Question

A political pollster is conducting an analysis of sample results in order to make predictions on election night. Assuming a​ two-candidate election, if a specific candidate receives at least 53% of the vote in the​ sample, that candidate will be forecast as the winner of the election. You select a random sample of 100 voters.

a.	What is the probability that a candidate will be forecast as the winner when the population percentage of her vote is 50.1​%?

The probability is ____ that a candidate will be forecast as the winner when the population percentage of her vote is

50.1​%.

​(Round to four decimal places as​ needed.)

b.	What is the probability that a candidate will be forecast as the winner when the population percentage of her vote is 57​%?

The probability is ____ that a candidate will be forecast as the winner when the population percentage of her vote is 57​%.

​(Round to four decimal places as​ needed.)

c.	What is the probability that a candidate will be forecast as the winner when the population percentage of her vote is 49​% ​(and she will actually lose the​ election)?

The probability is ____ that a candidate will be forecast as the winner when the population percentage of her vote is 49​%.

​(Round to four decimal places as​ needed.)

d.	Suppose that the sample size was increased to 400. Repeat process​ (a) through​ (c), using this new sample size. Comment on the difference.

The probability is ____ that a candidate will be forecast as the winner when the population percentage of her vote is 50.1​%.

​(Round to four decimal places as​ needed.)

The probability is ____ that a candidate will be forecast as the winner when the population percentage of her vote is 57​%.

​(Round to four decimal places as​ needed.)

The probability is ____ that a candidate will be forecast as the winner when the population percentage of her vote is 49​%.

​(Round to four decimal places as​ needed.)

Answer 1

Solution:- Given information :

a. n = 100, p = 0.501

p(p^ >= 0.53) = P((p^-p)/sqrt((pq/n) >= (0.53-0.501)/sqrt(0.501*0.499/100))
= P(Z > 0.58)
= 0.2810

b. n = 100 , p = 0.57

P(p^ >= 0.53) = P(Z >= (0.53-0.57)/sqrt(0.57*0.43/100))
= P(Z >= -0.8080)
= 0.7910

c. n = 100, p = 0.49

P(p^ >= 0.53) = P(Z >= (0.53-0.49)/sqrt(0.49*0.51/100))
= P(Z > 0.82)
= 0.2061

d. if n = 400

a) P(p > 0.53) = P(Z >= (0.53-0.501)/sqrt(0.501*0.499/400))
= P(Z >= 1.16)
= 0.1230

b) P(p > 0.53) = P(Z >= (0.53-0.57)/sqrt(0.57*0.43/400))
= P(Z >= -1.6159)
= 0.9474

c. P(p > 0.53) = P(Z >= (0.53-0.49)/sqrt(0.49*0.51/400))
= P(Z >= 1.60)
= 0.0548