a. Show that another expression for the chi-squared statistic is
Why is it more efficient to compute x2 using this formula?
b. When the null hypothesis is 
how does the formula of part (a) simplify? Use the simplified expression to calculate x2 for the pigeon/direction data in Exercise 4
Exercise 4 The article "Application of Methods for Central Statistical Monitoring in Clinical Trials" (Clinical Trials, 2013: 783-806) made a strong case for central statistical monitoring as an alternative to more expensive onsite data verification. It suggested various methods for identifying data characteristics such as outliers, incorrect dates, anomalous data patterns, unusual correlation structures, and digit preference. Exercise 3.21 of this book introduced Benford's Law, which gives a probability model for the first significant digit in many large data sets: 
The cited article gave the following frequencies for the first significant digit in a variety of variables whose values were determined in one particular clinical trial:
Carry out a test of hypotheses to see whether or not these frequencies are consistent with Benford's Law (the cited article gave P-value information).