Question

Suppose 56%of politicians are lawyers.If a random sample of size 564 is selected, what is the probability that the proportion of politicians who are lawyers will differ from the total politicians proportion by less than 4%

Answer 1

Answer: 0.9444

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 :-

`P(|q-p|<0.04)=P(-0.04<q-p<0.04)\\\\=P(\frac{-0.04}{\sqrt{((0.56)(1-0.56))/(564)}}<\frac{q-p}{\sqrt{(p(1-p))/(n)}}<\frac{0.04}{\sqrt{((0.56)(1-0.56))/(564)}})\\\\=P(-1.9137<z<1.9137) \ \ \ \ [\ Z=\frac{q-p}{\sqrt{(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`

Hence, the required probability = 0.9444