Question

A farmer focusing on the production of eco-friendly chicken eggs collects the following data about his output. In a sample of 50 eggs, the farmer finds the average egg to weigh 47 grams. The standard deviation of the egg weight is 2 grams and the distribution of weights resembles a normal distribution reasonably closely. The farmer can sell the eggs to a local distributor. However, they have to be in the interval between 44 grams and 50 grams (i.e., the lower specification limit is 44 grams and the upper specification limit is 50 grams). What is the capability score of the eco-friendly chicken egg operation? What percentage of the produced eggs fall within the specification limits provided by the local distributor? By how much would the farmer have to reduce the standard deviation of the operation if his goal were to obtain a capability score of Cp=2/3 (i.e., get 4.5% defects)?

Answer 1

Given	Average of data	47.000
	Standard deviation of Data	2.0000
	Upper Specification Limit (USL)	50.000
	Lower Specification Limit (LSL)	44.000
	Target Value	47.000
Part a	Process Capability score: Cp = (USL – LSL)/6σ	Cp = (50 – 44)/(6*2) = 0.500
Ans - a: The process capability score of the process is 0.500
Part b	Fraction of output within specification = output less than 50 and more than 44 Percentage of output within specification = P(LSL > X < USL) = P(X < USL) - P (X < LSL)
	P(X < LSL) = P( X < 44) z-score for X44 = z44 = (X – µ)/σ Z44= (44 – 47)/2 = -1.5	From excel, P(z <-1.5) = (=NORMDIST(-1.5)) = 0.0668 P(z <LSL) = 0.0668
	P(X < USL) = P( X < 50) z-score for X50 = z50 = (X – µ)/σ Z44= (50 – 47)/2 = 1.5	From excel, P(z <1.5) = (=NORMDIST(1.5)) = 0.9332 P(z <USL) = 0.9332
	P(LSL > X < USL) = P(X < USL) - P (X < LSL)	P(44 > X < 50) = 0.9332 – 0.0668 = 0.8664
Ans b.	Percentage of output within specification = 86.64%
Part c.	If the target capability is 2/3, then standard deviation (σ) will changed
	Cp = Cp = (USL – LSL)/6σ = 2/3	(50 – 44)/(6σ) = 2/3 σ
Ans - c	If the target capability is 2/3, then standard deviation (σ) should be reduced from 2 to 1.5