So we get to proper confidence intervals.

Now we need to construct these lower bounds and these upper bounds.

We call them limits, bound limits.

Now the sample statistic, remember whether it’s the mean of a group or

the difference between two means.

It's just one of countless means that could have been found.

So we're really talking central limit theorem.

So what we are going to do is,

we need to find out how many standard errors away from

that sample statistic we have

to be to find about 95% of all the values,

if we choose our confidence level to be 95%.

Now remember, the sample statistic would be at the middle, and the mean

and we just have to work out how many standard errors away from the mean,

we have to be for our specific confidence level.

Now remember the Z and the t-distributions?

If we knew and the population standard deviation,

we could use the Z distribution.

If not, we are going to be stuck with the t-distribution, and

this is exactly what we're going to see.

Now look at it, if we were to include 95% of all values under the curve,

an area that represents 95% of the area under the curve.

You would see what the values are there they are slightly further up.

If we know the depth, if we make our limit bounds a little bit smaller,

closer to the we drop our confidence.

Our confidence level decreases as we make our little gap,

our little range smaller and smaller.

Now what is really this true interpretation?

If I were to say, that the 95% confidence interval was from A to B.

What does it really mean?

Now take an example.

If I were to do a study on the exact same population parameter.

So we're choosing using a population parameter, and we are going to take

a sample and for that same parameter, we can work out a sample statistic.

Now, if I were to repeat this study with different individuals 100 times and

in each of those times, I work out the 95% confidence interval bounds.

It would really mean that this population parameter would

fall within those two bounds in 95 of those 100 cases.

So I do that study 100 times over.

Every time I'm going find slightly different 95% confidence intervals,

the lower bound and an upper bound.

The true population perimeter will fall within those 95 times.

It is not true to say that, there is a 95% chance that the true

population perimeter is within the bounds set by a single study.

It's an all or nothing thing.

Either that patient, that population parameter is within those limits or

they are not.

But if I could repeat it 100 times, in 95% of cases it would actually

create the true population parameter would fall in between those.

So it's a subtle little difference, but it's the proper understanding,

the proper explanation of confidence intervals.