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 550 voters and will use p̂, the sample proportion, to estimate p. what is the approximate probability that p̂ will be greater than 0.5, causing the polling organization to incorrectly predict the result of the upcoming election? (round your answer to four decimal places.)

Answer 1

Given that a particular candidate for public office is in fact favored by 48% of all registered voters in the district, thus p = 0.48

If a polling organization will take a random sample of 550 voters, i.e. n= 550.

The approximate probability that p̂ will be greater than 0.5 is given by:

$P(\hat{p}\ \textgreater \ 0.5)=1-P(\hat{p}\leq0.5) \\ \\ =1-P\left(z\leq \frac{0.5-0.48}{\sqrt{ \frac{0.48(1-0.48)}{550} }} \right) \\ \\ =1-P\left(z\leq \frac{0.02}{\sqrt{ \frac{0.2496}{550} }} \right)=1-P\left(z\leq \frac{0.02}{0.0213} \right) \\ \\ =1-P(z\leq0.9388)=1-0.82609=0.1739$

Therefore, the approximate probability that p̂ will be greater than 0.5, causing the polling organization to incorrectly predict the result of the upcoming election is 0.1739.