Question

Canine Gourmet Super Breath dog treats are sold in boxes labeled with a net weight of 12 ounces ​(340 ​grams) per box. Each box contains 8 individual​ 1.5-ounce packets. To reduce the chances of shorting the​ customer, product design specifications call for the​ packet-filling process average to be set at 43.0 grams so that the average net weight per box of 8 packets will be 344 grams. Tolerances are set for the box to weigh 344plus or minus10 grams. The standard deviation for the​ packet-filling process is 1.03 grams. The target process capability ratio is 1.33. One​ day, the ​packet-filling process average weight drifts down to 42.5 grams. Is the packaging process​ capable? Is an adjustment​ needed?

Since the process capability​ index, Upper C Subscript pk​, is

nothing​, the process

▼

is not capable

is capable

. ​(Enter your response rounded to three decimal places​.)

Answer 1

Process Capability details are as follows:

Cp = (USL – LSL)/6σ

Cpk = Min (Upper Cpk, Lower Cpk)

Cpk = Min ([(USL - µ) / (3σ)],[(µ - LSL) / (3σ)])

Specification	344.00 +/- 10 grams
Lower Specification Limit	LSL	344 – 10 = 334
Upper Specification Limit	USL	344 + 10 = 354
Process Mean weight of box	µ (new process mean for packet-filling process is 42.5, thus the mean weight of the box is 8*42.5 = 340 grams)	µ = 340 grams
Since the filling process standard deviation is 1.03, so the standard deviation of the weight of box will be calculated as follows: Variance of the filling process = (std. dev. of process)2 = (1.03)2 = 1.0609 Variance of 8 packets = 8 packets x variance = 8 x 1.0609 = 8.4872 Variance of weight of box = 8.4872 grams per box Standard deviation of box = √variance of weight per box = √8.4872 = 2.9133 grams per box
Standard deviation of weight per box	σ	2.9133
Upper Cpk	Upper Cpk = [(USL - µ) / (3σ)]	Upper Cpk = (354 – 340)/(3*2.9133) Upper Cpk = 1.6018
Lower Cpk	Lower Cpk = [(µ - LSL) / (3σ)]	Lower Cpk = (340 – 334)/(3*2.9133) Lower Cpk = 0.6865
Process Cpk	Cpk = Min (Upper Cpk, Lower Cpk) = Min ([(USL - µ) / (3σ)],[(µ - LSL) / (3σ)])	Cpk = min(1.6018, 0.6865) Cpk = 0.6865

Since the new Process capability index is 0.6865, which is less than 1, the process is not capable to produce the packets within tolerance.