posted
So my dad called me up with a statistics problem. If I get what he was telling me, someone is trying to pass off statistics that my dad thinks are dicey as valid. I know I should be able to figure this problem out, but my mind's just not getting it, so I wondered if someone else could give me a hand.
From what he told me, the standard they are using is that a test must have no more than a 7% bias (i.e. test values cannot be more than +/- 7% of the actual values) at 95% confidence and tests cannot have more than 7% deviation at 95% confidence. He wants to know, gven these conditions, what the probability that a reading would be 10% off of the actual value is?
This sounds like a basic problem to me, but I don't have any statistics books on hand for reference and I can't actually figure if what I'm doing is right. The percentage thing is really throwing me off.
I set it up as a two problem thing where the results of the first part (the bias thing) are used as the initial measurement for the second test (the deviation). Since we're dealing with percentages, I split the upper and lower problems. In the upper one, I got the upper bound of 95% confidence at u + .07u + (.07(u + .07u)) = 1.1449u. For the lower, I've got the lower bound at u - .07u - (.07(u - .07u)) = .8649u. From there, yeah, I don't really know where to go.
I did the (what I think is obviously wrong) thing of treating the bounds percentages as 2 * the standard deviations for the new problems, so that I've got the upper bound problem being the one-tailed probability of 1.1u with a standard deviation of .1449/2 = .07245 which gave me a probability of .0838 (using the z-charts). The same logic with the lower one gave me a standard deviation of .06755 which gave me a probability of .0694.
Add the two and dividing by two I get a total probability of 7.66%, which matches up with about what I'd figure it would be, but I'm pretty darn sure my steps somewhere are invalid. Could Bob or somebody check this out?
posted
I've heard that 100% of people who actually understand statistics also understand how powerful and important statistics are.
Posts: 15770 | Registered: Dec 2001
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quote:From what he told me, the standard they are using is that a test must have no more than a 7% bias (i.e. test values cannot be more than +/- 7% of the actual values) at 95% confidence and tests cannot have more than 7% deviation at 95% confidence. He wants to know, gven these conditions, what the probability that a reading would be 10% off of the actual value is?
This sounds like a basic problem to me, but I don't have any statistics books on hand for reference and I can't actually figure if what I'm doing is right. The percentage thing is really throwing me off.
I set it up as a two problem thing where the results of the first part (the bias thing) are used as the initial measurement for the second test (the deviation). Since we're dealing with percentages, I split the upper and lower problems. In the upper one, I got the upper bound of 95% confidence at u + .07u + (.07(u + .07u)) = 1.1449u. For the lower, I've got the lower bound at u - .07u - (.07(u - .07u)) = .8649u. From there, yeah, I don't really know where to go.
Okay... this is either a problem that is beyond me (i.e., I've never seen it before and I just packed all my stats books in anticipation of an upcoming move) or I am completely misunderstanding what it is you are asking.
If we can get some clarification, it might help.
1) Your use of the word "bias" You are really saying that the test is designed to give the real value +/- 7% at a 95% confidence level, right? If so, what you have here is a way of expressing the standard error of the estimate (s.e.e.). One way to think of this, I suppose, is that you have three distributions: the "real" one, and one that is inflated (shifted up the y axis) by 7% (making the peak of the bell curve just a bit taller) and one that is deflated (shifted down the y axis) by 7% (making the peak of the bell curve a little bit shorter).
2) Given the above graphic picture, you now want to specify that the distribution is generally shaped tall and skinny rather than flat and fat. That is, you are saying that the standard deviation around the mean value is pretty narrow (Another way to put this so that it reflects the 95% confidence limits is that values within +/- about 14% of the mean represent about 95% of all values).
3) Now you are asking what is the likelihood that a value is either 10% higher than the mean or 10% lower than the mean. I.e., you want to read a probability of that from a table or be able to calculate it.
If all of the above are correct characterizations, you can do this pretty easily, I think, as long as you are willing to assert that the thing you are measuring is, or should be, normally distributed. You just go to the tables for probability density under the normal curve and find that value that corresponds to (about) the mean + (10/14 * 1.96)
And you do that same calculation adding an inflator of 7% (so mean + 1.07(10/14 * 1.96)) or a deflator if you are working in the opposing direction (so, mean - 1.07(10/14 * 1.96).
But, you aren't quite done yet. This gives you the probability you asked for (the point probability for 10% off when you are allowing up to 14% off). But here's the thing, you might really want to know the probability of finding values that 10% or greater (up to 14%) off from the mean -- that is, the values you are going to call "okay" that are this extreme or more.
To do that, you would want to sum the probabilities between the point you just calculated and the point that is 14/14 * 1.96 on the normal curve. And you'd want to calculate the range for those by inflating and deflating by the 1.07 factor again.
Make sense?
WARNING -- This may be completely wrong, but it's what I've figured out would make sense from what I recall of probability density and the normal curve. A couple of the terms you presented (like "bias" mean something different to me than the way you've used them. So...I wouldn't necessarily adopt this methodology without consulting someone who really works with this stuff on a more frequent basis.
Good luck!
And thanks for waking my mind up.
Posts: 22497 | Registered: Sep 2000
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posted
Bob, Yeah, the bias thing threw me as well, but he insisted that that was the term they were using.
Maybe a hypothetic will explain it better. I talked with him in a little more depth and I think that it looks like this. You've got an instrument that for some reason, say someone bumping into the table it's on, over time has it's calibration messed with. However, 95% of the time, it is within 7% of accurate. This instrument is used to perform a test that has a variation of 7% 95% of the time.
To be honest, I didn't follow some of your explanation. Specifically, I didn't understand what you were doing with moving things on the y-axis and I didn't understand where you came up with 14%. I mean, I figure you added the two 7% together, but I didn't see why you did that.
In the situation I described, I'm pretty sure the way I did it (by breaking it into two vitual measurements) is sort of the way to do it but I don't feel comfortable with either the variation being expressed as a percentage or taking two 95%'s together and treating their result as 95%. If you gave me two numerical standard deviations, I think that what I did would give the right answer, but I don't know how you would treat it in this case.
Of course, statistics just doesn't jump out in my mind like many other types of math, probably because I resent it so much, so I could be partially or completely off. If you could explain your reasoning on those two steps a little more fully, maybe I could see what stupid line of thought I'm in. Either way, thanks for your time.
Posts: 10177 | Registered: Apr 2001
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posted
Oh, sorry. The 14% is roughly 2 standard deviations from the mean (assuming the standard deviation is 7% of the mean). I'm not sure this is a good assumption.
Oops, in fact I just realized it's wrong. You said 7% with 95% confidence, so the std deviation really amounts to 3.5%, roughly (it's 1.96 std deviations, not really 2).
Oops.
So, really mine is screwed up.
Sorry.
Everywhere it said 14, replace with 7 and that should work.
The example of trying to figure out "this deviant or greater" would have to change as well.
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