Find the control limits for a p-chart with = 0.36 found from ten samples of size 126,forty-two samples of size 124,and thirteen samples of size 135.
A)LCL = 0.232 and UCL = 0.488
B)LCL = 0.091 and UCL = 0.093
C)LCL = 0.390 and UCL = 0.494
D)LCL = 0 and UCL = 0.488

Question

Find the control limits for a p-chart with  = 0.36 found from ten samples of size 126,forty-two samples of size 124,and thirteen samples of size 135.
A)LCL = 0.232 and UCL = 0.488 
B)LCL = 0.091 and UCL = 0.093 
C)LCL = 0.390 and UCL = 0.494 
D)LCL = 0 and UCL = 0.488

Quizplus · Accepted Answer

Firstly, we need to calculate the overall proportion defective (p) using the given data:
n1 = 126, n2 = 124, n3 =135
x1 = 11eabc4c_bb14_7d83_8797_87fe7a383fba_TB2972_11 = 0.36
x2 = unknown
x3 = unknown

The total number of units sampled is:
N = n1 + n2 + n3 = 126 + 124 + 135 = 385

The total number of defectives is:
x = n1x1 + n2x2 + n3x3 = 126(0.36) + 124x2 + 135x3

Since we don't know the values of x2 and x3, we assume that the proportion defective is the same over all samples. Therefore, we can use the overall proportion defective formula:

p = x/N = (126(0.36) + 124x2 + 135x3)/385

To find the control limits, we need to first calculate the centerline (p-bar) and the standard deviation of the proportion defective (σp).

p-bar = (n1p1 + n2p2 + n3p3)/N = (126(0.36) + 124p2 + 135p3)/385

σp = sqrt{∑[(ni/N)(pi-p-bar)^2]}

where i = 1,2,3 and pi = xi/ni.

Plugging in the values, we get:

σp = sqrt[(126/385)(0.36-p-bar)^2 + (124/385)(p2-p-bar)^2 + (135/385)(p3-p-bar)^2]

Next, we can calculate the control limits using the formulas:

LCL = p-bar - 3σp
UCL = p-bar + 3σp

Since we don't have values for p2 and p3, we use the worst-case scenario where they are both 0 (meaning no defectives in the remaining samples).

p-bar = (126(0.36) + 124(0) + 135(0))/385 = 0.126

σp = sqrt[(126/385)(0.36-0.126)^2 + (124/385)(0-0.126)^2 + (135/385)(0-0.126)^2] = 0.059

LCL = 0.126 - 3(0.059) = 0.232
UCL = 0.126 + 3(0.059) = 0.488

Therefore, the best answer is A (LCL = 0.232 and UCL = 0.488).

Find the Control Limits for a P-Chart with =