Gan investigator analyzed the leading digits from 788 checks issued by seven suspect companies. the frequencies were found to be 0 , 15 , 3 , 59 , 219 , 431 , 9 , 19 , and 33 , and those digits correspond to the leading digits of 1, 2, 3, 4, 5, 6, 7, 8, and 9, respectively. if the observed frequencies are substantially different from the frequencies expected with benford's law shown below, the check amounts appear to result from fraud. use a 0.025 significance level to test for goodness-of-fit with benford's law. does it appear that the checks are the result of fraud?

Aprisoner is trapped in a cell containing three doors. the first door leads to a tunnel that returns him to his cell after two days of travel. the second leads to a tunnel that returns him to his cell after three days of travel. the third door leads immediately to freedom. (a) assuming that the prisoner will always select doors 1, 2 and 3 with probabili- ties 0.5,0.3,0.2 (respectively), what is the expected number of days until he reaches freedom? (b) assuming that the prisoner is always equally likely to choose among those doors that he has not used, what is the expected number of days until he reaches freedom? (in this version, if the prisoner initially tries door 1, for example, then when he returns to the cell, he will now select only from doors 2 and 3.) (c) for parts (a) and (b), find the variance of the number of days until the prisoner reaches freedom. hint for part (b): define ni to be the number of additional days the prisoner spends after initially choosing door i and returning to his cell.