0
$\begingroup$

I am currently working towards the Royal Statistical Society's Higher Diploma and have run across a strange result in one of their sample papers that I can't work out - wonder if anyone can help me.

In Q3 of this paper you are asked to work out the optimal (Neyman) allocation for strata sizes in a sample.

According to everything I've seen, the formula is:

$$n_h = n \cdot \frac{ N_h S_h}{ Σ_i N_i S_i}$$

However, in the sample solution the standard deviation seems to be calculated as √(nhph(1 – ph)) - in other words, the nh in the numerator rather than the denominator as is usual in the standard deviation formula for a proportion.

I'm sure I'm being dense and missing something obvious, but I have looked and can't seem to find an answer. Can anyone help me? Thanks so much in advance.

Improve this question
$\endgroup$

1 Answer 1

Reset to default
1
$\begingroup$

The denominator $D$ isn't being ignored. It is calculated as sum of the numerators. The solution in the document is $n_1 = 363$ and $n_2 = 137$. Here's how to reproduce these numbers.

From the Solutions Document:

The values of $N_hs_h$ are 8000√(300 × 0.3 × 0.7) = 63498.03 for urban areas and 4000√(150 × 0.6 × 0.4) = 24000.00 for rural areas. "

  1. Sum the two expressions to get the denominator. $$ D = N_1s_1 + N_2s_2 = 63498.03 + 24000 = 87,498.03 $$
  2. Calculate the proportion allocated to the first stratum: $$ \frac{N_1s_1}{D} = \frac{63498.03}{87498.03}= 0.725708 $$
  3. The sample size allocated to the first stratum $n_1 = 0.725708\times 500 = 363$, rounded to the nearest integer. Then $n_2 = 500-363$.
Improve this answer
$\endgroup$
2
  • $\begingroup$ Thanks Steve. My confusion though is in calculating sh - for a proportion I generally understood this was √(p)(1-p)/n, but here it seems to be √n(p)(1-p). Sure I'm missing something obvious, but just can't spot it! $\endgroup$
    – Cian Murphy
    Commented Apr 4, 2016 at 7:56
  • $\begingroup$ Sorry, the previous comment omitted some square roots. $s_h =\sqrt{p_h(1−p_h)}$ is the standard deviation of a binomial variable with success probability $p_h$. The numerator for stratum h in the document is $N_h\sqrt{ p_h(1−p_h}$ and is therefore correct. $N_h$ is the size of the population in stratum h. You are misreading the term as $n_h\sqrt{ p_h(1−p_h)}$. $\sqrt{p_h (1-p_h)/n_h}$ is the standard error or the estimated $p_h$, not its standard deviation. $\endgroup$
    – Steve Samuels
    Commented Apr 6, 2016 at 2:13

Your Answer

By clicking “Post Your Answer”, you agree to our terms of service and acknowledge you have read our privacy policy.

Not the answer you're looking for? Browse other questions tagged or ask your own question.