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 p₁,..., p₉analogously. What hypotheses about these proportions should be tested, and what is df for the chisquared 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 pij 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 chi-squared test of appropriate hypotheses about the pijklm's be based on the first 100,000 digits? Explain.

 d. The article "Are the Digits of p an Independent and Identically Distributed Sequence?" (The American Statis tician, 2000: 12-16) considered the first 1,254,540 digits of p, and reported the following P-values for group sizes of 1, ... , 5: .572, .078, .529, .691, .298. What would you conclude?