Answer: 0.9444
Step-by-step explanation:
Given: The proportion of politicians are lawyers : p =0.56
Sample size : n = 564
Let q be th sample proportion.
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(|q-p|<0.04)=P(-0.04<q-p<0.04)\\\\=P(\dfrac{-0.04}{\sqrt{\dfrac{(0.56)(1-0.56)}{564}}}<\dfrac{q-p}{\sqrt{\dfrac{p(1-p)}{n}}}<\dfrac{0.04}{\sqrt{\dfrac{(0.56)(1-0.56)}{564}}})\\\\=P(-1.9137<z<1.9137) \ \ \ \ [\ Z=\dfrac{q-p}{\sqrt{\dfrac{p(1-p)}{n}}}\ ]\\\\=2P(Z<1.9137)-1\ \ \ \ [P(-z<Z<z)=2(Z<z)-1]\\\\=2(0.9722)-1\ \ \ [\text{by p-value table}]\\\\=0.9444[/tex]
Hence, the required probability = 0.9444
