Questions

The width of a confidence interval is influenced by the sample size ($n$), among other factors. Generally, larger sample sizes result in narrower confidence intervals because the standard error decreases as the sample size increases.

The formula for the margin of error (E) in a confidence interval for the mean is:

$E = z_{\alpha/2} \left(\frac{\sigma}{\sqrt{n}}\right)$

where:

• $z_{\alpha/2}$ is the z-value corresponding to the desired confidence level,
• $\sigma$ is the population standard deviation,
• $n$ is the sample size.

Comparison of Sample Sizes:

• For $n = 100$: $E = z_{\alpha/2} \left(\frac{\sigma}{\sqrt{100}}\right) = z_{\alpha/2} \left(\frac{\sigma}{10}\right)$

• For $n = 400$: $E = z_{\alpha/2} \left(\frac{\sigma}{\sqrt{400}}\right) = z_{\alpha/2} \left(\frac{\sigma}{20}\right)$

Analysis:

The margin of error is inversely proportional to the square root of the sample size ($\sqrt{n}$). Therefore, as the sample size increases, the margin of error decreases.

• When $n = 100$, the standard error is $\frac{\sigma}{10}$.
• When $n = 400$, the standard error is $\frac{\sigma}{20}$.

Since $\frac{\sigma}{20}$ is smaller than $\frac{\sigma}{10}$, the margin of error for $n = 400$ is smaller than the margin of error for $n = 100$.

A sample size of $n = 100$ would result in a wider confidence interval compared to a sample size of $n = 400$.