Runs and Autocorrelation test

For the runs test example on page 182-183, partition the sequence into blocks and gaps without a block appearing next to a block and without a gap appearing next to a gap. Thus

111 000 11 000 1 000 1 0 1 00 111 0 1111 00 1 00 1 00 1 00 1111 000 11...

there are 81 groupings in total. Of these $B_1=25$ of them are 1, $B_2=4$ of them are 11, $B_3=5$ of them are 111, $G_1=8$ of them are 0, $G_2=20$ of them are 00 and $G_3=12$ of them are 000. There are also 7 instances of 1111 which we do not count.

In the autocorrelation test we look at equation (5.5) on page 182: $$X_5=2\frac{A(d)-\frac{n-d}2}{\sqrt{n-d}}$$ then with $n=160$, $d=8$ we have $A(8)=100$ and so $$X_5=2\times\frac{100-\frac{160-8}2}{\sqrt{160-8}}=3.89331.$$

The $d=8$ here is chosen arbitrarily for the example, $A(8)$ is calculated by lining up the first 152 terms of the sequence with the last 152. Thus

11100011000100010100111011110010010010011110001100... 00010001010011101111001001001001111000110001000100...

and counting where the entries are different. There are 100 out of 152 positions in total where this is the case.