Question

which of the following estimates at 95% confidence level most likely comes from a small sample?
a: 71%(-+4%)
b:60%(-+18%)
c:65%(-+2%)
d:62%(-+6%)

Answer 1

A confidence interval has the general form
\[(\text{estimate}-\text{margin of error},~\text{estimate}+\text{margin of error})\]
The margin of error is a value that depends on a constant (determined by the confidence level) and the standard error (which depends on the estimated statistic).

For example, for a normally distributed population, a 95% confidence interval for the mean would be
\[\left(\bar{x}-1.96\frac{\sigma}{\sqrt n},~\bar{x}+1.96\frac{\sigma}{\sqrt n}\right)\]
where \(\bar{x}\) is the sample (estimate) mean, 1.96 is the critical value for a 95% confidence level (the constant I mentioned), \(\sigma\) is the standard deviation, and \(n\) is the sample size.

Notice that if \(n\) is small, the fraction \(\dfrac{\sigma}{\sqrt n}\) tends to be large, whereas if \(n\) is large, the fraction approaches 0.

From this you can gather that a smaller sample size generally increases the error margin.