Stats homework please help!

Question

Suppose that a customer wants us to manufacture a product that has a lower spec limit of 59.5 and an upper spec limit of 60.5. What do we have to lower the process standard deviation to in order to be in spec most of the time? (3 decimal places)

Answer 1

Let's assume the production variations are a normal distribution around the mean. If their specs call for a range of 59.5 to 60.5, then we can also assume the mean is right in between - 60.

So what the question is asking is how small do you have to make the standard deviation to make sure that "most" of the products produced are within .5 of that mean. Remember that in a normal distribution, we can use the empiricle rule to estimate how much of the population falls between standard deviation ranges.

68 percent of a population are within 1 standard deviation of the mean (i.e. between -1 standard deviation and +1 standard deviation). 95 percent are within 2 standard deviations, and 99.7 percent are within 3 standard deviations. Any of these would qualify as "most," but usually in a business where a product needs to be within a spec, we want 2 or more standard deviations. In fact, the famous quality control methodology known as 6 sigma is named for the 3 standard deviation rule - sigma is the symbol used for standard deviation of a population.

So lets say we want 99.7 percent of the product to be within 59.5 to 60.5. That means we want the standard deviation to be such that 3 standard deviations away from the mean (60) is plus or minus .5. So the standard deviation has to be 0.5 / 3 or about 0.1667. If you only needed 95 percent to be in the range, you would only need 2 standard deviations, so one standard deviation would be 0.5 / 2 or 0.25.

So, to reiterate, if you lower the standard deviation of your production to 0.1667, then 59.5 and 60.5 would each be 3 standard deviations away from the mean, and thus 99.7 percent of your product would be within spec.

Answer 2

This is a little tricky because the problem tells us they want parts in spec "most of the time". I would say that 95% of the time is most of the time. If we use that as our definition for "most of the time" and realize that 2 standard deviations make up 95% of all parts, then the standard deviation would be the (max - mean)/2 = 0.5/2

That means a mean of 60 and standard deviation of 0.250

If 95% is not enough, and we want to say we need say good parts 99.7% of the time, then we need to use 3 standard deviations meaning the mean is 60 and the standard deviation is 0.167 (0.5/3)