The Tale of the Long-Tailed Penguin
Some penguins are known to have the trait of a “long-tail.” Some Population Geneticists claim that this trait serves as a benefit to a penguin when swimming, allowing them to escape from the capture of sea predators, such as killer whales and lion seals, and also capture more of their own prey. Others claim that this may actually serve as an impediment while on land, so as to slow the penguins retreat from land predators, such as the lion seal and polar bears. Therefore, it would be important to know whether or not this particular trait is showing signs of either growing or diminishing in the penguin population. In order to study the propensity of this trait in this population, the geneticists gathered samples from the population.
Assume that the container of poker chips represents the entire population of the particular species of penguins being studied. Let the red poker chips represent the penguins with the regular tail size and the black poker chips represent the penguins with an elongated tail.
1) Take a sample of size 5 from the population and record the proportion of penguins in your sample with an elongated tail.
2) Compile your data with the data of the entire class and create a sampling distribution for the population proportion of penguins with a “long-tail.”
3) Calculate the mean and standard deviation of this sampling distribution.
4) Repeat the procedure above for samples of size 10 and 25.
5) Use your results from above to answer the following questions.
A. What do you notice about the shapes of each of the three distributions above?
B. What do you notice about the means of each of the three distributions above?
C. What do you notice about the standard deviations of the three distributions above?
D. How do the answers of A through C above relate to the Central Limit Theorem?
E. Based on the above answers, comment on the relationship between sample size and Power of a Test.
F. Based on the above answers, comment on the relationship between sample size and the margin of error.
Sampling Distribution for n=5
Sampling Distribution for n=10
Sampling Distribution for n=25