Question

Get on the boat! A small ferry runs every half hour from one side of a large river to the other. The probability distribution for the random variable Y= money collected on a randomly selected ferry trip is shown here. From Exercise 7, $\mu Y=\$19.35$.

(a) Find the median of Y.

(b) Compare the mean and median. Explain why this relationship makes sense based on the probability distribution.

Short answer

Part (a) $20.

Part (b) Distribution is left-skewed.

Step-by-step solution

Part (a) Step 1. Given information.

The given information is:

Part (a) Step 2. Find the median of Y. 

Money Collected	Probability	Cumulative Probability
0	0.02	0.02
5	0.05	0.07
10	0.08	0.15
15	0.16	0.31
20	0.27	0.58
25	0.42	1

The median will be the amount of money collected for the category with a cumulative probability of at least 0.5 and a probability of less than 0.5 for the prior category.

The $20 collected category has a cumulative probability of 0.58, while the $ 15 collected category has a cumulative probability of 0.31.

As a result, the median is set at $20.

Part (b) Step 1. Compare the mean and median.

The given mean is $19.35. $20 is the median that we calculated.

We can see that the mean is less than the median, indicating that the distribution is skewed to the left. This is because the mean is influenced by unusual values more strongly than the median, and thus there appear to be unusually small values in the distribution which are affecting the mean.

This is consistent with the conclusion we reached in the previous exercise about the form of the histogram.