Question

Explain why the margin of error determines the accuracy with which a sample mean estimates a population mean.

Short answer

If the margin of error is small, population mean is estimated using $\overline{x}$and precision of estimate is high.

Step-by-step solution

Given Information

Confidence level is $100(1-\alpha )\%$

Sample mean is $\stackrel{—}{x}$

Population mean is $\mu$

Margin of error is$E$

Explanation

As margin of error is equal to the half of the length of the confidence interval. $\stackrel{-}{x}$is the midpoint of the confidence interval.

With particular level of confidence $100(1-\alpha )\%$, $\stackrel{-}{x}$is deviated from $\mu$by $E$units.

The maximum possible error is $E$(margin of error).

Hence, if the margin of error is small, population mean is estimated using $\stackrel{-}{x}$results in high precision.

This is how the margin of error determines the precision with which sample mean estimates a population mean.