Question

Skee BallAna is a dedicated Skee Ball player (see photo in Exercise 4) who always rolls for the 50-point slot. The probability distribution of Ana’s score Xon a randomly selected roll of the ball is shown here. From Exercise 8, $\mu X=23.8$.

(a) Find the median of X.

(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 right-skewed.

Step-by-step solution

Part (a) Step 1. Given information.

The given information is:

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

Score	Probability	Cumulative Probability
10	0.32	0.32
20	0.27	0.59
30	0.19	0.78
40	0.15	0.93
50	0.07	1

The median will be the score 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 category 20 has a cumulative probability of 0.59, while the category 10 has a cumulative probability of 0.32.

As a result, the median is set at 20.

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

The given mean is 23.8. 20 is the median that we calculated.

We can see that the mean is more than the median, indicating that the distribution is skewed to the right. This is because the mean is influenced by unusual values more strongly than the median, and thus there appear to be unusually large 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.