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Answer :
The product standard deviation control limits for the M observations are approximately 20.76 (LCL) and 44.51 (UCL).The none of the samples are out of control based on the given data.
To find the control limits for M observations and identify any samples that are out of control, to calculate the mean (x) and standard deviation (S) of the given data. The control limits are typically set at ±3 standard deviations from the mean.
Given the data: 36.3, 28.6, 32
Calculate the mean (x):
x = (36.3 + 28.6 + 32) / 3 = 97.9 / 3 = 32.6333
Calculate the standard deviation (S):
S = √((∑(Xi - x)²) / (n - 1))
= √(( (36.3 - 32.6333)² + (28.6 - 32.6333)² + (32 - 32.6333)² ) / (3 - 1))
= √((14.684 + 16.304 + 0.4056) / 2)
= √(31.3936 / 2)
= √(15.6968)
≈ 3.959
Calculate the control limits:
Upper Control Limit (UCL) = x + 3S
= 32.6333 + (3 ×3.959)
≈ 32.6333 + 11.877
≈ 44.51
Lower Control Limit (LCL) = x - 3S
= 32.6333 - (3 × 3.959)
≈ 32.6333 - 11.877
≈ 20.76
Identify samples that are out of control:
To determine if a sample is out of control, we compare the hardness measurements to the control limits.
The given data points are: 36.3, 28.6, 32.
Checking each data point:
36.3 is within the control limits (20.76 to 44.51).
28.6 is within the control limits (20.76 to 44.51).
32 is within the control limits (20.76 to 44.51).
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