Do the successive digits in the decimal expansion of π behave as though they were select...

Do the successive digits in the decimal expansion of π 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 p₁, …., p₉ 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 012….901……901…… . Consider nonoverlapping groups of two digits, and let p_ij denote the long run 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 p_ijklm’s be based on the first 100,000 digits? Explain.

d. The paper “Are the Digits of π an Independent and Identically Distributed Sequence?” (The American Statistician, 2000: 12– 6) considered the first 1,254,540 digits of π, and reported the following P-values for group sizes of 1, …, 5: .572, .078, .529, .691, .298. What would you conclude?