Question

9 multiple choice 1 point
select the most appropriate response.

as sample size increases (other things being equal), what tends to happen to 95% confidence intervals?

they become wider.
they become narrower.
they will not change.

Explanation:

To determine the effect of increasing sample size on a 95% confidence interval, we recall the formula for a confidence interval (e.g., for a mean: $\bar{x} \pm z_{\alpha/2}\frac{\sigma}{\sqrt{n}}$ or $\bar{x} \pm t_{\alpha/2}\frac{s}{\sqrt{n}}$). The standard error term is $\frac{\sigma}{\sqrt{n}}$ (or $\frac{s}{\sqrt{n}}$ for sample standard deviation). As the sample size $n$ increases, the standard error $\frac{\sigma}{\sqrt{n}}$ decreases (since $n$ is in the denominator under a square root). A smaller standard error leads to a narrower confidence interval because the margin of error (the part after $\pm$) decreases. So, as sample size increases, 95% confidence intervals become narrower.

Answer:

B. They become narrower.