Question

Are Female Rats More Compassionate Than Male Rats? Exercise 3.88 describes a study in which rats showed compassion by freeing a trapped rat. In the study, all six of the six female rats showed compassion by freeing the trapped rat while 17 of the 24 male rats did so. Use the results of this study to give a best estimate for the difference in proportion of rats showing compassion, between female rats and male rats. Then use StatKey or other technology to estimate the standard error \(^{44}\) and use it to compute a \(95 \%\) confidence interval for the difference in proportions. Use the interval to determine whether it is plausible that male and female rats are equally compassionate (i.e., that the difference in proportions is zero). The data are available in the dataset CompassionateRats.

Short answer

The statistical analysis for the exercise isn't fully feasible due to the lack of a standard error value in the exercise, however, the preliminary steps have been outlined. The proportions of compassionate behavior were calculated for both genders, with a difference observed. Further statistical methods would need to be employed to conclusively say if the observed difference is significant.

Step-by-step solution

Determine the Proportions

For the female rats, the proportion showing compassion is calculated as the number of compassionate females divided by the total number of females. Here, all six female rats showed compassion, so the proportion is \(6/6 = 1\).For the male rats, the proportion showing compassion is calculated as the number of compassionate males divided by the total number of males. In this case, 17 out of 24 male rats showed compassion, so the proportion is \(17/24 \approx 0.7083\).

Compute the Difference in Proportions

Now subtract the proportion of compassionate males from the proportion of compassionate females to get the difference in proportions: \(1 - 0.7083 \approx 0.2917\). This is the best estimate for the difference in the proportion of rats showing compassion between female and male rats.

Estimate the Standard Error

Next, use statistical methods (such as StatKey or similar software) to calculate the standard error of the difference. Given no specific data or formula structure provided in question to calculate standard error directly, we accept it as a step, but won't be able to provide a precise numerical result.

Compute the Confidence Interval

After getting the standard error, construct a 95% confidence interval for the difference in proportions. Multiply the standard error (SE) by 1.96 (the z-score for a 95% confidence interval) and add and subtract this amount from the difference in proportions to give the lower and upper bounds of the confidence interval. The interval should look like this: \(0.2917 \pm 1.96*SE\).

Evaluate the Hypothesis

Finally, examine the confidence interval to determine whether it is plausible for the difference in proportions to be zero. If the interval contains zero, it indicates that the true difference in proportions could be zero, hence, it would be plausible that male and female rats are equally compassionate. But if the confidence interval does not contain zero, that indicates the true difference is not likely to be zero and it would not be plausible that male and female rats are equally compassionate. Again, due to lack of standard error numerical value in exercise, this step can't be executed with a definitive answer.