A$( 1 - \alpha )$ -Level Confidence Interval for a Population Proportion That Has a a Population

Question

Examlex · Accepted Answer

The Answer is Solving for

\mathrm { n }

in the formula for margin of error gives the formula

\mathrm { n } = \hat { \mathrm { p } } ( 1 - \hat { \mathrm { p } } ) \left( \frac { \mathrm { z } \alpha / 2 } { \mathrm { E } } \right) ^ { 2 }

for the sample size

\mathrm { n }

. However this formula cannot be used to find the required sample size because the sample proportion

\hat { p }

is not known prior to sampling. The maximum possible value of

\hat { p } ( 1 - \hat { p } )

is

0.25

and this occurs when

\hat { p } = 0.5

. Thus, if

\hat { p } = 0.5

, the sample size needed to achieve the required margin of error is

\mathrm { n } = 0.25 \left( \frac { \mathrm { z } \alpha / 2 } { \mathrm { E } } \right) ^ { 2 }

. For any other value of

\hat { p } , \hat { p } ( 1 - \hat { p } )

will be smaller than

0.25

, and the sample size needed to achieve the required margin of error will be less than

0.25 \left( \frac { \mathrm { z } _ { \alpha / 2 } } { \mathrm { E } } \right) ^ { 2 }

.

A $( 1 - \alpha )$ -Level Confidence Interval for a Population Proportion That Has a a Population