Question

Suppose that two random samples of sizes n1, and n2 are selected without replacement from two binomial populations with means = \(\mu_1= n_1p_1, \,\,\,\mu_2= n_2p_2,\) and variances \(σ_1 ^2 = n_1p_1q_!, \,\,\,σ_2^2 = n_2p_2q_2\) , respectively. Let the difference of sample proportions $\overline{P_1} $ and $\overline{P_2} $ approximate a normal distribution with mean ($p_1 - p_2$). Then the standard deviation of the difference of sample proportions  $\overline{P_1} $ and $\overline{P_2} $, is

• A. \(\sqrt{(\frac{p_1q_1}{n_1})(\frac{N_1-n_1}{N_1-1})+ (\frac{p_2q_2}{n_2})(\frac{N_2-n_2}{N_2-1})}\)
• B. \(\sqrt{(\frac{p_1q_q}{n_1})+(\frac{p_2q_2}{n_@})}\)
• C. \(\sqrt{(\frac{p_1q_1-p_2q_2}{n_1+n2}})\)
• D. \(\sqrt{(\frac{p_1q_1}{n_1+n_2})(\frac{N_1-n_1}{N_1-1})+ (\frac{p_2q_2}{n_1+n_2})(\frac{N_2-n_2}{N_2-1})}\)

Answer 1

The Correct Option is A

To solve the question regarding the standard deviation of the difference between two sample proportions from binomial distributions, let's review the relevant concepts and apply the appropriate formulae.

Given:

• Two binomial populations with means \(\mu_1 = n_1p_1\) and \(\mu_2 = n_2p_2\).
• Variances \(\sigma_1^2 = n_1p_1q_1\) and \(\sigma_2^2 = n_2p_2q_2\), where \(q_1 = 1-p_1\) and \(q_2 = 1-p_2\).

The difference in sample proportions \(\overline{P_1}\) and \(\overline{P_2}\) is normally distributed with a mean of \(p_1 - p_2\). We are asked to find the standard deviation of this difference.

The standard deviation of the difference between two proportions is calculated using the following formula:

\[\sqrt{\left(\frac{p_1q_1}{n_1}\right)\left(\frac{N_1-n_1}{N_1-1}\right) + \left(\frac{p_2q_2}{n_2}\right)\left(\frac{N_2-n_2}{N_2-1}\right)}\]

This forms the correct answer because:

• The formula incorporates the finite population correction, which is necessary when sampling without replacement. The terms \(\frac{N_1-n_1}{N_1-1}\) and \(\frac{N_2-n_2}{N_2-1}\) account for this correction.
• Each part of this formula represents the contribution of each sample's variance to the overall variance of the difference of proportions.

Let's assess the other options and identify why they are incorrect:

1. The option \(\sqrt{\left(\frac{p_1q_1}{n_1}\right) + \left(\frac{p_2q_2}{n_2}\right)}\) does not account for the finite population correction.
2. The option \(\sqrt{\left(\frac{p_1q_1-p_2q_2}{n_1+n_2}\right)}\) incorrectly mixes the deviations of the two independent proportions, which is not mathematically justified.
3. The option \(\sqrt{\left(\frac{p_1q_1}{n_1+n_2}\right)\left(\frac{N_1-n_1}{N_1-1}\right) + \left(\frac{p_2q_2}{n_1+n_2}\right)\left(\frac{N_2-n_2}{N_2-1}\right)}\) improperly combines numerator terms in the variance formula, affecting the correction and proportion terms.

Conclusively, the formula with finite population correction, ensuring accurate variance calculation for samples without replacement, corresponds to the first option:

\(\sqrt{\left(\frac{p_1q_1}{n_1}\right)\left(\frac{N_1-n_1}{N_1-1}\right) + \left(\frac{p_2q_2}{n_2}\right)\left(\frac{N_2-n_2}{N_2-1}\right)}\)