Question

Suppose that a particular candidate for public office is in fact favored by 48%of all registered voters in the district. A polling organization will take a random sample of 500voters and will use p, the sample proportion, to estimate p. What is the approximate probability that p will be greater than 0.5, causing the polling organization to incorrectly predict the result of the upcoming election?

Answer 1

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 $\hat{p}$ 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 :

$P(\hat{p}>0.5)=P(\dfrac{\hat{p}-p}{\sqrt{\dfrac{p(1-p)}{n}}}>\dfrac{0.5-0.48}{\sqrt{\dfrac{0.48(0.52)}{500}}})\\\\=P(z>0.8951)\ \ [\because\ z=(\dfrac{\hat{p}-p}{\sqrt{\dfrac{p(1-p)}{n}}}]\\\\=1-P(z\leq0.8951)\ \ [\because\ P(Z>z)=1-P(Z\leq z)]\\\\ = 1-0.8146=0.1854$

∴ Required probability = 0.1854