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environmental impact. Our data supports this claim because the 28 mpg is inside of
this 95% confidence interval for the population mean.
You are probably asking yourself: “Where did that 1.96 in the formula come
from?”
3.1.4 Where Did the Number “1.96” Come From?
A detailed mathematical answer to that question is beyond the scope of this topic,
but here is the basic idea.
We make an assumption that the data in the population are “normally
distributed” in the sense that the population data would take the shape of a “normal
curve” if we could test all of the people in the population. The normal curve looks
like the outline of the Liberty Bell that sits in front of Independence Hall in
Philadelphia, Pennsylvania. The normal curve is “symmetric” in the sense that if
we cut it down the middle, and folded it over to one side, the half that we folded
over would fit perfectly onto the half on the other side.
A discussion of integral calculus is beyond the scope of this topic, but essentially
we want to find the lower limit and the upper limit of the population data in the
normal curve so that 95% of the area under this curve is between these two limits. If
we have more than 40 people in our research study , the value of these limits is plus
or minus 1.96 times the standard error of the mean (s.e.) of our sample. The number
1.96 times the s.e. of our sample gives us the upper limit and the lower limit of our
confidence interval. If you want to learn more about this idea, you can consult a
good statistics book (e.g. Salkind, 2010 or van Emden, 2008).
The number 1.96 would change if we wanted to be confident of our results at a
different level from 95% as long as we have more than 40 people in our research
study.
For example:
1. If we wanted to be 80% confident of our results, this number would be 1.282.
2. If we wanted to be 90% confident of our results, this number would be 1.645.
3. If we wanted to be 99% confident of our results, this number would be 2.576.
But since we always want to be 95% confident of our results in this topic, we will
always use 1.96 in this topic whenever we have more than 40 people in our research
study.
By now, you are probably asking yourself: “Is this number in the confidence
interval about the mean always 1.96 ?” The answer is: “No!”, and we will explain
why this is true now.
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