Exam 12: Inferences for Population Proportions

Correct Answer:

Verified

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