FISH DEMOCRACIES onsider 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.57 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. 1. Training Fish to Pick a Color Fish can be trained quite easily. With just seven days of training, golden shiner fish learn to pick a color (yellow or blue) to receive a treat, and the fish will swim to that color immediately. On the first day of training, however, it takes them some time. In the study described under Fish Democracies above, the mean time for the fish in the study to reach the yellow mark is with 51 seconds with a standard error for this statistic of 2.4. a. Find and interpret a 95% confidence interval for the mean time it takes a golden shiner fish to reach the yellow mark. b. Is it plausible that the average time it takes fish to find the mark is 60 seconds? c. Is it plausible that it is 55 seconds? 2. 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. a. Interpret this confidence interval. b. Is it plausible that fish in this situation are equally likely (50/50) to go for either of the two options? 3. What Is the Effect of Including Some Indifferent Fish? In the experiment described above under Fish Democracies, the schools of fish in the study with an opinionated minority and a less passionate majority picked the majority option only about 17% of the time. However, when groups also included 10 fish with no opinion, the schools of fish picked the majority option 61% of the time. We want to estimate the effect of adding the fish with no opinion to the group, which means we want to estimate the difference in the two proportions. We learn from the study that the standard error for estimating this difference is about 0.14. a. Define the parameter we are estimating, give the best point estimate, and find and interpret a 95% confidence interval. b. Is it plausible that adding indifferent fish really has no effect on the outcome