Statistical Analysis and Commentary
Goldfish behavioral response to magnetic field direction was measured for possible use in the pilot metacognition experiment. The evidence suggests goldfish detect magnetic field direction. However, behavioral response to different colors was used instead of field direction in the pilot metacognition experiment because (1) goldfish are already known to respond to color differences, and (2) it was easier to present a range of intermediate colors than to present a range of variation of magnetic field direction that was testable.
Description of the chi-square distribution test:
This indicates that with no magnetic field or color stimulus, the goldfish selected the left target 23 times, the middle target 11 times, and the right target 14 times.
To find the ꭓ2 statistic, we must compute the normalized squared difference between observed values and expected values. To do so, for each column, we take o, the observed number of target hits, and e, the expected number of hits, and compute (o-e)2/e. The sum of these values is the ꭓ2 statistic. This is the bottom row (the ꭓ2 statistic is highlighted):
Finally, the p-value of this statistic is computed, given 2 degrees of freedom. (This can be done with a variety of online calculators or with the spreadsheet function CHIDIST(ꭓ2, 2).) This gives a computed p-value of 0.087, which is greater than 0.05. This means there is an 8.7% chance that the results observed in this trial are due to random chance and not a dominating influence on the behavior of the goldfish. Because this is larger than 0.05, we fail to reject the null hypothesis, as expected. The goldfish were not given any indications of a correct target in this control trial and so are expected to only randomly hit a target.
Adding up all the values in each column of each test, the table for our chi square matrix looks like this :
The results for the chi - squared test are: