statistics

An investigator analyzed the leading digits of the amounts from 200 checks issued by three suspect companies. The frequencies were found to be 68, 40, 18, 19, 8, 20, 6, 9, 12 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, the check amounts appear to be the result of fraud. Use a 0.05 significance level to test for goodness-of-fit with Benford's law.

1. Calculate the χ2 test statistic.

2. Calculate the χ2 critical value.

3.  Is there sufficient evidence to conclude that the checks are the result of fraud?

Alert nurses at the Veteran's Affairs Medical Center in Northampton, Massachusetts, noticed an unusually high number of deaths at times when another nurse, Kristen Gilbert, was working. Kristen Gilbert was arrested and charged with four counts of murder and two counts of attempted murder. When seeking a grand jury indictment, prosecutors provided a key piece of evidence consisting of the table below. Use a 0.01 significance level to test the defense claim that deaths on shifts are independent of whether Gilbert was working.

4.  Calculate the χ2 test statistic.

5.  Calculate the χ2 critical value.

6.  Is there sufficient evidence to reject the defense claim that deaths on shifts are independent of whether Gilbert was working?