Question

You want to obtain a sample to estimate a population proportion. At this point in time, you have no reasonable preliminary estimation for the population proportion. You would like to be 95% confident that you estimate is within 3.5% of the true population proportion. How large of a sample size is required?

Answer 1

Answer:

n  = 400

Step-by-step explanation:

The formula for the error in our estimate is given by:

Standard Error :  √ ( p(1-p)/ n)

Error =  SE = Zα/2   √ ( p(1-p)/ n)  where

Zα/2= critical  value for 95% confidence level = 1.96

and we know our error is 3.5 %

But we do not the sample proportion p. Then what we can do is give an estimate of p in the absence of any other information.

In this case we will use p= 0.5 which is the value that maximizes the expression for the standard error :

if p = 0.8 then SE= 0.040

  p = 0.3 then SE =0.036

  p = 0.1 then  SE = 0.030

  p = 0.5 then SE = 0.050

Substituting

3.5/100  = 1.96 x  √ (( 0.5 x 0.5 ) /n )

3.5/ (100 x 1.96 x 0.5 ) = 1/ √n

0.0357 = 1 /√n

n = 20²

n = 400