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
Cp = process capable
b)Compute and interpret the machine Cpk
Cpk = capability index
c)What % of the bottles will be considered under-filled?
I got a lower distribution limit of -1.67 and upper distribution limit of 5.
How can I find area corresponding to Z=5 on the Standardized Normal Distribution Table. I need to calculate the % of bottles that will be considered under-filled?
To find the area corresponding to Z = 5 on the Standardized Normal Distribution Table, you can follow these steps:
1. Look up the Z-score of 5 in the Standardized Normal Distribution Table. It is important to note that most Standardized Normal Distribution Tables only go up to a Z-score of 3.99. Z-scores beyond that value are extremely rare and usually considered outliers. Thus, a Z-score of 5 will be exceptionally unusual.
2. If you still want to estimate the area corresponding to Z = 5, you can make use of the properties of the Normal Distribution curve. The total area under the curve is 1 (or 100%). Since the Normal Distribution is symmetric around the mean, the area to the left of the mean is 0.5 (or 50%). Therefore, the area to the right of the mean is also 0.5 (or 50%).
3. Since Z = 5 is so far from the mean, it represents an extreme and unlikely event. Therefore, the area under the curve corresponding to Z = 5 (or any value beyond Z = 3.99) is essentially zero. Thus, you can assume that the percentage of bottles considered under-filled based on a Z-score of 5 is negligible.
If you need a more accurate estimate for the percentage of bottles that will be considered under-filled, you should consider using the specification limits given in the question. The lower specification limit is 15.8 ounces, which corresponds to a Z-score of -0.17 (calculated as (15.8 - 15.9) / 0.06). The upper specification limit is 16.2 ounces, which corresponds to a Z-score of 0.33 (calculated as (16.2 - 15.9) / 0.06).
To find the percentage of bottles that will be considered under-filled, you need to find the area to the left of the lower specification limit (Z = -0.17) and to the right of the upper specification limit (Z = 0.33), and then subtract these two values from 100%. You can use the Standardized Normal Distribution Table or a statistical calculator to find these values and perform the calculation.
Keep in mind that these calculations assume a normal distribution of the bottle filling process.