# Quality Control/Probability Problem

#### accstats

##### New Member
A production engineer has designed a system to remove sugars from a liquid product. Each product must contain an amount of sugar less than or equal to 12g. The engineer randomly samples 3 products and finds the sugar contents to be 4.5, 3.7, and 3.2g respectively. The sample mean is 3.8g and the sample standard deviation is 0.6557. How certain can the engineer be that no product in the population of products will contain a sugar content greater than 12g?

#### Karabiner

##### TS Contributor
Hello, and welcome to Talkstats! This seems like a homework problem. Our homework policy you can find here.

#### accstats

##### New Member
Hello, and welcome to Talkstats! This seems like a homework problem. Our homework policy you can find here.
This is not a homework problem. I created this hypothetical example to help depict a real world problem.

I am looking at this problem from a quality control perspective with the goal being to ensure that a certain parameter of a given product will not exceed a given limit. I am just not sure how to properly express how confident I am that none of the products will exceed that limit.

I had considered a 95% CI, however that will only give me the range of values in which the population mean can be expected to be found with 95% confidence. This is unhelpful since the degree to which products deviate from the population mean could lead to some unknown number of products exceeding my parameter limit.

Another approach I had considered was somehow performing a 95% CI for the standard deviation (not sure how to do that), taking the upper value from that CI and adding it to the upper value for the mean 95% CI to account for both mean and standard deviation uncertainty. I found no precedent for this technique, however.

Finally, I came across the Empirical Rule and had thought that finding the range of values across 2 standard deviations of the mean would allow me to be 95% certain that none of my products would fall outside that range. What cause me to doubt this approach is that I would be using my calculated sample mean and standard deviation rather than the population mean and standard deviation.

Ultimately, the aim of all of these approaches would be to just give me a range of values which I could be 95% certain none of my products would fall out of (which is fine), but what I would really like to be able to do is point to a process and say, "I can say with X% certainty that this none of the products made using this process will exceed the maximum limit." (12g is the maximum limit in the example I conjured up in my original post)

#### Karabiner

##### TS Contributor
So your sample size really is n=3?
what I would really like to be able to do is point to a process and say, "I can say with X% certainty that this none of the products made using this process will exceed the maximum limit."
A question for the Bayesian approach, IMHO, but unfortunately
beyond my skills.

With kind regards

Karabiner

#### katxt

##### Active Member
A "tolerance interval" will let you make statements like "we can be 95% sure that 99% of the product is less than 12g" but you'll need a lot more data.
kat