Dear wrench234,
Thanks for your question. Just to restate the question, a company has
a bag. How much it holds is irrelevant to the question. The strength
of the current bags is measured as a breaking strength of 50 pounds.
The new bag, presumably of some cheaper material, is being tested. To
be called ?better? than the current bag, the new bag will have to have
a breaking strength of at least 50 pounds, like the old bags. In this
sense, better means just as strong, but cheaper. It may also be that
the new bags are a different color, etc., but the key thing is that
they?re at least as strong as the current bags.
That being said, when one measures any quantity, there is some
variation in the measurement. This gives rise to a distribution or
?spread? of values around an average value. This is true for the
current bags and for the new bags. How much these distributions
overlap tells us to what degree the new bags are similar to or
different from the current bags. This is a basic idea in statistics,
and ways of measuring the overlap between these distributions give
rise to a number of statistical tests.
The idea of a ?confidence interval? means that, statistically, one is
N% confident that the true value (for example, the mean) lies within
the range specified. For example, for a distribution of breaking
strengths, the 95% confidence interval may be 45-55 pounds, meaning
that we are 95% confident that the real mean (u, the value we would
get if we measured every bag in the world) lies between those 45-55.
We?re not 100% certain, because we?ve only measured a sample of the
bags, not every one in existence. It?s possible (with a 5%
probability in the example I?ve given) that the real mean lies below
45 or above 55.
On to the problem you posed:
This website from the folks that developed Mathematica (a computer
program for mathematical computation and manipulation) gives a good
definition of the terms and the specific mathematical terminology for
each of them. It also derives how to calculate the confidence
intervals based on the standard deviation of the distribution.
http://mathworld.wolfram.com/StandardDeviation.html
Here?s a table from that page:
Confidence Interval Range
0.800 +/- 1.28155 x SD
0.900 +/- 1.64485 x SD
0.950 +/- 1.95996 x SD
0.990 +/- 2.57583 x SD
0.995 +/- 2.80703 x SD
0.999 +/- 3.29053 x SD
So, for the 95% Confidence Interval, the range would be:
95% CI = 50.575 +/- 1.95996 x 1.6438
= 50.575 +/- 3.22178
= 47.353 to 53.797
For the 99% CI:
99% CI = 50.575 +/- 2.57583 x 1.6438
= 50.575 +/- 4.23415
= 46.341 to 54.809
What this means is that we?re 95% certain that the mean (u) for the
breaking strength of all the new bags is between 47.353 and 53.797.
We?re 99% certain that it?s between 46.341 and 54.809. To narrow
these ranges, we would need to test more bags or make the new bags
more uniformly. Based on this, the breaking strength of the old bags
lies within these intervals, so we would say that the new bags are
likely to be as good as the old bags in terms of breaking strength.
I hope this answer was helpful. Feel free to ask for clarification.
-welte-ga |
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