Question

Use the formula for the standard error of \(\hat{p}\) to explain why increasing the sample size decreases the standard error.

Short answer

Increasing the sample size decreases the standard error because the sample size (\(n\)) is in the denominator of the standard error formula. Thus, increasing \(n\) makes the whole fraction smaller, hence reducing the standard error.

Step-by-step solution

Understanding the concept of Standard Error and Sample Size

The standard error of an estimate is a measure of how much we expect it to vary if we take multiple random samples from the population. In our case, the estimate is \(\hat{p}\) - the sample proportion. So, the standard error of \(\hat{p}\) tells us how much we expect \(\hat{p}\) to vary from sample to sample. Sample size (\(n\)) refers to the number of observations or replicates in a sample.

Analysis of the formula for Standard Error of \(\hat{p}\)

The formula for standard error of \(\hat{p}\) is \(\sqrt{(\hat{p}(1-\hat{p})/n)}\). In this formula, the denominator under the square root symbol is the sample size \(n\). Since standard error measures the expected variability of \(\hat{p}\) and \(n\) appears in the denominator of this formula, it tells that standard error is inversely related to the square root of the sample size.

Reasoning why increasing the sample size decreases the Standard Error

Since sample size (\(n\)) is in the denominator of the standard error formula, as we increase the sample size, the value of the whole denominator increases. For any fraction, if the denominator gets larger while the numerator stays the same, the overall value of the fraction gets smaller. The square root doesn't change this relationship. Thus, increasing the sample size (\(n\)) will decrease the standard error of \(\hat{p}\).