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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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