Which is MORE LIKELY: a.) a single butterfly has wings that are shorter than 0.8 cm, or b.) a sample of 50 butterflies has an average wing length that is shorter than 0.8 cm.

Question

For a certain species of butterfly, the length of a wing (at its longest point) is normally distributed with a mean of 1 cm and a standard deviation of 0.2 cm.

Which is MORE LIKELY: a.) a single butterfly has wings that are shorter than 0.8 cm, or b.) a sample of 50 butterflies has an average wing length that is shorter than 0.8 cm.

Explain your reasoning. 

Answer 1

With the population mean and standard deviation held constant, the z-stat for the mean is always going to smaller in magnitude (absolute value) than the z-stat for a individual value, so the probability will always be greater for the mean.