Question

Days before a presidential​ election, a nationwide random sample of registered voters was taken. Based on this random​ sample, it was reported that​ "52% of registered voters plan on voting for Robert Smith with a margin of error of plus or minus±​3%." The margin of error was based on a​ 95% confidence level. Can we say with​ 95% confidence that Robert Smith will win the election if he needs a simple majority of votes to​ win? Choose the correct answer below.

A. ​No, because the margin of error can never be more than​ 1%.
B. ​Yes, because​ 50% is within the bounds of the confidence interval.
C. ​Yes, since over​ 50% of the voters in the sample say they will vote for Robert Smith.
D. ​No, because​ 50% is within the bounds of the confidence interval.

Answer 1

D. ​No, because​ 50% is within the bounds of the confidence interval.

Explanation:

"52% of registered voters plan on voting for Robert Smith with a margin of error of plus or minus±​3%." means that

There is 95% probability that Robert Smith is voted between 49% and 55%. Majority of votes is needed to win the election, but this confidence interval doesn't guarantee 50+% vote.

Therefore we cannot say Robert Smith will win the election with this 95% confidence interval.

Answer 2

D. ​No, because​ 50% is within the bounds of the confidence interval.

Explanation:

"52% of registered voters plan on voting for Robert Smith with a margin of error of plus or minus±​3%"

This means that the 95% confidence interval for the percentage of people that plan on voting for Robert Smith is

(52-3 = 49%, 52+3 = 55%).

We can't say that he will win the election, because at the 95% confidence level, the proportion is not guaranteed to be above 50%.

So the correct answer is:

D. ​No, because​ 50% is within the bounds of the confidence interval.