Question

Exercises 3.71 to 3.73 consider the question (using fish) of whether uncommitted members of a group make it more democratic. It has been argued that individuals with weak preferences are particularly vulnerable to a vocal opinionated minority. However, recent studies, including computer simulations, observational studies with humans, and experiments with fish, all suggest that adding uncommitted members to a group might make for more democratic decisions by taking control away from an opinionated minority. \({ }^{36}\) In the experiment with fish, golden shiners (small freshwater fish who have a very strong tendency to stick together in schools) were trained to swim toward either yellow or blue marks to receive a treat. Those swimming toward the yellow mark were trained more to develop stronger preferences and became the fish version of individuals with strong opinions. When a minority of five opinionated fish (wanting to aim for the yellow mark) were mixed with a majority of six less opinionated fish (wanting to aim for the blue mark), the group swam toward the minority yellow mark almost all the time. When some untrained fish with no prior preferences were added, however, the majority opinion prevailed most of the time. \({ }^{37}\) Exercises 3.71 to 3.73 elaborate on this study. How Often Does the Fish Majority Win? In a school of fish with a minority of strongly opinionated fish wanting to aim for the yellow mark and a majority of less passionate fish wanting to aim for the blue mark, as described under Fish Democracies above, a \(95 \%\) confidence interval for the proportion of times the majority wins (they go to the blue mark) is 0.09 to \(0.26 .\) Interpret this confidence interval. Is it plausible that fish in this situation are equally likely to go for either of the two options?

Short answer

The given 95% confidence interval shows that the proportion of times the majority of fish (who want to aim for blue mark) would win is between 9% to 26%. It is not plausible that fish are equally likely to aim for either mark, as the value of 0.5, indicating equal likelihood, is not within this confidence interval.

Step-by-step solution

Interpret the Confidence Interval

The given confidence interval for the proportion of times the majority wins (going to the blue mark) is between 0.09 and 0.26. The interpretation of a 95% confidence interval is that if we would repeat the experiment many times (an infinite number ideally) then 95% of the time the observed proportion (of the fish swimming towards blue) would be within this interval.

Assessing Plausibility

Now, we need to assess whether it's plausible that fish in this situation are equally likely to aim for either the blue or yellow mark (which would indicate a proportion of 0.5). Here, we see that the confidence interval (0.09 to 0.26) does not include this value. Thus, it is not plausible that the fish are equally likely to swim to the blue mark as to yellow mark as per the given confidence interval.

Key concepts

Statistical Significance

Statistical significance is a term that is used to indicate whether the results of an experiment or study are unlikely to have occurred by chance. In other words, it's a measure of certainty -- or uncertainty -- about the findings from a statistical analysis.

For an outcome to be considered statistically significant, the results would have to be extreme enough that they are not likely to be the result of random variation. This is usually determined using a p-value, which calculates the probability that the observed data, or something more extreme, would occur if the null hypothesis (usually the hypothesis that there is no effect or no difference) were true.

In general, if the p-value is below a predetermined threshold, such as 0.05, then the results are considered statistically significant. This threshold is called the significance level, and it represents the risk of concluding that an effect exists when, in fact, it does not (a false positive). In the context of the fish experiment discussed in the exercise, the concept of statistical significance would be pertinent when we want to determine whether the presence of uncommitted members could really make the group decisions more democratic, as opposed to those outcomes happening purely by chance.

95% Confidence Interval

A 95% confidence interval (CI) is a range of values that is used to estimate the true population parameter with a certain level of confidence. In other words, it is the interval within which we are 95% confident that the true value lies. When we say a 95% confidence interval, we imply that if we were to take 100 equally-random samples and compute a confidence interval from each of those samples, we would expect about 95 of those intervals to contain the true population parameter.

In the context of our fish scenario, the confidence interval from 0.09 to 0.26 tells us that, based on the sample data, we are 95% confident that the true proportion of times the majority of fish aim for the blue mark lies between 9% and 26%. This does not mean that there is a 95% probability that the true proportion is within this range; rather, the interval itself is a fixed range, and the true value is either within it or it isn't. The fact that 0.5 (equal likelihood of aiming for yellow or blue) is not within this interval is critical as it provides strong evidence that one color is favored over the other.

Assessing the Interval

To better understand the confidence interval, it is important to consider not only the range but also what is excluded from it. Since the 95% CI is from 0.09 to 0.26 and doesn't include 0.5, we can infer that there is a statistically significant difference in the preferences of the fish, with them not equally likely to swim towards either mark as initially proposed in the null hypothesis.

Proportion of Times the Majority Wins

The 'proportion of times the majority wins' refers to the frequency with which a specified outcome occurs over a series of events or trials. This proportion is the ratio of the number of times an event occurs to the total number of trials or times the event could have happened.

In our fish example, the majority refers to the group of less opinionated fish that preferred the blue mark over the yellow mark. The proportion of times the majority wins is an indication of how often the group’s collective action leads to the more preferred outcome despite the presence of a vocal minority. This proportion is a central point of interest in social behavior studies because it sheds light on the dynamics of group decision-making and the impact of individuals with varying levels of conviction on the overall group's choices.

For the school of fish in the experiment, the data shows a confidence interval from 0.09 to 0.26, which implies that the proportion of times the majority's choice (the blue mark) wins is estimated to be between 9% and 26%. It hints at the idea that while uncommitted or less passionate fish are added to the group, they seemingly contribute to a balance that allows the majority's preference to surface more often than if those members were not part of the decision-making process. The strength of this conclusion would depend on sample size and consistency of findings across multiple trials.
Understanding the implications of this proportion helps to explain social and behavioral phenomena not just in fish but potentially in larger animal groups, including humans, in the context of the influence of group members on collective outcomes.