Answer :
Answer: 0.0238
Step-by-step explanation:
Given : Proportion of politicians are lawyers: p= 0.56
Sample size : n= 787
Let [tex]\hat{p}[/tex] be th sample proportion.
The the probability that the proportion of politicians who are lawyers will differ from the total politicians proportion by greater than 4% will be :-
[tex]P(|\hat{p}-0.56|>0.04)=P(-0.04>\hat{p}-0.56>0.04)=\\\\ P(-0.04+0.56>\hat{p}>0.04+0.56)\\\\=P(0.52>\hat{p}>0.60)\\\\=P(\dfrac{0.52-0.56}{\sqrt{\dfrac{0.56(1-0.56)}{787}}}>\dfrac{\hat{p}-p}{\sqrt{\dfrac{p(1-p)}{n}}}>\dfrac{0.60-0.56}{\sqrt{\dfrac{0.56(1-0.56)}{787}}})\\\\=P(-2.26>z>2.26)\ \ \ [Z=\dfrac{\hat{p}-p}{\sqrt{\dfrac{p(1-p)}{n}}}]\\\\=1-P(-2.26<z<2.26)\\\\=1-(2P(Z > 2.26)-1)\ \ \ [P(-z<Z<z)=2P(Z > |z|)-1]\\\ =2-2P(Z > 2.26)\\\\=2-2(0.9881)=0.0238[/tex]
Hence, the required probability = 0.0238