Calculating Standard Error Ratio ($e_P / e_Q$)

The standard error of the mean ($e$) measures the variability of sample means around the population mean. It is calculated using the population standard deviation ($\sigma$) and the sample size ($n$) with the formula:

Defining Variables and Relationships

• Let $\sigma_P$ and $\sigma_Q$ be the population standard deviations for P and Q.
• Let $n_P$ and $n_Q$ be the sample sizes for P and Q.
• Given: $\sigma_P = 2 \times \sigma_Q$
• Given: $n_P = 4 \times n_Q$
• Let $e_P$ and $e_Q$ be the standard errors of the means for samples from P and Q.

Step-by-Step Calculation

1. Express the standard error for population P:

   $ e_P = \frac{\sigma_P}{\sqrt{n_P}} $

2. Express the standard error for population Q:

   $ e_Q = \frac{\sigma_Q}{\sqrt{n_Q}} $

3. Calculate the ratio $ e_P / e_Q $:

   $ \frac{e_P}{e_Q} = \frac{\sigma_P / \sqrt{n_P}}{\sigma_Q / \sqrt{n_Q}} $

4. Rearrange the terms:

   $ \frac{e_P}{e_Q} = \left( \frac{\sigma_P}{\sigma_Q} \right) \times \left( \frac{\sqrt{n_Q}}{\sqrt{n_P}} \right) $

5. Substitute the given relationships ($\sigma_P = 2\sigma_Q$ and $n_P = 4n_Q$):

   $ \frac{e_P}{e_Q} = \left( \frac{2\sigma_Q}{\sigma_Q} \right) \times \left( \frac{\sqrt{n_Q}}{\sqrt{4n_Q}} \right) $

6. Simplify the expression:

   $ \frac{e_P}{e_Q} = (2) \times \left( \frac{\sqrt{n_Q}}{2\sqrt{n_Q}} \right) $

   $ \frac{e_P}{e_Q} = 2 \times \frac{1}{2} $

7. Final Result:

   $ \frac{e_P}{e_Q} = 1 $

The ratio of $e_P$ to $e_Q$ is 1.