Answer: The sample size of 315 is needed so that the confidence interval will have a margin of error of 0.052 .
Step-by-step explanation:
The Margin of error for population proportion is given by :-
[tex]E=z_{\alpha/2}\times\sqrt{\dfrac{p(1-p)}{n}}[/tex]
Given : Significance level : [tex]\alpha = 1-0.95=0.05[/tex]
[tex]z_{0.025}=1.96[/tex] [By standard normal distribution table]
Proportion of adults who believe that economic conditions are getting better : p=0.33.
Margin of error : E= 0.052
Substitute all the value in the above formula, we get
[tex]0.052=1.96\times\sqrt{\dfrac{0.33(0.67)}{n}}\\\\\Rightarrow\0.0265=\sqrt{\dfrac{0.2211}{n}} [/tex]
Squaring both sides , we get
[tex]0.00070225=\dfrac{0.2211}{n}\\\\\Rightarrow\ n=\dfrac{0.221}{0.00070225}=314.702741189\approx315[/tex]
Hence, the required sample size = 315
