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 55​% 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. Complete parts​ (a) through​ (c) below.

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 nothing 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 55​%?

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)?

d.

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

Answer 1

(a)

Here we have

n=100 ,p = 0.501

The sampling distribution of sample proportion will be approximately normal with mean

and standard deviation

The z-score for is

Using z table, the probability that a candidate will be forecast as the winner is

(b)

Here we have

n=100 ,p = 0.55

The sampling distribution of sample proportion will be approximately normal with mean

and standard deviation

The z-score for is

Using z table, the probability that a candidate will be forecast as the winner is

(c)

Here we have

n=100 ,p = 0.49

The sampling distribution of sample proportion will be approximately normal with mean

and standard deviation

The z-score for is

Using z table, the probability that a candidate will be forecast as the winner is

(d)

Here we have

n=400 ,p = 0.501

The sampling distribution of sample proportion will be approximately normal with mean

and standard deviation

The z-score for is

So  the probability that a candidate will be forecast as the winner is

------------------------------------

Here we have

n=400 ,p = 0.55

The sampling distribution of sample proportion will be approximately normal with mean

and standard deviation

The z-score for is

So  the probability that a candidate will be forecast as the winner is

-----------------------------------

Here we have

n=400 ,p = 0.49

The sampling distribution of sample proportion will be approximately normal with mean

and standard deviation

The z-score for is

So  the probability that a candidate will be forecast as the winner is