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Answer :
Based on the calculations, the process is not considered capable according to the Cpk value.
To calculate the Cpk (Process Capability Index), we need to gather the necessary data and perform a series of calculations. Let's go step by step:
Step 1: Calculate the standard deviation for the 12-ounce chocolate bar:
The variance for a single 12-ounce bar is (0.80 grams)^2 = 0.64 grams^2.
Therefore, the standard deviation for a single 12-ounce bar is sqrt(0.64 grams^2) = 0.8 grams.
Step 2: Calculate the standard deviation for the six 1-ounce chocolate bars:
Since the variance for a single 1-ounce bar is six times that of the 12-ounce bar, the variance for a single 1-ounce bar is 6 * 0.64 grams^2 = 3.84 grams^2.
Therefore, the standard deviation for a single 1-ounce bar is sqrt(3.84 grams^2) = 1.96 grams.
Step 3: Convert the specifications for the six 1-ounce bars to grams:
The lower specification is 160 grams and the upper specification is 180 grams.
Step 4: Calculate the process mean:
The process mean is equal to the target specification, which is (160 grams + 180 grams) / 2 = 170 grams.
Step 5: Calculate the process variation:
The process variation is equal to the sum of the variances of the individual bars, which is (6 * 1.96 grams)^2 = 138.24 grams^2.
Step 6: Calculate the process standard deviation:
The process standard deviation is equal to the square root of the process variation, which is sqrt(138.24 grams^2) = 11.75 grams.
Step 7: Calculate the process capability:
Cpk is calculated using the formula: Cpk = min((USL - process mean) / (3 * process standard deviation), (process mean - LSL) / (3 * process standard deviation)).
In this case, the upper specification limit (USL) is 180 grams and the lower specification limit (LSL) is 160 grams.
Cpk = min((180 grams - 170 grams) / (3 * 11.75 grams), (170 grams - 160 grams) / (3 * 11.75 grams))
= min(10 grams / (3 * 11.75 grams), 10 grams / (3 * 11.75 grams))
= min(0.283, 0.283)
= 0.283
Step 8: Determine process capability:
A process is considered capable if Cpk is greater than or equal to 1. A Cpk value of 0.283 indicates that the process capability is less than 1.
Therefore, based on the calculations, the process is not considered capable according to the Cpk value.
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