Answer:
The sample size must be greater than or equal to 756
Step-by-step explanation:
The formula to calculate the error of the proportion is the following
[tex]E=z_{\alpha/2}*\sqrt{\frac{p(1-p)}{n}}[/tex]
where p is the proportion, n the sample size, E is the error and z is the z-score for a confidence level of 95%
For a confidence level of 95% [tex]z_{\alpha/2}=1.96[/tex]
We know that for this case [tex]p=0.23[/tex]
We require that the error be 0.03 as maximum
Therefore we solve for the variable n
[tex]z_{\alpha/2}*\sqrt{\frac{p(1-p)}{n}}\leq0.03\\\\1.96*\sqrt{\frac{0.23(1-0.23)}{n}}\leq0.03\\\\\sqrt{\frac{0.23(1-0.23)}{n}}\leq \frac{0.03}{1.96}\\\\(\sqrt{\frac{0.23(1-0.23)}{n}})^2\leq (\frac{0.03}{1.96})^2\\\\\frac{0.23(1-0.23)}{n}\leq (\frac{0.03}{1.96})^2\\\\\frac{0.23(1-0.23)}{(\frac{0.03}{1.96})^2}\leq n\\\\n\geq\frac{0.23(1-0.23)}{(\frac{0.03}{1.96})^2}\\\\n\geq756[/tex]
