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. 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 \(\bar{x}=51\) seconds with a standard error for this statistic of 2.4 . Find and interpret a \(95 \%\) confidence interval for the mean time it takes a golden shiner fish to reach the yellow mark. Is it plausible that the average time it takes fish to find the mark is 60 seconds? Is it plausible that it is 55 seconds?

Short answer

The 95% confidence interval for the mean time it takes a golden shiner fish to reach the yellow mark is computed by plugging in the given values into the formula for a confidence interval. The plausibility of the average time for the fish to find the mark being 60 seconds or 55 seconds is then assessed by checking if these values fall within the computed confidence interval.

Step-by-step solution

Compute the Confidence Interval

A 95% confidence interval is computed using the formula: \(\bar{x} ± Z * SE\), where \(\bar{x}\) is the mean, \(Z\) is the Z-score (which is 1.96 for a 95% confidence interval), and \(SE\) is the standard error. In this problem, \(\bar{x}=51\), \(Z=1.96\), and \(SE=2.4\). Substitute these values into the formula: \(51 ± 1.96 * 2.4\). Compute to get the confidence interval.

Interpret the Confidence Interval

The computed confidence interval defines a range of values within which the true population mean is expected to lie with a 95% probability. If a given value falls inside this range, it is considered plausible for it to be the true population mean.

Evaluate Plausibility of Mean Times

Compare 60 seconds and 55 seconds with the computed confidence interval. If these times fall inside the interval, they are plausible values for the average time it takes a fish to find the mark. If they fall outside the interval, they are not considered plausible.