Question

In the last election, a state representative received 52% of the votes cast. One year after the election, the representative organized a survey that asked a random sam- ple of 300 people whether they would vote for him in the next election. If we assume that his popularity has not changed, what is the probability that more than half of the sample would vote for him?

Answer 1

The probability that more than half of the sample would vote for him is P=0.7549.

Explanation:

With a sample size of n=300, we can approximate this to the normal distribution.

The parameters will be

`\mu=p\cdot n=0.52\cdot 300=156\\\\\sigma=√(np(1-p)) =√(300\cdot 0.52(1-0.52)) =√(74.88)= 8.65`

We have to calculate the probability that half or more of the sample vote for him. This is P(x>150).

To calculate this probability, first we calculate the z-value:

`z=(x-\mu)/(\sigma)=(150-156)/(8.65)=(-6)/(8.65)=-0.69`

Then

`P(x>150)=P(z>-0.69)=0.7549`

The probability that more than half of the sample would vote for him is P=0.7549.