Question

Predicting Election Results Throughout the US presidential election of \(2016,\) polls gave regular updates on the sample proportion supporting each candidate and the margin of error for the estimates. This attempt to predict the outcome of an election is a common use of polls. In each case below, the proportion of voters who intend to vote for each of two candidates is given as well as a margin of error for the estimates. Indicate whether we can be relatively confident that candidate A would win if the election were held at the time of the poll. (Assume the candidate who gets more than \(50 \%\) of the vote wins.) \(\begin{array}{lll}\text { (a) Candidate A: } 54 \% & \text { Candidate }\end{array}\) B: \(46 \%\) Margin of error: \(\pm 5 \%\) (b) Candidate A: \(52 \%\) Candidate B: \(48 \%\) Margin of error: \(\pm 1 \%\) \(\begin{array}{ll}\text { (c) Candidate A: } 53 \% & \text { Candidate }\end{array}\) B: \(47 \%\) Margin of error: \(\pm 2 \%\) \(\begin{array}{lll}\text { (d) Candidate A: } 58 \% & \text { Candidate }\end{array}\) B: \(42 \%\) Margin of error: \(\pm 10 \%\)

Short answer

For options (b) and (c), we can be relatively confident that candidate A would win if the election were held at the time of the poll. For options (a) and (d), we cannot be confident about that.

Step-by-step solution

Predicting the outcome for option (a)

For option (a), the proportion for candidate A is 54% and the margin of error is ±5%. This means that candidate A's true proportion could be as low as 49% (54% - 5%) and as high as 59% (54% + 5%). Since the lowest limit for candidate A (49%) is above candidate B's highest limit (46% + 5% = 51%), we cannot be confident that candidate A would win.

Predicting the outcome for option (b)

For option (b), the proportion for candidate A is 52% and the margin of error is ±1%. This means that candidate A's true proportion could be as low as 51% (52% - 1%) and as high as 53% (52% + 1%). Since the lowest limit for candidate A (51%) is above candidate B's highest limit (48% + 1% = 49%), we can be relatively confident that candidate A would win.

Predicting the outcome for option (c)

For option (c), the proportion for candidate A is 53% and the margin of error is ±2%. This means that candidate A's true proportion could be as low as 51% (53% - 2%) and as high as 55% (53% + 2%). Since the lowest limit for candidate A (51%) is above candidate B's highest limit (47% + 2% = 49%), we can be relatively confident that candidate A would win.

Predicting the outcome for option (d)

For option (d), the proportion for candidate A is 58% and the margin of error is ±10%. This means that candidate A's true proportion could be as low as 48% (58% - 10%) and as high as 68% (58% + 10%). Since the lowest limit for candidate A (48%) is not above candidate B's highest limit (42% + 10% = 52%), we cannot be confident that candidate A would win.