Answer: 0.1854
Step-by-step explanation:
Given : Suppose that a particular candidate for public office is in fact favored by 48% of all registered voters in the district.
Let [tex]\hat{p}[/tex] be the sample proportion of voters in the district favored a particular candidate for public office .
A polling organization will take a random sample of n=500 voters .
Then, the probability that p will be greater than 0.5, causing the polling organization to incorrectly predict the result of the upcoming election :
[tex]P(\hat{p}>0.5)=P(\dfrac{\hat{p}-p}{\sqrt{\dfrac{p(1-p)}{n}}}>\dfrac{0.5-0.48}{\sqrt{\dfrac{0.48(0.52)}{500}}})\\\\=P(z>0.8951)\ \ [\because\ z=(\dfrac{\hat{p}-p}{\sqrt{\dfrac{p(1-p)}{n}}}]\\\\=1-P(z\leq0.8951)\ \ [\because\ P(Z>z)=1-P(Z\leq z)]\\\\ = 1-0.8146=0.1854[/tex]
∴ Required probability = 0.1854
