4. A candidate is running for election. Her supporters conduct a poll, which they hope incorporate the same biases as will be found in the sample of voters who actually vote on election day, to determine if she is likely to win. Out of 200 voters contacted, 105 indicate that they will vote for her, and the other 95 indicate that they will vote for her opponent.

- a) Find a 95% confidence interval for the proportion of the population who intend to vote for her.
- b) What is the probability that she will win, based on this sample?
- c) How many voters should be included in the sample so that the margin of error is within 3%?

a) For a 95% confidence interval we use the formula

First we need to establish the sample proportion

The sample size is

and the critical value of z for a 95% confidence interval is

This gives us

This gives us an upper confidence limit of

and a lower confidence limit of

b) At this point her supporters wonder what her chances of winning are. 45.6% is below 50%, so she might lose. To find her chances of losing, consider the bell shaped curve

Her probability of winning is the area to the right of 50%. We need to turn the 50% into a z score

We look up a z-score of -.71 in Table A and get an area of .2389. This is the probability that she will lose. the probability that she will win is

c) The candidate's supporters would like to be more sure that she will win. They decide to increase the sample size until the 95% confidence interval is only plus or minus 3%. They want to know how large of a sample they will need to get that kind of accuracy. We use the formula for the margin of error.

To solve for n, first square both sides to get rid of the radical.

Multiply both sides by n and divide both sides by m^{2}.

For a 95% confidence interval z* = 1.96. We put in our other numbers

= 1064.4433

so they would have to survey 1065 people in order to get a 95% confidence interval that would only stretch up and down 3%.