Question

Is a tall president better than a short one? Do Americans tend to vote for the taller of the two candidates in a presidential selection? In 33 of our presidential elections between 1856 and \(2006,\) 17 of the winners were taller than their opponents. Assume that Americans are not biased by a candidate's height and that the winner is just as likely to be taller or shorter than his opponent. Is the observed number of taller winners in the U.S. presidential elections unusual? a. Find the approximate probability of finding 17 or more of the 33 pairs in which the taller candidate wins. b. Based on your answer to part a, can you conclude that Americans might consider a candidate's height when casting their ballot?

Short answer

In 33 US presidential elections, 17 of the winners were taller than their opponents. Based on this data and using a binomial distribution, the probability of observing 17 or more taller-winning pairs is approximately 0.305. Since this value is not considered unusual (greater than a 5% significance level), we cannot conclude that Americans might consider a candidate's height when casting their ballot.

Step-by-step solution

Identify the Binomial Distribution Parameters

Since we are considering the number of taller-winning election pairs, we can use a binomial distribution. In a binomial distribution, we have two parameters: n (number of trials) and p (probability of success). In this case, n = 33 (number of presidential elections) and p = 0.5 (assuming there is no bias and the probability of a taller candidate winning is equal to the probability of a shorter candidate winning).

Compute the Probability of 17 or More Taller-Winning Pairs Using Cumulative Binomial Probability

We want to find the probability of observing 17 or more taller-winning pairs, P(x >= 17). Since we know the cumulative binomial probability formula, we can find the P(x >= 17) by subtracting the cumulative probability of x = 16 from 1: P(x >= 17) = 1 - P(x <= 16) Using a cumulative binomial probability calculator or statistical software, we get: P(x <= 16) = 0.695 So, P(x >= 17) = 1 - 0.695 = 0.305.

Draw a Conclusion about the Observed Number of Taller Winners

The probability of observing 17 or more taller-winning pairs, assuming there is no height bias among voters, is 0.305. Since this value is greater than a commonly-used threshold of 0.05 (5% significance level), it is not considered unusual. Therefore, based on this analysis, we cannot conclude that Americans might consider a candidate's height when casting their ballot. #a.# The approximate probability of finding 17 or more of the 33 pairs in which the taller candidate wins is 0.305. #b.# Based on the analysis, we cannot conclude that Americans might consider a candidate's height when casting their ballot, as the observed number of taller winners is not unusual.

Key concepts

Probability

Probability is a fundamental concept that helps us understand how likely an event is to occur. Imagine flipping a coin; if the coin is fair, there is an equal likelihood of landing heads or tails. Similarly, in the context of the exercise about presidential elections, each candidate has a 50% probability of being the taller one since we're assuming no bias. This probability is represented by the parameter \( p = 0.5 \) in a binomial distribution, indicating no preference toward taller candidates.

In our specific example, the probability question focuses on the outcome of 17 or more elections won by the taller candidate out of 33. The probability of this event, assuming each election is an independent trial, provides insight into whether such an outcome could occur by chance. The resulting probability of 0.305 from the binomial distribution tells us there is about a 30.5% chance for this result, implying it can happen relatively frequently. Thus, while it might seem like height plays a role, statistically, it's not an unusual outcome if no real preference exists.

Statistical Inference

Statistical inference is the process of using data from a sample to make generalizations about a larger population. In our exercise, the sample data consists of 33 presidential elections, where the height of each election winner is recorded to see if there was a pattern favoring taller candidates.

To make an inference, we rely on statistical tools such as the binomial distribution to analyze and interpret the data. By calculating probabilities, like the 30.5% chance of having 17 or more elections won by the taller candidate, we assess whether it deviates significantly from what is expected. This analysis helps us infer whether there might be an underlying pattern in voter behavior, such as a bias for taller candidates.

While statistical inference provides a path to understanding and interpreting data, it is crucial to note that it involves some uncertainty. No conclusions are made with absolute certainty, and in this case, the analysis suggests that no significant height preference is detected in voter behavior.

Hypothesis Testing

Hypothesis testing is a method used to assess a theory or hypothesis based on sample data. It provides a framework for making decisions about whether an observed effect or pattern is statistically significant. In our example, the hypothesis might be that voters do not have a bias based on a candidate's height.

To test this, we use a threshold known as the significance level, often set at 0.05. This level represents the probability of rejecting the null hypothesis (no bias) when it is, in fact, true. The observed probability of 0.305 is much larger than our threshold, leading us not to reject the null hypothesis.

Through hypothesis testing, we determine whether our sample data provides enough evidence to support the alternative hypothesis, which in this case would be a bias toward taller candidates. Since the probability of observing our data is common under the assumption of no bias, we conclude that there is insufficient evidence to claim that Americans prefer taller presidential candidates.