Consider a machine that fills soda bottles. The process has a mean of 15.9 ounces and a standard deviation of 0.06 ounces.
The specification limits are set between 15.8 and 16.2 ounces.
a)Compute and interpret the machine’s Cp
b)Compute and interpret the machine Cpk
c)What % of the bottles will be considered under-filled?
I don't know what "Cp" and "Cpk" mean. It would help to spell out these concepts. However,
Z = (score-mean)/SD
Find table in the back of your statistics text labeled something like "areas under normal distribution" to find the proportions related to the Z scores.
For example:
c) Z = (15.8-15.9)/.06
To compute the machine's Cp and Cpk, we need to calculate the process capability indices. These indices provide a measure of how well the process is able to meet the specification limits.
a) To compute Cp, we use the formula:
Cp = (USL - LSL) / (6 * standard deviation), where USL is the upper specification limit and LSL is the lower specification limit.
In this case, USL = 16.2 ounces and LSL = 15.8 ounces. The standard deviation is given as 0.06 ounces.
Cp = (16.2 - 15.8) / (6 * 0.06)
Cp = 0.4 / 0.36
Cp = 1.11
Interpretation:
Cp describes the potential capability of the process to meet the specifications. A Cp value greater than 1 indicates that the process has the potential to meet the specification limits.
b) To compute Cpk, we use the formula:
Cpk = min((USL - mean) / (3 * standard deviation), (mean - LSL) / (3 * standard deviation)), where mean is the process mean.
Cpk = min((16.2 - 15.9) / (3 * 0.06), (15.9 - 15.8) / (3 * 0.06))
Cpk = min(0.5 / 0.18, 0.1 / 0.18)
Cpk = min(2.78, 0.56)
Cpk = 0.56
Interpretation:
Cpk measures the actual capability of the process to meet the specifications, taking into account any deviation from the target mean. A Cpk value greater than 1 indicates that the process is capable of meeting the specifications.
c) To calculate the percentage of bottles considered under-filled, we need to determine the proportion of the distribution that falls below the lower specification limit (LSL).
Using a standard normal distribution table or statistical software, we can find the z-score corresponding to the LSL and then find the area to the left of that z-score. This area represents the percentage of bottles below the LSL.
For example, assuming a normal distribution, a z-score table shows that a z-score of -0.87 corresponds to approximately 19.24% in the lower tail of the distribution. Therefore, approximately 19.24% of the bottles will be considered under-filled.
Note: The normal distribution assumption may need to be verified using actual data or a larger sample size.