solving for k in Spearman-Brown Prophecy Formula

[abent]

I have tried for a long time to find out how you solve for k in the Spearman-Brown Prophecy (or "Prediction") Formula (used to find out how many items is required in a test to get a certain reliability). Tt is very frustrating not knowing how to do this. If anyone could please help me that would be awesome! (I know the end result just not how to get there).

That is, I have this:

. . .rxx' = (k rxx) / [1 + (k-1)rxx']

I need to get to this:

. . .k = [rxx' (1 - rxx)] / [rxx' (1 - rxx')]

How do you describe this solving process?

Also seen as pxx* = N pxx' / 1+ (N-1) pxx'
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Edited by stapel -- Reason for edit: Fixing formatting, capitalization.

 

[bobbi.umbra]

So I know this request is pretty old, but I just had the same problem and couldnt find an answer anywhere.

[math]R_k = \frac{k*R}{1+(k-1)*R}[/math] 1. multiply by numerator
[math]R*k = R_k*(1+(k-1)*R)[/math] 2. divide by R_k
[math] \frac{R*k}{R_k} = 1+(k-1)*R[/math] 3. divide by R
[math]\frac{k}{R_k} = \frac{1}{R}+k-1[/math] 4. subtract k
[math]\frac{k}{R_k} - \frac{k}{1} = \frac{1}{R}-\frac{1}{1}[/math] 5. expand fractions on both sides
[math]\frac{k}{R_k} - \frac{k*R_k}{R_k} = \frac{1}{R}-\frac{R}{R}[/math][math]\frac{k-k*R_k}{R_k} = \frac{1-R}{R}[/math] 6. extract k
[math]\frac{k(1-R_k)}{R_k} = \frac{1-R}{R}[/math] 7. divide by (1-R_k) and multiply by R_k

[math]k = \frac{R_k*(1-R)}{R*(1-R_k}[/math]
its my first time creating a post here so unfortunately I couldn't figure out how to align the functions but I hope its alright still

 

[Steven G]

Looks fine to me.

 

[Otis]

I couldn't figure out how to align the functions

Hi bobbi. If you're talking about aligning the equals signs, the LaTeX command for that environment is broken here. If you're talking about aligning statements on the left margin, then invoke LaTeX with [tex] and [/tex] tags, instead of the other options. The options are listed in a LaTeX thread on the News board. If it's some other type of alignment, then please explain.

PS: I note that you didn't repeat abent's transcription error (using both rxx and rxx'). Good job.
[imath]\;[/imath]