Do the successive digits in the decimal expansion of p behave as though they were selected from a random number table (or came from a

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Do the successive digits in the decimal expansion of p behave as though they were selected from a random number table (or came from a computer’s random number generator)?

a. Let p0 denote the long run proportion of digits in the expansion that equal 0, and define analogously. What hypotheses about these proportions should be tested, and what is df for the chi-squared test?

b. H0 of part

(a) would not be rejected for the nonrandom sequence . Consider nonoverlapping groups of two digits, and let pij denote the long run 012c901c901c proportion of groups for which the first digit is i and the second digit is j. What hypotheses about these proportions should be tested, and what is df for the chi-squared test?

c. Consider nonoverlapping groups of 5 digits. Could a chisquared test of appropriate hypotheses about the be based on the first 100,000 digits? Explain.

d. The paper “Are the Digits of p an Independent and Identically Distributed Sequence?” (The American Statistician, 2000: 12–16) considered the first 1,254,540 digits of p, and reported the following P-values for group sizes of . What would you conclude?
1,

c, 5: .572, .078, .529, .691, .298 pijklm’s