The chocolate manufacturer makes chocolate candies, including a 12-ounce chocolate bar (340 grams) and six 1-ounce chocolate bars (170 grams).

b) The machine that fills the bar molds for the 1-ounce bars has a standard deviation of 0.80 grams. The filling is set to deliver 1.01 ounces per bar. Specifications for the six-bar pack are 160 to 180 grams.

Calculate the Cpk. Is the process capable?

Hint: The variance for a six-bar pack is equal to six times the bar variance; 1 ounce = 28.33 grams.

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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