Answer: a. 170 b. 752
Step-by-step explanation:
The formula to find the sample size is given by :-
![n= p(1-p)(\dfrac{z_{\alpha/2}}{E})^2](https://tex.z-dn.net/?f=n%3D%20p%281-p%29%28%5Cdfrac%7Bz_%7B%5Calpha%2F2%7D%7D%7BE%7D%29%5E2)
, where p is the prior estimate of population proportion.
= Two-tailed z-value for
(significance level).
E= Margin of error .
Given : Margin of error = 0.03
Confidence level = 90%=0.90
By z-value table : ![z_{\alpha/2}=z_{0.05}=1.645](https://tex.z-dn.net/?f=z_%7B%5Calpha%2F2%7D%3Dz_%7B0.05%7D%3D1.645)
a) The president's political advisors estimated the proportion supporting the current policy to be : p= 0.06.
Required sample size :
![n= 0.06(1-0.06)(\dfrac{1.645}{0.03})^2](https://tex.z-dn.net/?f=n%3D%200.06%281-0.06%29%28%5Cdfrac%7B1.645%7D%7B0.03%7D%29%5E2)
![n=0.0564(3006.6944)](https://tex.z-dn.net/?f=n%3D0.0564%283006.6944%29)
![n=169.57756416\approx170](https://tex.z-dn.net/?f=n%3D169.57756416%5Capprox170)
∴ Required sample size = 170
b) If no prior estimate of population proportion is given , then we assume
p= 0.5
Required sample size :
![n= 0.5(1-0.5)(\dfrac{1.645}{0.03})^2](https://tex.z-dn.net/?f=n%3D%200.5%281-0.5%29%28%5Cdfrac%7B1.645%7D%7B0.03%7D%29%5E2)
![n=0.25(3006.6944)](https://tex.z-dn.net/?f=n%3D0.25%283006.6944%29)
![n=751.6736\approx752](https://tex.z-dn.net/?f=n%3D751.6736%5Capprox752)
∴ Required sample size = 752