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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

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JeanaShupp

Answer:  40000

Step-by-step explanation:

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

[tex]n=p(1-p)(\dfrac{z_{\alpha/2}}{E})^2[/tex], 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. [tex]n\ \alpha\ \dfrac{1}{E^2}[/tex]

By the equation inverse variation, we have

[tex]n_1E_1^2=n_2E_2^2[/tex]

Given : [tex]E_1=0.05[/tex]   [tex]n_1=1000[/tex]

[tex]E_2=0.025[/tex]

Then, we have

[tex](1000)(0.05)^2=n_2(0.025)^2\\\\\Rightarrow\ 2.5=0.000625n_2\\\\\Rightarrow\ n_2=\dfrac{2.5}{0.000625}=4000[/tex]

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

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