Answer: Do the successive digits in the decimal expansion

Chapter 14, Problem 48

(choose chapter or problem)

Do the successive digits in the decimal expansion of pbehave as though they were selected from a random number table (or came from a computers random numbergenerator)?a. Let p0 denote the long run proportion of digits inthe expansion that equal 0, and define p1,, p9 analogously. What hypotheses about these proportionsshould be tested, and what is df for the chisquaredtest?b. H0 of part (a) would not be rejected for the nonrandomsequence 012901901. Consider nonoverlappinggroups of two digits, and let pij denote the longrun proportion of groups for which the first digit isi and the second digit is j. What hypotheses aboutthese proportions should be tested, and what is df forthe chi-squared test?c. Consider nonoverlapping groups of 5 digits. Could achi-squared test of appropriate hypotheses about thepijklms bebased on the first 100,000 digits? Explain.d. The article Are the Digits of p an Independent andIdentically Distributed Sequence? (The AmericanStatis tician, 2000: 1216) considered the first 1,254,540digits of p, and reported the following P-values forgroup sizes of 1, , 5: .572, .078, .529, .691, .298. Whatwould you conclude?