Chapter 12.1, Problem 21E

Is the lottery fair? Mega Millions is a multistate lottery in which players try to guess the numbers that will tum up in a drawing of numbered balls. One of the balls drawn is the Mega Ball. Matching the number drawn on the Mega Ball increases ones winnings. During a recent three-year period, the Mega Ball was drawn from a collection of 15 balls numbered 1 through 15 and a total of 344 drawings were made. For the purposes of this exercise, we grouped the numbers into five categories: 1-3, 4-6, and so on. If the lottery is fair, then the winning number is equally likely to occur in any category. Following are the observed frequencies.

1. Compute the expected frequencies.
2. Compute the value of ${\chi }^2.$
3. How many degrees of freedom are there?
4. Test the hypothesis that each of the categories is equally likely. Use the $\alpha =0.05$ level of significance.