 177
PART-II
Q5A. Quality Control Chart

Answer- A
Observations (Coil spring length, cm), n
Sam-
ple k 1 2 3 4 5 X-BAR R
1 10.2 10 9.94 9.99 9.96 9.982 0.08
2 10.01 10.02 10.07 9.95 9.96 10.002 0.12
3 9.99 10 9.93 9.92 9.99 9.966 0.08
4 10.03 9.91 10 9.98 9.89 9.962 0.14
5 9.95 9.92 10.03 10.05 10.01 9.992 0.13
6 9.97 10.06 10.05 9.96 10.02 10.012 0.1
7 10.05 10.01 10.1 8.96 9.98 9.82 1.14
8 10.09 10.1 10 9.99 10.08 10.052 0.11
9 10.14 10.1 9.99 10.08 10.09 10.08 0.15
10 10.01 9.98 10.08 10.07 9.99 10.026 0.1
99.894 2.15
9.9894 0.215
Observations (Coil spring length, cm), n
Sam-
ple k 1 2 3 4 5 X-BAR R
1 10.2 10 9.94 9.99 9.96 =(SUM(B3:F3))/
5 =B3-D3
2 10.01 10.02 10.07 9.95 9.96 =(SUM(B4:F4))/
5 =D4-E4
3 9.99 10 9.93 9.92 9.99 =(SUM(B5:F5))/
5 =C5-E5
4 10.03 9.91 10 9.98 9.89 =(SUM(B6:F6))/
5 =B6-F6
5 9.95 9.92 10.03 10.05 10.01 =(SUM(B7:F7))/
5 =E7-C7
6 9.97 10.06 10.05 9.96 10.02 =(SUM(B8:F8))/
5 =C8-E8