Answer: a) 683 b) 1067
Step-by-step explanation:
The confidence interval for population proportion is given by :-
[tex]p\pm z_{\alpha/2}\sqrt{\dfrac{p(1-p)}{n}}[/tex]
a) Given : Significance level :[tex]\alpha=1-0.95=0.05[/tex]
Critical value : [tex]z_{\alpha/2}}=\pm1.96[/tex]
Margin of error : [tex]E=0.03[/tex]
Formula to calculate the sample size needed for interval estimate of population proportion :-
[tex]n=p(1-p)(\dfrac{z_{\alpha/2}}{E})^2\\\\=0.2(0.8)(\dfrac{1.96}{0.03})^2=682.951111111\approx683[/tex]
Hence, the required sample size would be 683 .
b) If no estimate of the sample proportion is available then the formula to calculate sample size will be :-
[tex]n=0.25(\dfrac{z_{\alpha/2}}{E})^2\\\\=0.25(\dfrac{1.96}{0.03})^2=1067.11111111\approx1067[/tex]
Hence, the required sample size would be 1067 .
