Question

A) If you want to be 95​% confident of estimating the population mean to within a sampling error of plus or minus 2 and the standard deviation is assumed to be 16​, what sample size is​ required?

The sample size required is _____

​(Round up to the nearest​ integer.)  

B) If you want to be 95​% confident of estimating the population proportion to within a sampling error of plus or minus 0.06, what sample size is​ needed?

A sample size of ____ is needed.

​(Round up to the nearest​ integer.)  

Answer 1

Solution :

Given that,

A) standard deviation = $\sigma$ = 16

margin of error = E = 2

At 95% confidence level the z is ,

$\alpha$ = 1 - 95% = 1 - 0.95 = 0.05

$\alpha$ / 2 = 0.05 / 2 = 0.025

Z_$\alpha$/2 = Z_0.025 = 1.96

Sample size = n = ((Z_$\alpha$/2 * $\sigma$ ) / E)²

= ((1.96 * 16 ) / 2)²

= 245.8

Sample size = 24

B)  $\hat p$ = 1 - $\hat p$ = 0.5

margin of error = E = 0.06

At 95% confidence level the z is ,

$\alpha$ = 1 - 95% = 1 - 0.95 = 0.05

$\alpha$ / 2 = 0.05 / 2 = 0.025

Z_$\alpha$/2 = Z_0.025 = 1.96

sample size = n = (Z_{$\alpha$ / 2} / E )² * $\hat p$ * (1 - $\hat p$ )

= (1.96 / 0.06)² * 0.5 * 0.5

= 266.7

sample size = 267