Question

Assume that sample mean x overbar and sample standard deviation s remain exactly the same as those you just calculated but that are based on a sample of nequals25 observations. Repeat part a. What is the effect of increasing the sample size on the width of the confidence​ intervals?

Answer 1

Answer:

As we increase the number of observations the width of the confidence interval decreases.

Step-by-step explanation:

The general formula for the confidence interval for the mean is:

$IC[\mu, (1 - \alpha)\%] = \overline{x} \pm k\sqrt{Var(\overline{x})}$

And we know that the variance of $\overline{x}$ is:

$Var(\overline{x}) = Var(\frac{1}{n} \sum_{i} x_i) = \frac{1}{n^2} Var(\sum_{i} x_i)$

As it was informed the mean and the standard deviation remains the same during the process, so the second term $\frac{k}{n^2} Var(\sum_{i} x_i)$ depends only on the number of observations, and the relationship is inverse. So if we increase the n the second term becomes smaller, and so the width of the interval decreases.