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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: \(\pm 10 \%\)

Step by step solution

01

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.

02

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.

03

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.

04

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.

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Key Concepts

These are the key concepts you need to understand to accurately answer the question.

Margin of Error

Polls often include a margin of error, a term that indicates the range within which the true value of a survey's result is expected to fall. When polling data is shared, it's crucial to communicate this variability to predict election outcomes more accurately. The margin of error represents a bit of wiggle room or flexibility. It's introduced because we survey a sample, not the entire population.

• An important point to remember is that the larger the margin, the less precise the poll's estimate.
• In election polls, the margin of error tells us how much a candidate's proportion could realistically go up or down.

For example, if a candidate has 54% support in a poll with a ±5% margin of error, the candidate's actual support could realistically be between 49% and 59%. Understanding the margin helps assess how strongly the poll predicts an election's result. A smaller margin confers higher certainty to the prediction.

Sample Proportion

When discussing polling predictions, the term 'sample proportion' frequently arises. It's vital to understand what it signifies in the context of election polling. The sample proportion is essentially the percentage of survey respondents indicating their support for a candidate.

• This figure is based on the sample taken from a larger population.
• Polls give us information on how many, out of this sample group, support a specific candidate.

For example, if in a poll of 1,000 voters, 540 indicated support for Candidate A, the sample proportion for Candidate A is 54%. Although this number is a useful snapshot, considering the margin of error is crucial, as the actual proportion in the whole population might slightly differ from what the sample shows.

Confidence Levels

Confidence levels are crucial when interpreting polling predictions because they indicate the likelihood that the poll's results reflect the true opinions of the total population. In simpler terms, a confidence level tells you how sure you can be about the poll's accuracy.

• A standard confidence level is usually set at 95% in polling.
• This means there's a 95% chance that the poll results are correct, or that they fall within the given margin of error.

For example, if a poll indicates that Candidate A has 52% support with a confidence level of 95%, it suggests a high probability that if all voters were surveyed, Candidate A's actual support would be very close to 52%. Confidence levels are essential for evaluating how reliable a poll is and, therefore, how much weight should be given to its predictions.