The fairness of the South African lottery was recently challenged by one of the country's political parties. The lottery publishes historical statis¬tics at its website (www.nationallottery.co.za). Here is a table of the number of times each number appeared in the lottery and as the "Powerball" number as of June 2007:

We wonder if all the numbers are equally likely to be the "Powerball."
a) What kind of test should we perform?
b) There are 655 Powerball observations. What are the appropriate expected value(s) for the test?
c) State the null and alternative hypotheses.
d) How many degrees of freedom are there?
e) The value of x2 is 34.5. What's the P-value?
f) State your conclusion.

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Number Lotto Powerbal Number Lotto Powerball 26 12 12 67 12 90 12 12 103 42 19 67 46 90