Question

After calculating the sample size needed to estimate a population proportion to within 0.05, you have been told that the maximum allowable error (E) must be reduced to just 0.025. If the original calculation led to a sample size of 1000, the sample size will now have to be

Answer 1

Answer: 40000

Explanation:

The formula to find the sample size is given by :-

`n=p(1-p)((z_(\alpha/2))/(E))^2`, where p is the prior estimate of the population proportion.

Here we can see that the sample size is inversely proportion withe square of margin of error.

i.e.
`n\ \alpha\ (1)/(E^2)`

By the equation inverse variation, we have

`n_1E_1^2=n_2E_2^2`

Given :
`E_1=0.05`
`n_1=1000`

`E_2=0.025`

Then, we have

`(1000)(0.05)^2=n_2(0.025)^2\\\\\Rightarrow\ 2.5=0.000625n_2\\\\\Rightarrow\ n_2=(2.5)/(0.000625)=4000`

Hence, the sample size will now have to be 4000.