So how would we do this?

For A, what we're going to do is compute the simple random sampling variance, or

I suppose we could compute the design effect first, and

then the simple random sampling variance, but we're going to have a new sample size,

a proportion that we're going to have to make an assumption about.

In step two, a design effect, in which we're going to use a past value of roh and

our new sub-sample size or old sub-sample size, in our case and in step two,

for alternative A, B is exactly the same as before, it's 40.

And then we're going to compute the product of the simple random sampling

variance and

the design effect to give us our projected sampling variance under our new design.

And for alternative B,

we'd repeat these steps as well replacing B = 40 with B = 20.

With the same roh value, the same proportion and

that would allow us to compare sampling variances between these two designs.

For design A, for example, with 1,200 as our sample size and

30 clusters of 40 elements each, our design effect in this

particular case is 2.1795, it's the same one that we had before.

We haven't changed anything before, we're using the same value of roh,

the same value of b, so it's the same.

So all we need to do in this particular case then

is compute a new simple random sampling variance.

And ignoring the finite population correction, which we actually did anyway

in that prior calculation in the illustration we had done before.

We have p(1-p), 0.4, the value we had before,

x 0.6 divided by 1,200, or

a sampling variance of the proportion of 0.0002.

The product of that with our design effect gives us a sampling variance of 0.0004358.

We would take a square root to get a standard error of course, but we'll stop

there, because we're going to compare this variance to the one under design B.

Under design B we have a design effect now that is different.

B has changed from 40 to 20, that design effect goes

from 2.17, 2.18, down to 1.575.

It's not cut in half because we cut the sample size in half,

the effect has been cut in half.

And so now in this particular case then, when we multiply together this

projected design effect, 1.575 under this new design,

by the simple random sampling variance, which hasn't changed.

It's still the same sample size and the same proportions,

we get a variance of 0.000315.

Well we can compare these.

And here's the table comparing what we had before in our original design, 2,400,

that's that first row of numbers, where we had 60 clusters of 40 elements each and

a design effect of 2.17, 2.18 and a variance of 0.000218.

In our projected A version, with 1,200 cases in the sample and

30 clusters of 40 elements each, same design effect as we've noted.

But our variance is now twice as large, it's doubled.

That is, if we take the sample size and cut it in half,

by cutting the number of clusters in half, we double the sampling variance.

It goes the other way too.

If we were to take the sample size and double it, by doubling the number of

clusters, our sampling variance would decrease by one-half.